UNIVERSIDADE ESTADUAL DE CAMPINASINSTITUTO DE FILOSOFIA E CIÊNCIAS HUMANAS

ANA CLAUDIA DE JESUS GOLZIO

NON-DETERMINISTIC MATRICES:THEORY AND APPLICATIONS TO ALGEBRAIC

SEMANTICS

MATRIZES NÃO-DETERMINÍSTICAS:TEORIA E APLICAÇÕES À SEMÂNTICA

ALGÉBRICA

CAMPINAS2017

ANA CLAUDIA DE JESUS GOLZIO

NON-DETERMINISTIC MATRICES:THEORY AND APPLICATIONS TO ALGEBRAIC SEMANTICS

MATRIZES NÃO-DETERMINÍSTICAS:TEORIA E APLICAÇÕES À SEMÂNTICA ALGÉBRICA

Tese apresentada ao Instituto de Filosofia e

Ciências Humanas da Universidade Estadual de

Campinas como parte dos requisitos exigidos

para a obtenção do título de Doutora em

Filosofia.

Thesis presented to the Institute of Philosophy

and the Humanities of the University of

Campinas in partial fulfillment of the

requirements for the degree of Doctor in

Philosophy.

Supervisor: Prof. Dr. Marcelo Esteban Coniglio

ESTE EXEMPLAR CORRESPONDE À VERSÃO

FINAL DA TESE DEFENDIDA PELA ALUNA ANA

CLAUDIA DE JESUS GOLZIO E ORIENTADA PELO

PROF. DR. MARCELO ESTEBAN CONIGLIO.

CAMPINAS

2017

Agência(s) de fomento e nº(s) de processo(s): FAPESP, 2013/04568-1

Ficha catalográfica

Universidade Estadual de Campinas

Biblioteca do Instituto de Filosofia e Ciências Humanas

Cecília Maria Jorge Nicolau - CRB 8/3387

Golzio, Ana Claudia de Jesus, 1985-

G584n GolNon-deterministic matrices : theory and applications to algebraic semantics /

Ana Claudia de Jesus Golzio. – Campinas, SP : [s.n.], 2017.

GolOrientador: Marcelo Esteban Coniglio.

GolTese (doutorado) – Universidade Estadual de Campinas, Instituto de

Filosofia e Ciências Humanas.

Gol1. Lógica algébrica. 2. Lógica matemática não-clássica. 3. Categorias

(Matemática). 4. Álgebra universal. I. Coniglio, Marcelo Esteban,1963-. II.

Universidade Estadual de Campinas. Instituto de Filosofia e Ciências

Humanas. III. Título.

Informações para Biblioteca Digital

Título em outro idioma: Matrizes não-determinísticas : teoria e aplicações à semântica

algébrica

Palavras-chave em inglês:Algebraic logic

Non-classical mathematical logic

Categories (Mathematics)

Universal algebra

Área de concentração: Filosofia

Titulação: Doutora em Filosofia

Banca examinadora:Marcelo Esteban Coniglio [Orientador]

Hércules de Araújo Feitosa

Ciro Russo

Hugo Luiz Mariano

Newton Marques Peron

Data de defesa: 30-03-2017

Programa de Pós-Graduação: Filosofia

UNIVERSIDADE ESTADUAL DE CAMPINASINSTITUTO DE FILOSOFIA E CIÊNCIAS HUMANAS

A Comissão Julgadora dos trabalhos de Defesa de Tese de Doutorado, composta pelosProfessores Doutores a seguir descritos, em sessão pública realizada em 30 de Março de2017, considerou a candidata ANA CLAUDIA DE JESUS GOLZIO aprovada.

Prof. Dr. Marcelo Esteban Coniglio

Prof. Dr. Hércules de Araújo Feitosa

Prof. Dr. Ciro Russo

Prof. Dr. Hugo Luiz Mariano

Prof. Dr. Newton Marques Peron

A Ata de Defesa, assinada pelos membros da Comissão Examinadora, consta no processode vida acadêmica da aluna.

To my parents: Altacir and João,To my husband: Claudecir,

In memory of Mozart Luiz Carbonieri.

Acknowledgements

I would like to express my sincere gratitude:

To my parents Altacir and João and my husband Claudecir, for their support, encourage-ment and comprehension.

To my supervisor Marcelo Esteban Coniglio, whom I admire, I respect, and I am immenselygrateful for all his help and dedication during these years of work.

To professors Itala M. L. D’Ottaviano and Walter A. Carnielli, that always offered goodworking conditions for them students.

To professors Hércules de Araújo Feitosa and Luiz Henrique da C. Silvestrini, for thesupport, encouragement and friendship through my academic career so far.

To professor Aldo Figallo-Orellano for his valuable contribution in this work.

To professor Rodolfo Ertola Biraben for several suggestions.

To longtime friends and study partners Kleidson Êglicio C. da S. Oliveira and Angela P.Rodrigues Moreira for the friendship and affection.

To closer working friends: Thiago, Felipe, João, Alexandre and Edson for the relaxedcompanionship and for many suggestions.

To other colleagues of Centre for Logic, Epistemology and History of Science (CLE) forthe excellent companionship.

To Mr. Travis D. Warwick (Head of the Kleene Mathematics Library at University ofWisconsin, in Madison, United States), and to professors Thomas Vougiouklis and GeorgesHansoul by sending me bibliographical items that were fundamental to the writing of thefirst chapter.

To FAPESP for the financial support (scholarship grant 2013/04568-1).

To employees of CLE, for their good job that made this work possible.

To everyone that in any way contributed to my personal and professional progress alongthese years.

“We hear within us the perpetual call:There is the problem.

Seek its solution.”(David Hilbert)

ResumoChamamos de multioperação qualquer operação que retorna, para cada argumento, umconjunto de valores ao invés de um único valor. Através das multioperações podemosdefinir uma estrutura algébrica munida com pelo menos uma multioperação. Esta estruturaé chamada de multiálgebra. O estudo delas começou em 1934, com a publicação de umartigo de Marty. No âmbito da Lógica, as multiálgebras foram consideradas por Avron eseus colaboradores sob o nome de matrizes não-determinísticas (ou Nmatrizes) e utilizadascomo ferramenta semântica para a caracterização de algumas lógicas que não podem sermodeladas por uma única matriz finita. Carnielli e Coniglio introduziram a semânticade estruturas swap para LFIs (Lógicas da Inconsistência Formal), que são Nmatrizesdefinidas sobre ternas em uma álgebra booleana, que generaliza a semântica de Avron.Nesta Tese iremos apresentar um novo método de algebrização de lógicas, baseado emmultiálgebras e em estruturas swap, que é similar ao método clássico de algebrização deLindenbaum-Tarski, porém mais abrangente, porque podemos aplicá-lo a sistemas emque alguns operadores não são congruenciais. Em particular, este método será aplicado àuma família de lógicas modais não-normais e à algumas LFIs que não são algebrizáveispor nenhum método bem conhecido, incluindo a teoria geral de Blok e Pigozzi. Tambémobteremos teoremas de representação para alguns LFIs e provaremos que, sob a nossaabordagem, as classes de estruturas de swap para algumas extensões axiomáticas de mbCsão subclasses da classe das estruturas swap para a lógica mbC.

Palavras-chave: semântica algébrica; multiálgebra; matriz não-determinística; estruturaswap.

AbstractWe call multioperation any operation that return for even argument a set of valuesinstead of a single value. Through multioperations we can define an algebraic structureequipped with at least one multioperation. This kind of structure is called multialgebra.The study of them began in 1934 with the publication of a paper of Marty. In therealm of Logic, multialgebras were considered by Avron and his collaborators underthe name of non-deterministic matrices (or Nmatrices) and used as semantics tool forcharacterizing some logics which cannot be characterized by a single finite matrix. Carnielliand Coniglio introduced the semantics of swap structures for LFIs (Logics of FormalInconsistency), which are Nmatrices defined over triples in a Boolean algebra, generalizingAvron’s semantics. In this thesis we will introduce a new method of algebraization oflogics based on multialgebras and swap structures that is similar to classical algebraizationmethod of Lindenbaum-Tarski, but more extensive because it can be applied to systemssuch that some operators are non-congruential. In particular, this method will be appliedto a family of non-normal modal logics and to some LFIs that are not algebraizable bymethod as Blok and Pigozzi general theory. We also will obtain representation theoremsfor some LFIs and we will prove that, under out approach, the classes of swap structuresfor some axiomatic extensions of mbC are a subclass of the class of swap structures forthe logic mbC.

Keywords: algebraic semantics, multialgebra; non-deterministic matrix; swap structure.

Contents

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

0 BASIC CONCEPTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150.1 Notions of set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150.2 Notions of logic and universal algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 200.3 Notions of category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1 ALGEBRAIC HYPERSTRUCTURES: ORIGINS . . . . . . . . . . . . . . . . . . . 301.1 Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.2 Hyperlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.3 Multialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.3.1 Homomorphisms of multialgebras and other concepts . . . . . . . . . . . . . . 381.3.2 Representation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.4 Hyperrings and hyperfields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.5 Hv-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.6 Non-deterministic matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2 SOME CONCEPTS IN UNIVERSAL MULTIALGEBRA . . . . . . . . . . . . . . . 482.1 Multialgebras and homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.2 Submultialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.3 Interpretation of formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.4 Multicongruences and quotient multialgebras . . . . . . . . . . . . . . . . . . . . . 572.5 The category of multialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3 NON-DETERMINISTIC SEMANTICS FOR NON-NORMAL MODAL LOGICS . 653.1 The Systems Tm, T4m, T45m, TBm and Dm . . . . . . . . . . . . . . . . . . . . . 653.2 Swap structures for Tm, T4m, T45m, TBm and Dm . . . . . . . . . . . . . . . . . 70

4 AN ALGEBRAIC STUDY OF LFIS BY MEANS OF SWAP STRUCTURES . . . 894.1 Swap structures for CPL+

e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.2 Swap structures for mbC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.3 Swap structures for some extensions of mbC . . . . . . . . . . . . . . . . . . . . . 106

FINAL CONSIDERATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Introduction

We call algebraic hyperstructures the class of algebras in which the operations(called multioperations) return, for each argument, a set of values instead of a single valueas in ordinary algebras. The study of hyperstructures began with the paper “Sur unegénéralisation de la notion de groupe” in 1934, by the French mathematician FrédéricMarty (MARTY, 1934). In that paper, the author introduces a definition of hypergroup,that is a generalization of usual notion of group by use of a multioperation. After Marty,several hyperstructures have emerged, including the multialgebras.

A multialgebra (also known as hyperalgebra) is defined as a set provided with atleast one multioperation. This concept was studied from many points of view and appliedto several areas of Mathematics, Science Computer and Logic. In the realm of the Logic,multialgebras were used as semantis for logical systems. More recently, matrix semanticsbased on multialgebras were considered by Avron and his collaborators under the nameof non-deterministic matrix (or Nmatrix) and used to characterize logics, in particular,some Logics of Formal Inconsistency - LFIs (see for instance (AVRON, 2005)). Semanticsbased on multialgebras for LFIs called swap structures were also proposed by Carnielliand Coniglio (CARNIELLI; CONIGLIO, 2016).

The main motivation this Thesis is the study of multialgebras as a semanticaltool for algebrization of logical systems and as an alternative algebraic structure (in thesense of the universal algebra). From the algebraic perspective, it is not so immediate thatthe generalization of basic concepts such as homomorphism, subalgebras and congruencesworks fine. Several different alternatives were proposed, but little was done with thepurpose of being used in logic as semantics tool. So, the aim of this work is to presentan algebraic study of multialgebras theory and to introduce a new algebraization methodbased on multialgebras and its properties, similar to the classical algebraization method ofLindenbaum-Tarski. Finally, we will apply this method to some LFIs and to a family of non-normal modal systems that lie outside the scope of the usual techniques of algebraizationof logics such as Blok and Pigozzi’s method.

In order to achieve the objective proposed, this Thesis begins with a Chapter0 where we will specify the notation and the basic concepts that will be used. We startshowing basics notions of set theory and then we present some definitions, results andexamples in the scope of universal algebra and logic. The reader who is already familiarwith these concepts can pass quickly or even ignore the existence of this introductorychapter.

T dev lop ult in ultialgebras, it to in estigate what

already done in the literature. However, this task was not very easy due to the different clas-sifications used to treat the same concepts. For instance, hyperlattice, that is a lattice withmultioperations, was introduced by Benado with the name of multistructure (BENADO,1953) and by Morgado (MORGADO, 1962) as reticuloide, however most of the authors,use the terms hyperlattice or multilattice. The names “multialgebra” and “hyperalgebra”,often are used with the same significance, but in 1950 Pickert called them “structures”.Already, a different approach, closer to relational systems, was given by Jonsson and Tarskithat used the name of complex algebra (JONSSON; TARSKI, 1951; JONSSON; TARSKI,1952). Many these authors have developed their concepts independently and so buildinga bridge between them is not a trivial task. The application of these structures to thecontext of the algebraic logic is still very little explored and authors such as Avron thatuses these concepts in his definition of non-deterministic matrices, probably developed hiswork independently and so he does not refer to any of the works developed by authors suchas Marty, Benado, Morgado, and so on. Since we consider important to establish a bridgebetween the logic and algebra in the framework of hyperstructures, then in Chapter 1 wewill present a historical and analytical background of the main hyperstructures studied inliterature until the case of the non-deterministic matrices used in logic.

Our motivation to the study of multialgebras started before we knew the existingalgebraic literature, even before we knew them with the name “multialgebras”. We only hadinformation about the non-deterministic matrices developed by Avron and its collaborators.Initially, the idea of the project was to generalize concepts of universal algebra to non-deterministic matrices, because we believed in the possibility of obtaining advantages interms of algebraization of logics. Thus we started our research with developments along thisway, most of them are presented in Chapter 2. During the process of drafting that chapter,we discovered that the concept of multioperations used in the non-deterministic matriceswas already applied in the literature of hyperstructures, and after an intensive investigationwe found a broad literature about this subject. If, on the one hand, a large part of ouroriginal results was “lost” (in the sense that they were nothing new to compound a thesis),on the other hand we verified that we were on the correct way, since most of our conceptswere in accordance with the published papers. Then, after we “walked over troubled water”,we started a new stage of the research with the goal of finding out what had already beendone in this area (as much as possible). This research generated the Chapter 1.

Of course, we do not change the initial goal of using multialgebras as a kind ofalgebraic semantic and in Chapters 3 and 4 we developed new algebraization tools using themultialgebras theory. Note that making developments in the framework of hyperstructuresis not a simple task because there are different alternatives definitions of multialgebra,homomorphism, submultialgebra and so on. Thus, to make appropriate choices is notalways obvious. In Chapter 2, we present which were our choices, by showing what are

the definitions of multialgebra, homomorphism between multialgebras, submultialgebra,homomorphic image of multialgebra, multicongruence and quotient multialgebra suitablefor our purposes. In this chapter, we initiate a study of category for multialgebras which isused especially in Chapter 4. In particular it is shown that the category of multialgebrashas arbitrary products. Note that other authors such as Nolan (NOLAN, 1979) has alreadydone a categorial study of multialgebras, but our goal in this Thesis is not purely algebraicas in the works of Nolan. In Chapter 4, these results are applied to obtain representationtheorems for some LFIs. In the future, we intend to do something similar with the familyof non-normal modal systems studied in Chapter 3 of this Thesis.

Chapter 3 is other important chapter of this Thesis. There we present somenon-normal modal systems proposed by Ivlev in (IVLEV, 1988) in Hilbert-style deductionsystem. Later on algebraic semantics is proposed for these modal systems, consisting ofswap structures (as introduced in (CARNIELLI; CONIGLIO, 2016) for LFIs), a kind ofthese swap structures constitute the original ϕ-valued non-deterministic matrices proposedby Avron for these systems, which were additionally studied by (CONIGLIO; CERRO;PERON, 2015). Then, we will present the so called Lindenbaum-Tarski swap structuresfor these logics and we will use a canonical valuation to show that the swap structuresfor these systems are correct and complete with respect to their respective Hilbert-styleversions. This method is more simpler and natural than the proposed by Blok-Pigozziand it is similar to the classical algebraization method of Lindenbaum-Tarski with thepeculiarity that the use of multialgebras eliminates the requirement that all the operatorsbe congruential.

The Chapter 4 was developed with the collaboration of the Professor AldoFigallo Orellano (Universidad Nacional del Sur, Bahía Blanca, Argentina) who is currentlydeveloping a postdoctoral at the CLE/Unicamp1. Part of the results presented in thatchapter are based on the pre-print (CONIGLIO; ORELLANO; GOLZIO, 2016). We startedstudying the (purely linguistic) expansion of CPL+, denoted by CPL+

e . This expansion isnothing more than CPL+ defined by adding ¬ and ◦ to the language of CPL+ withoutany axioms or rules for them. In order to prove completeness, we will apply, once againthe method of the Lindenbaum-Tarski swap structure already used in the Chapter 3 formodal systems. After this step, we will concentrate our efforts on the algebraic theory ofKmbC, the class of swap structures for the logic mbC, the weakest system in the hierarchyof LFIs and on some extensions of mbC. Algebraic semantic of swap structures for thesesystems will be reintroduced in a slightly more general form, in order to define a hierarchyof classes of multialgebras associated to the corresponding hierarchy of logics. This is inline with the traditional approach of algebraic logic, in which hierarchies of classes of1 The Centre for Logic, Epistemology and the History of Science (CLE) at the State University of

Campinas (UNICAMP) is the place were this Thesis was developed.

algebraic models are associated to hierarchies of logics. We will show that the class KmbC

is closed under sub-swap-structures and products, but it is not closed under homomorphicimages, hence it is not a variety in the usual sense. And using the results of Chapter 2, wewill present Birkhoff-style representation theorems for the class of each one of these logicalsystems.

At last in the Final Considerations, we highlight the most important results ofthis Thesis and discuss the possible future works in the interaction between multialgebrasand logics.

0 Basic concepts

This chapter is the theoretical background for the remainder of the work. Theconcepts will be presented briefly and they will be used as a way to standardize thenotation.

0.1 Notions of set theoryWe assume that the reader is familiar with the basics notions of set theory. If

it is not the case, we suggest specific books of set theory, as (HRBACEK; JECH, 1999;FEITOSA; NASCIMENTO; ALFONSO, 2011; PRISCO, 1997).

We adopt the intuitive notion that a set is a collection of objects and call theseobjects elements of the set. Let A be a set. If a is an element of A, we write a ∈ A and,otherwise, a �∈ A. Also, we assume that the basic operations in set theory have the usualmeaning, they are: inclusion, union, intersection and difference, they are denoted by ⊆, ∪,∩ and −, respectively.

We use the symbols “=” to denote the equality and “ �=” to denote the factthat two sets are not equal. If A and B are sets, then we define the operation of propersubset, in symbols A ⊂ B, meaning A ⊆ B and A �= B. When A is a subset of a set B, weabbreviate B − A by A�.

To denote the empty set, that is, a set which has no elements, we use the symbol∅. We say that two sets A and B are disjoints, when A ∩ B = ∅.

We will use P(A) to denote the set of all subsets of a set A and we use theexpression P(A)+ to denote the set of non-empty subsets of a set A, that is P(A) − {∅}.

Definition 0.1.1 (Cartesian product). Let A and B be two sets. The cartesian productof A and B is the set:

A × B = {(a, b) : a ∈ A and b ∈ B},

such that (a, b) are ordered pairs.

If A and B are sets, then a binary relation (or just a relation on A × B) is anysubset of A × B.

Definition 0.1.2 (Finite cartesian product). Let n be a positive integer and A1, A2, . . . , An

be sets. The Cartesian product A1 × A2 × . . . × An, of A1, A2, . . . , An is defined by:

A1 × A2 × . . . × An =n�

1Ai = {(a1, a2, . . . , an) : a1 ∈ A1, a2 ∈ A2, . . . , an ∈ An},

such that (a1, a2, . . . , an) are ordered n-tuples.

If, in the above definition, A1 = A2 = . . . = An, we represent A1 ×A2 × . . .×An

by An and take A0 as {∅}.

Let A be a set and n a positive integer. An n-ary relation R on A is defined asa subset of An and R is called a relation of type n.

If an n-tuple is an element of an n-ary relation R, we denote this by (a1, a2, . . . , an)∈ R. In the case that S is a binary relation we use the notations (a1, a2) ∈ S or a1Sa2

with the same meaning.

We can eventually write �a to denote the n-tuple (a1, . . . , an) of elements in An.

Definition 0.1.3 (Inverse relation, image and inverse image).

If S is a binary relation on A × B, then

(i) The inverse relation of S is a relation:

S−1 = {(b, a) ∈ B × A : (a, b) ∈ S}.

(ii) The image of C ⊆ A by S is the set:

S[C] = {b ∈ B : ∃a ∈ C, such that (a, b) ∈ S}.

In particular, the image of S is im(S) = S[A].

(iii) The inverse image of D ⊆ B by S is the set:

S−1[D] = {a ∈ A : ∃b ∈ D, and (a, b) ∈ S}.

In particular, the domain of S is dom(S) = S−1[B].

Definition 0.1.4 (Composition of relations).

If S is a binary relation on A × B and R is a binary relation on B × C, thenthe composition of S and R is the relation:

R ◦ S = {(a, c) ∈ A × C : ∃b ∈ B, such that (a, b) ∈ S and (b, c) ∈ R}.

Definition 0.1.5 (Relation properties).

(i) reflexive when, for all a ∈ A, we have aSa;

(ii) symmetric when, for all a, b ∈ A, if aSb, then bSa;

(iii) transitive when, for all a, b, c ∈ A, if aSb and bSc, then aSc;

(iv) antisymmetric when, for all a, b ∈ A, if aSb and bSa, then a = b;

Definition 0.1.6 (Equivalence relation).

Let A be a set. We say that S is an equivalence relation on A, if S is a reflexive,symmetric and transitive relation.

Definition 0.1.7 (Equivalence class). Let A be a set and S be an equivalence relationon A and a ∈ A. The equivalence class of a w.r.t.1 S is the set:

[a]S = {b ∈ A : bSa}.

Notation 0.1.8. When the equivalence relation is clear from the context, we can denotethe class above only by [a].

Definition 0.1.9 (Quotient set). Let A be a set and S be an equivalence relation on A.The quotient set of A by the relation S, denoted by A/S, is the set of the equivalenceclasses of S, that is:

A/S = {[a] : a ∈ A}.

Definition 0.1.10 (Ordering relation). Let A a set. We say that S is an ordering relation(or partial ordering relation) on A, if S is a reflexive, antisymmetric and transitive relation.

Definition 0.1.11 (Poset). A partially ordered set (or poset2) is a pair �A, S� such thatA is a nonempty set and S is a partial ordering relation on A.

Definition 0.1.12. If �A, ≤� is a poset and B ⊆ A, then we say that:

(i) an element a1 of A is an upper bound of B, when for all b ∈ B, we have that b ≤ a1;

(ii) an element a2 of A é a lower bound of B, when for all b ∈ B, we have that a2 ≤ b;

(iii) an element b1 of B is a maximum of B, when for all b ∈ B, we have that b ≤ b1;

(iv) an element b2 of B is a minimum of B, when for all b ∈ B, we have that b2 ≤ b;

(v) an element b1 of B is a maximal of B, when for all b ∈ B if, b1 ≤ b, then b = b1;

(vi) an element b2 of B is a minimal of B, when for all b ∈ B if, b ≤ b2, then b2 = b.1 Abbreviation for “with respect to”.2 That is the usual abbreviation for partially ordered set

Definition 0.1.13 (Supremum and infimum). Let �A, ≤� be a poset and B ⊆ A. Thesupremum of B (denoted by supB), if it exists, is the minimum of the set of all upperbounds of B and the infimum of B (denoted by infB), if it exists, is the maximum of theset of all lower bounds of B.

Definition 0.1.14 (Quasi ordering relation). Let A a set. We say that S is a quasiordering relation on A, if S is a reflexive and transitive relation.

Definition 0.1.15 (Chain). A chain C in a poset A = �A, ≤� is a subset of A such that,for all a, b ∈ C, a ≤ b or b ≤ a.

Definition 0.1.16 (Function). Let A and B be sets. A function f from A to B, denotedby f : A → B is an binary relation on A × B, such that for all a ∈ A, there is exactly oneb ∈ B with (a, b) ∈ f . In this case, we write f(a) = b.

In a function f : A → B, the sets A and B are called of domain and range off , respectively.

The set of all functions from A to B will be denoted by BA. If A �= ∅, then∅A = ∅. But, if A = ∅, then B∅ = {∅}, because there is a single function ∅ : ∅ → B withempty domain and, in particular, ∅∅ = {∅}.

Definition 0.1.17 (Image and inverse image). Let f : A → B a function. If C ⊆ A

and D ⊆ B, then the sets f [C] = {f(c) : c ∈ C} and f−1[D] = {a ∈ A : f(a) ∈ D}represent the image of C by the function f and the inverse image of D by the function f ,respectively.

Remark 0.1.18. If f : A → B is a function, �b = (b1 . . . , bn) ∈ Bn (for n > 0) then f−1(�b)will stand for {�a ∈ An : (f(a1), . . . , f(an)) = �b}.

Definition 0.1.19 (Constant function). A function f : A → B is called a constantfunction if, f [A] is a singleton.

Definition 0.1.20 (Identity function). A function f : A → A is called the identityfunction if, f(a) = a for all a ∈ A.

As well known, the composition of functions is still a function:

Proposition 0.1.21 (Composition function). If f : A → B and g : B → C are twofunctions, then g ◦ f : A → C is a function, such that (g ◦ f)(a) = g(f(a)), for all a ∈ A.

Definition 0.1.22 (Injection, surjection and bijection). A function f : A → B is called:

(i) an injection (or one-to-one) when, for all a1, a2 ∈ A, if f(a1) = f(a2), then a1 = a2.

(ii) a surjection (or onto) when, for all b ∈ B, there is a ∈ A, such that f(a) = b.

(iii) a bijection when it is an injection and a surjection.

Definition 0.1.23 (Projection). Let A1 and A2 be sets, such that, a1 ∈ A1 and a2 ∈ A2.The function πi : A1 × A2 → Ai defined by πi((a1, a2)) = ai, is called projection on the ithcoordinate of A1 × A2, for i = 1, 2.

Definition 0.1.24 (Indexed family of sets). Let I and A be sets. An indexed family ofsets I, denoted by {fi}i∈I , is a function f : I → A. If i ∈ I , then we can denote f(i) by fi.

The set I is called of index set of family {fi}i∈I .

Definition 0.1.25 (Generalized union and intersection). If {Ai}i∈I family of subsets ofa certain set A, then the union and the intersection of these sets are denoted, respectively,by:

�

i∈I

Ai = {a : a ∈ Ai for some i ∈ I} and

�

i∈I

Ai = {a : a ∈ Ai for all i ∈ I �= ∅}.

Definition 0.1.26 (Infinite cartesian product). Let I be an index set and {Ai}i∈I be afamily of sets indexed by I. The cartesian product of the sets in A is defined by:

�

i∈I

Ai = {f : I →�

i∈I

Ai : f(i) ∈ Ai for all i}.

Definition 0.1.27 (General projections). Let I be an index set, such that i, j ∈ I and{Ai}i∈I be a family of sets indexed by I. The function πj : �

i∈I Ai → Aj, defined byπj(f) = f(j), is called of projection on the ith coordinate of �

i∈I Ai.

Axiom of choice: For any set A there is a function f : P(A) → A, such that if∅ �= B ∈ P(A), then f(B) ∈ B. This function is called of choice function.

In relation to the sets of numbers, we will denote the usual sets of natural,integers, rational, irrational and real numbers by N,Z,Q, I and R, respectively.

We will present in the following some more specific properties of sets in general.

Definition 0.1.28 (Filter). Let S be a set. A filter on S is a collection F of subsets ofS that satisfies the following three conditions:

(i) S ∈ F ;

(ii) If A ∈ F and A ⊆ B ⊆ S, then B ∈ F ;

(iii) If A F and B F then A B F

Definition 0.1.29 (Generated filter). Let X be a subset of S. A generated filter by X,on S is F = {A ⊆ S : X ⊆ A}.

Definition 0.1.30 (Proper filter). A proper filter on S is a filter F , such that F �= P(S).

Definition 0.1.31 (Ultrafilter). Let S a set. A filter U on S is an ultrafilter on S if U isa proper filter and for all set A ⊆ S, we have that or A ∈ U or (S − A) ∈ U .

0.2 Notions of logic and universal algebra

We will introduce the basic notions of logic in general, as well as some definitionsand properties of the propositional calculus. For the reader who wants to know more aboutthese topics, we suggest the books (SHOENFIELD, 1967; MENDELSON, 1987; FEITOSA;PAULOVICH, 2005).

Definition 0.2.1 (Variables). The set of variables, denoted by Ξ, is a countable set ofsymbols defined by:

Ξ = {ξn : n ∈ N}.

Notation 0.2.2. We can denote a variable in Ξ by p or q with or without subscripts.

Definition 0.2.3 (Signature). Let Ξ be a set of variables. A signature Θ is a family{Θn}n∈N, such that for all Θn, Ξ ∩ Θn = ∅ and if n �= m, then Θn ∩ Θm = ∅.

Elements of Θn are called operator symbols of arity n. Elements of Θ0 are calledconstants. The support of Θ is the set:

|Θ| =�

Θn = {c : c ∈ Θn for some n ∈ N}.

When there is no risk of confusion, we write Θ instead of |Θ|.

Definition 0.2.4 (Propositional language). A propositional language with signature Θon Ξ, denoted by L(Θ, Ξ), is the algebra defined by:

(i) If ξ ∈ Ξ, then ξ ∈ L(Θ, Ξ);

(ii) If c ∈ Θn and {α1, α2, . . . , αn} ∈ L(Θ, Ξ), then c(α1, α2, . . . , αn) ⊆ L(Θ, Ξ);

(iii) The set L(Θ, Ξ) is generated only by items (i) and (ii).

If L(Θ, Ξ) satisfies the three conditions above, we say that the propositional

Definition 0.2.5 (Formulas). If Θ is a signature and L(Θ, Ξ) is a propositional languageover Θ, then the elements of L(Θ, Ξ) are called formulas over Θ.

We can use the notation LΘ or For(Θ) instead of L(Θ, Ξ) to denote the set ofthe formulas over Θ, omitting the set of variables Ξ, whenever it is clear in the context.

We will use Greek lowercase letters, such as ϕ, ψ, α, β and γ, with or withoutindex, to denote arbitrary formulas in L(Θ, Ξ).

The set of variables that occurs in a formula α of L(Θ, Ξ) will be denoted byV ar(α).

Definition 0.2.6. Let Θ be a signature and Ξ� ⊂ Ξ. We denote by L(Θ, Ξ�) the subsetof L(Θ, Ξ) formed by the formulas α, such that V ar(α) ⊆ Ξ�. In particular, if Ξn = {ξi :0 ≤ i ≤ n}, for n ≥ 0, then L(Θ, Ξn) is the subset of L(Θ, Ξ) formed by formula schemesα, such that V ar(α) ⊆ {ξ0, . . . , ξn}.

Definition 0.2.7 (Consequence relation). Let Θ be a signature, L(Θ, Ξ) be a propositionallanguage over Θ and Γ, Δ ⊆ P(L(Θ, Ξ)). A consequence relation in L(Θ, Ξ) is a relation� ⊆ P(L(Θ, Ξ)) × L(Θ, Ξ) that satisfies the following conditions:

(i) If α ∈ Γ, then Γ � α (strong reflexivity);

(ii) If Γ � α and Γ ⊆ Δ, then Δ � α (monotonicity);

(iii) If Γ � α and Δ � β, for all β ∈ Γ, then Δ � α (transitivity).

Definition 0.2.8 (Propositional logic). A propositional logic is a pair L = �Θ, ��, suchthat Θ is a signature and � is a consequence relation in L(Θ, Ξ).

Remark 0.2.9. The classical propositional calculus (or classical propositional logic)is a particular case of propositional logic. We use CPC as abbreviation for ClassicalPropositional Calculus or Classical Propositional Logic and in a deduction we will use thisabbreviation whenever we are referring to a result that hold in the CPC.

Definition 0.2.10 (Theory). Let L = �Θ, �� be a propositional logic. A theory in L is aset T ⊆ L(Θ, Ξ) such that if T � α, then α ∈ T .

We denote the set of all the theories in L by Th(L).

Definition 0.2.11 (Algebra). Let Θ be a signature. An algebra A over Θ (or of type Θ)is a pair �A, σA� such that A is a non-empty set, the universe (or support) of A, and σA isa function that assigns, for each n ∈ N and c ∈ Θn, an n-ary operation in A, cA : An → A.

In the sequel, sometimes we will refer to an algebra A = �A, σA� by means ofits support A and the support of A will be frequently denoted by |A|. When there is norisk of confusion, we write σ instead of σA.

Definition 0.2.12 (Similar algebras). Two algebras A = �A, σA� and B = �B, σB� aresimilar if they share the same type Θ

Definition 0.2.13 (Logic and algebra of same type). If L = �Θ, �� is a propositionallogic over Θ and A = �A, σ� is an algebra, also over Θ, then we say the logic L and thealgebra A have the same type.

Definition 0.2.14 (Multifunction). Let A and B be nonempty sets. A multifunction (ormultioperation) g from B to A, denoted by g : B →M A, is a function g : B → (P(A)−{∅}).

Definition 0.2.15 (Partial multifunction). Let A and B be nonempty sets. A functiong : B → P(A) is a partial multifunction g from B to A.

Definition 0.2.16 (Multifunction composition). Let A, B and C nonempty sets, and g1 :A →M B, g2 : B →M C multifunctions. The multifunction composition is a multifunctiong2◦g1 : A →M C, given by (g2◦g1)(a) = �{g2(b) : b ∈ g1(a)}, for every a ∈ A.

Proposition 0.2.17. The composition operation between multifunctions is associative,i.e. If g1 : A →M B, g2 : B →M C and g3 : C →M D are multifunctions, then (g3◦g2)◦g1 =g3◦(g2◦g1).

Proof. We must show that, for all a ∈ A, ((g3◦g2)◦g1)(a) = (g3◦(g2◦g1))(a). First of all:given a ∈ A, (g2◦g1)(a) = �{g2(b) : b ∈ g1(a)}, while

(g3◦(g2◦g1))(a) =�

{g3(c) : c ∈ (g2◦g1)(a)}.

So, d ∈ (g3◦(g2◦g1))(a) if, and only if there is c ∈ (g2◦g1)(a), such that d ∈ g3(c) if, andonly if

there is b ∈ g1(a) and there is c ∈ g2(b) such that d ∈ g3(c). (∗)

On the other hand, if b ∈ B, (g3◦g2)(b) = �{g3(c) : c ∈ g2(b)}, while

((g3◦g2)◦g1)(a) =�

{(g3◦g2)(b) : b ∈ g1(a)}.

So, d ∈ ((g3◦g2)◦g1)(a) if, and only if there is b ∈ g1(a), such that d ∈ (g3◦g2)(b), if, andonly if there is b ∈ g1(a) and there is c ∈ g2(b), such that, d ∈ g3(c) i. e. if and only if thecondition (∗) is true. This shows that the sets ((g3◦g2)◦g1)(a) and (g3◦(g2◦g1))(a) are thesame, for all a ∈ A. Therefore, (g3◦g2)◦g1 = g3◦(g2◦g1).

Definition 0.2.18 (Homomorphism). Let A = �A, σ� and B = �B, σ�� be two similaralgebras over Θ. A homomorphism h : A → B from A to B over Θ is a function h : A → B

such that for all n ≥ 0, c ∈ Θn and a1, . . . , an ∈ A,

h(cA(a1, . . . , an)) = cB(h(a1), . . . , h(an)).

In particular, h(cA) = cB, if c ∈ Θ0.

Definition 0.2.19 (Valuation). Let A = �A, σ� an algebra over Θ. The homomorphismsh : L(Θ, Ξ) → A are called of valuations of L(Θ, Ξ) in A.

Definition 0.2.20 (Subuniverse). Let A = �A, σ� be an algebra over Θ. A subuniverseof A over Θ is a non-empty subset B of A which is closed under the operations of A, i. e.,for all n ≥ 0, if c ∈ Θn and b1, . . . , bn ∈ B, then cA(b1, . . . , bn) ∈ B.

Definition 0.2.21 (Subalgebra). If A = �A, σ� and B = �B, σ�� are two similar algebrasover Θ, such that B is a subuniverse of A, then we say that B is a subalgebra of A over Θand we denote this by B ⊆ A.

Definition 0.2.22 (Subuniverse generated). Let A = �A, σ� be an algebra over Θ andX ⊆ A. The subuniverse of A over Θ generated by X, denoted by sg(X), is defined asfollows:

sg(X) =�

{B : B is subuniverse of A over Θ and X ⊆ B}.

Remark 0.2.23. Note that A is a subuniverse of A that contains X. So, the set {B :B is subuniverse of A over Θ and X ⊆ B} is non-empty and therefore sg(X) is welldefined.

Definition 0.2.24 (Subalgebra generated). Let �A, σ� an algebra and X ⊆ A. We saythat �A, σ� is generated by X, if sg(X) = A.

Definition 0.2.25 (Congruence). Let A = �A, σ� be an algebra over a signature Θ andθ ⊆ A × A. Then θ is a congruence in A if:

1. θ is an equivalence relation;

2. for all n > 0, c ∈ Θn and a1, ..., an, b1, ..., bn ∈ A, if aiθbi, for all 1 ≤ i ≤ n, thencA(a1, . . . , an)θcA(b1, . . . , bn).

Remark 0.2.26. In the last definition (and throughout the text) we use the symbols Θand θ to denote distinct things: Θ is used denoting signatures and θ denoting congruenceor equivalence relations.

Definition 0.2.27 (Quotient algebra). If A = �A, σ� is an algebra over a signature Θand θ ⊆ A × A is a congruence in A, then the quotient algebra of A by θ, denoted byA/θ, is the algebra over Θ with support A/θ with the operations cA/θ(a1/θ, . . . , an/θ) =cA(a1, . . . , an)/θ, such that a1, . . . , an ∈ A, n ≥ 0 and c ∈ Θn. In particular, cA/θ = cA/θ

if c ∈ Θ0.

The operations in A/θ are well defined, because θ is a congruence in A.

In what follows, we will present some examples of well known algebras in theliterature, since they will be used throughout the thesis.

Example 0.2.28 (Semigroup). A semigroup is an algebra �G, ·� such that · is a binaryoperation that satisfies the following identity:

(G1) For all a, b, c ∈ G, a · (b · c) = (a · b) · c [associativity];

Example 0.2.29 (Group). A group is an algebra �G, ·, −1, 1� such that the operations·, −1 and 1 are, binary, unary and nullary, respectively and the following identities aresatisfied:

(G1) For all a, b, c ∈ G, a · (b · c) = (a · b) · c [associativity];

(G2) For all a ∈ G, a · 1 = a = 1 · a [identity element];

(G3) For all a ∈ G, a · a−1 = 1 = a−1 · a [inverse element].

Example 0.2.30 (Abelian group). A group �G, ·, −1, 1� is abelian (or commutative) if thefollowing identity is satisfied:

(G4) For all a, b ∈ G, a · b = b · a [commutativity].

Example 0.2.31 (Ring). A ring is an algebra �A, +, ·, −, 0� such that the operations +,·, − and 0 are binary, binary, unary and nullary, respectively and the following conditionsare satisfied:

(A1) �A, +, −, 0� is an abelian group;

(A2) �A, ·� is a semigroup;

(A3) For all a, b, c ∈ A, a · (b + c) = (a · b) + (a · c) and (a + b) · c = (a · c) + (b · c)[distributivity].

Example 0.2.32 (Field). A field is an algebra �C, +, ·, 1, 0� such that the operations+ and · are binary, the operations 1 and 0 are nullary, and the following conditions are

(C1) �C, +, −, 0� is an abelian group;

(C2) �C − {0}, ·, −1, 1� is an abelian group;

(C3) For all a, b, c ∈ C, a · (b + c) = a · b + a · c [distributivity].

Example 0.2.33 (Semilattice). A semigroup �G, ·� is called of semilattice if it satisfiesthe following identities:

(G4) For all a, b ∈ G, a · b = b · a [commutativity];

(S2) For all a ∈ G, a · a = a [idempotency].

Example 0.2.34 (Lattice). A lattice is an algebra �L, ∧, ∨� such that ∧ and ∨ are binaryoperations and the following identities are satisfied:

(L1) For all a ∈ L, a ∧ a = a and a ∨ a = a [idempotency]

(L2) For all a, b, c ∈ L, (a ∧ b) ∧ c = a ∧ (b ∧ c) and (a ∨ b) ∨ c = a ∨ (b ∨ c) [associativity]

(L3) For all a, b ∈ L, a ∧ b = b ∧ a and a ∨ b = b ∨ a [commutativity]

(L4) For all a, b ∈ L, (a ∧ b) ∨ b = b and (a ∨ b) ∧ b = b [absorption].

Example 0.2.35 (Distributive lattice). A lattice �L, ∧, ∨� is distributive if the followingidentities are satisfied:

(L4) For all a, b, c ∈ L, (a ∨ b) ∧ c = (a ∧ c) ∨ (b ∧ c) and (a ∧ b) ∨ c = (a ∨ c) ∧ (b ∨ c)[distributivity].

Example 0.2.36 (Bounded lattice). A lattice �L, ∧, ∨� is said to be bounded from belowif there is an element 0 ∈ L, such that 0 ≤ a for all a ∈ L and L is said to be boundedfrom above if there is an element 1 ∈ L, such that a ≤ 1 for all a ∈ L. A bounded lattice isone that is bounded both from above and below.

Example 0.2.37 (Boolean algebra). A Boolean algebra is an algebra �B, ∧, ∨,� , 0, 1� suchthat ∧ and ∨ are binary operations, � is an unary operation, 0 and 1 are nullary operationsand the following identities are satisfied:

(B1) �B, ∧, ∨� is a distributive lattice;

(B2) For all a ∈ B, a ∧ 0 = 0 and a ∨ 1 = 1;

(B3) For all a ∈ B, a ∧ a� = 0 and a ∨ a� = 1 [complement].

Example 0.2.38 (Heyting algebra). A Heyting algebra is an algebra �H, ∧, ∨, →, 0, 1�,such that ∧, ∨ e → are binary operations, 0 and 1 are nullary operations and the followingidentities are satisfied:

(H1) �H, ∧, ∨� is a distributive lattice;

(H2) For all a ∈ H, a ∧ 0 = 0 and a ∨ 1 = 1;

(H3) For all a ∈ H, a → a = 1;

(H4) For all a, b ∈ H, (a → b) ∧ b = b and a ∧ (a → b) = a ∧ b;

(H5) For all a, b, c ∈ H , a → (b∧c) = (a → b)∧(a → c) and (a∨b) → c = (a → c)∧(b → c).

Example 0.2.39 (Heyting algebra). A Heyting algebra is a bounded lattice (whosemaximum and minimum elements are denoted 1 and 0, respectively) with a binaryoperation → such that a ∧ c ≤ b if and only if c ≤ (a → b).

Proposition 0.2.40. The two definitions of Heyting algebras above (Examples 0.2.38 and0.2.39) are equivalents.

The two following results are well known in the literature. A proof can be foundin (RASIOWA; SIKORSKI, 1963).

Proposition 0.2.41. Let �H, ∧, ∨, 0, 1� be a lattice satisfying (H1) and (H2) of the Defi-nition 0.2.38, if there exists → such that (H3) - (H5) of the Definition 0.2.38 are true,then → is unique.

Proposition 0.2.42. If A = �A, ∧, ∨, →, 0, 1� is a Heyting algebra, then A is a Booleanalgebra if and only if a ∨ (a → b) = 1, for any a, b ∈ A.

Definition 0.2.43 (Order lattice). An order lattice is a poset �L, ≤� such that for everya, b ∈ L both sup{a, b} and inf{a, b} exist.

Remark 0.2.44. The two lattice definitions (see Example 0.2.34 and Definition 0.2.43)are equivalent. To show it, if L is a lattice by Example 0.2.34, then we define ≤ over L bya ≤ b iff 3 a = a ∧ b (or by a ≤ b iff b = a ∨ b) and when L is an order lattice, then wedefine the operations ∧ and ∨ by a ∧ b = inf{a, b} and a ∨ b = sup{a, b}. For more detailsabout this proof, see (BURRIS; SANKAPPANAVAR, 1981, p. 8).

Definition 0.2.45. An implicative lattice is an algebra A = �A, ∧, ∨, →, 1� where�A, ∧, ∨, 1� is a lattice with the greatest element 1, such that �{c ∈ A : a∧c ≤ b} exists for3 Abbreviation for “if and only if”.

every a, b ∈ A,4 and → is the induced implication given by a → b = �{c ∈ A : a ∧ c ≤ b}for every a, b ∈ A (note that 1 def= a → a is the top element of A, for any a ∈ A). If,additionally, a ∨ (a → b) = 1 for every a, b then A is said to be a classical implicativelattice.5

The following well known results connect implicative lattice with Heytingalgebra and classical implicative lattice with Boolean algebra:

Proposition 0.2.46. Let A be an implicative lattice. Then:(1) If A has a bottom element 0, then it is a Heyting algebra.(2) If A is a classical implicative lattice and it has a bottom element 0, then it is a Booleanalgebra.

0.3 Notions of category theory

In this section, we will introduce the basic notions of category theory thatwill be used in this Thesis, mainly in the Section 2.5 and in the Chapter 4. The subjectpresented here can be found in any introductory book about category theory. In particular,we suggest the book “Category Theory” (AWODEY, 2010).

Definition 0.3.1 (Category). A category C is given by a collection Ob(C) of elementsA, B, C, D, . . ., called objects of C and a collection Mor(C) of elements f, g, h, . . ., calledarrows or morphisms of C that satisfy the following conditions:

(i) For each morphism f of C is assigned a pair of objects dom(f), cod(f) called domainof f and codomain of f , respectively. If A = dom(f) and B = cod(f), we say that f

is a morphism from A to B and we denote it by f : A → B;

(ii) For each pair of morphisms f : A → B, g : B → C of C is assigned a morphismg ◦ f : A → C called composition of f and g;

(iii) For each object A of C is assigned a morphism 1A : A → A, called identity morphismof A;

(iv) If f : A → B, g : B → C and h : C → D are morphisms, then h ◦ (g ◦ f) = (h ◦ g) ◦ f

(associativity);

(v) For each object B of C, let f : A → B and g : B → A be morphisms, we have1B ◦ f = f and g ◦ 1B = g (identity);

4 Here,�

X denotes the supremum of the set X ⊆ A w.r.t. partial order associated with the lattice ≤,provided that it exists.

(vi) For any objects A, B of C, the collection MorC(A, B) of the morphisms from A to B

is a set.

Definition 0.3.2 (Subcategory). The category D is a subcategory of the category C if:

(i) Ob(D) ⊆ Ob(C);

(ii) If A, B ∈ Ob(D), then MorD(A, B) ⊆ MorC(A, B).

Definition 0.3.3 (Full subcategory). The category D is a full subcategory of the categoryC if:

(i) D is a subcategory of C;

(ii) If A, B ∈ Ob(D), then MorD(A, B) = MorC(A, B).

Definition 0.3.4 (Diagram). A diagram in a category C is a (possibly empty) set ofobjects of C with a (possibly empty) set of morphisms between these objects. The objectsof a diagram C are called vertices of C and the morphisms are called edges of C.

Definition 0.3.5 (Branch). Let C be a category. A branch in C is a finite sequencef1, . . . , fn of edges of C, such that dom(fi+1) = cod(fi) for i ∈ {1, 2, . . . , n − 1}.

Definition 0.3.6 (Commutative diagram). Let C be a category. A diagram is commutative(or the diagram commutes) if, for any branches f1, . . . , fn and g1, . . . , gm in C, with n ≥ 2or m ≥ 2, we have fn ◦ fn−1 ◦ . . . ◦ f1 = gm ◦ gm−1 ◦ . . . ◦ g1.

Definition 0.3.7 (Monomorphism). The morphism f : A → B of a category C is amonomorphism if for every pair of morphisms g : C → A, h : C → A of C, if f ◦ g = f ◦ h,then g = h.

Definition 0.3.8 (Epimorphism). The morphism f : A → B of a category C is anepimorphism if for every pair of morphisms g : B → C, h : B → C of C, if g ◦ f = h ◦ f ,then g = h.

Definition 0.3.9 (Isomorphism). The morphism f : A → B of a category C is anisomorphism if there is a morphisms g : B → A in C, such that g ◦ f = 1A and f ◦ g = 1B.

Definition 0.3.10 (Initial object). Let C be a category. Um object 0 is initial if for eachobject B ∈ Ob(C) there is a unique morphism f : 0 → B in C.

Definition 0.3.11 (Terminal object). Let C be a category. Um object 1 is terminal if foreach object C ∈ Ob(C) there is a unique morphism g : C → 1 in C.

Definition 0.3.12 (Product). Let C be a category and A, B ∈ Ob(C). A product ofthe objects A and B in C is �A × B, {pA, pB}� where A × B is an object of C andpA : A × B → A, pB : A × B → B are morphisms of C such that, for every pair ofmorphisms f : C → A, g : C → B of C, there is a unique morphism from C to A × B suchthat the diagram below commutes:

C

f

��

�� g

��

A × B

pA��

pB��

A B

Definition 0.3.13 (Product of a family). Let C be a category and {Ai}i∈I a family ofobjects of C. A product of the family {Ai}i∈I in C is ��

i∈I Ai, {pi}i∈I� where �i∈I Ai is

an object of C and pi : �i∈I Ai → Ai are morphisms of C such that if fi : C → Ai is a

family of morphisms of C, then there is a unique morphism from C to �i∈I Ai such that

the diagram below commutes (for any i ∈ I):

C

��

fi

���i∈I Ai pi

�� Ai

Definition 0.3.14 (Product in the category). A category C has (finite) products if thereis, in C, the (finite) product of any set of objects of C.

Definition 0.3.15 (Functor). Let C and D be two categories. A functor F from C to D,denoted F : C → D, is a map that assigns:

(i) For each object A ∈ Ob(C), an object F (A) ∈ Ob(D);

(ii) For each morphism f ∈ MorC(A, B), a morphism F (f) ∈ MorD(F (A), F (B)) thatsatisfies:

(a) F (f ◦g) = F (f)◦F (g), if dom(f) = cod(g) for every pair of morphisms f, g ∈ C;

(b) F (1A) = 1F (A), for every A ∈ Ob(C).

1 Algebraic Hyperstructures: Origins

This chapter is exclusively focused on the historical chronology, no technicaldevelopments (relationship between several definitions of the same concept, for instance)will be done here. Doing this would constitute the subject of a research project of a differentnature than the present thesis.

An operation is basically a function1 that manipulate elements of a set andreturns a value that is in another set. And if we think about a more general definition?Something that returns a set of values instead of a single value? This concept has alreadybeen thought and named multioperation (or hyperoperation).

“Operation” is a fundamental concept in algebraic theory. It is used to definealgebras, that are structures such as groups, lattices, rings2, composed by a set andat least one operation. In a similar way, we can have structures composed by at leastone multioperation and the class of this kind of structures is what we call of algebraichyperstructure. The elements in the class of hyperstructures are multialgebras, that isalgebras equipped with at least one multioperation.

In this chapter we will present some examples of multialgebras, such as multi-groups, multilattices and so on, from a historical point of view and we will relate thesehyperstructures with non-deterministic matrices.

The concept of non-deterministic matrices will be presented with more rigor inthe last section, but in essence, non-deterministic matrices are multialgebras and they havebeen used as semantics for logical systems that do not have usual deterministic semantics.

As we will see, hyperstructures have been studied by several authors, butwith different nomenclatures and this hinders the development of a detailed and precisechronology.

The intrinsic relationship between multialgebras and non-deterministic matricesis not evident in the literature. In this sense, we believe that our work will give an importantcontribution.1 The mathematical concept in the previous chapter.2 For a formal definition of these concepts, see the previous chapter.

1.1 HypergroupsThe study of hyperstructures began with the presentation of the paper entitled

“Sur une généralisation de la notion de groupe”, in 1934 by the French mathematicianFrédéric Marty in the 8º Congress of Scandinavian Mathematicians (MARTY, 1934). Inthis paper, Marty presents the notion of hypergroups (or multigroups) from the analysisof their properties. But, due to his premature death, Marty only published two papersrelated to his concept of hypergroups.

1934 was the year that Frederic Marty defined the hypergroup (MARTY, 1934).This happened in connection with his thesis on meromorphic functions, whichwas written under the direction of Paul Montel. Unfortunately F. Marty diedyoung, during the Second World War, when his airplane was shot down overthe Baltic Sea, while he was going on a mission to Finland. In the duration ofhis short life (1911-1940), F. Marty studied properties and applications of thehypergroups in two more communications (MARTY, 1935; MARTY, 1936).(MASSOUROS; MASSOUROS, 2006, p. 19)

In 1937, H. S. Wall (WALL, 1937) and M. Krasner (KRASNER, 1937) alsogave their respective definitions of hypergroup.

As R. Bayon and N. Lygeros highlight in (BAYON; LYGEROS, 2008, p. 821),the origin of the hyperstructures is still not completely known but the Marty’s hypergroupdefinition took to emergence of several works related to multialgebras. We will quote someof them in the course of this chapter.

In (MARTY, 1934), Frédéric Marty introduced the following definition ofhypergroup:

Let be a set of elements, non-empty, with four combination laws: AB, BA,AB| and A

|B , each of which may have several determinations; the first two areassociative. If C is a determination of AB we write AB ⊃ C, (AB containsC). We will say that the family is a hypergroup if the two divisions arerelated to multiplication by the following relations: A

B| ⊃ C � BC ⊃ A andA|B ⊃ C � CB ⊃ A (MARTY, 1934, p. 46, our translation)3

In the definition of F. Marty, AB| represents the division on the right and A

|Brepresents the division on the left.

The definition of Marty is equivalent to the following definition:

Definition 1.1.1. (KRASNER, 1937) A set H organized by a composition law a.b ofeach pair a, b ∈ H is called hypergroup with regard to this composition law if3 Soit un ensemble d’éléments, non vide, possédant quatre lois de combinaison AB, BA, A

B| et A|B ;

chacune d’elles pouvant avoir plusieurs déterminations; les deux premières étant associatives. Si Cest une détermination de AB nous écrirons AB ⊃ C, (AB contient C). Nous dirons que la familleconstitue un hypergroupe si les deux divisions sont liées à la multiplication par les relations suivantes:

(i) a.b is a non-empty subset of H;

(ii) (a.b).c = a(b.c) (associative law);

(iii) For each pair a, c ∈ H there is x ∈ H such that c ∈ a.x and there is x� ∈ H suchthat c ∈ x�.a.

Also Christos G. Massouros, in his paper (MASSOUROS, 1989, p. 7) showshow to derive the property of regenerativity4 from original definition of F. Marty and lateron, in (MASSOUROS; MASSOUROS, 2006; MASSOUROS; MASSOUROS, 2014) theauthors present the following definition of hypergroup equivalent to the definition of F.Marty.

Remark 1.1.2. Many mathematicians in several countries contributed to the studies ofthe hypergroups, a bibliographical list can be found online on the site of A.H.A. (AlgebraicHyperstructures and Applications)5, a scientific group of Democritus University of Thracein Greece. One of the first books dedicated to hypergroups was written by P. Corsini in1993 and it is named “Prolegomena of hypergroup theory” (CORSINI, 1993).

In the following, we will present another important definition in the history ofthe hyperstructures: the hyperlattice definition.

1.2 HyperlatticesThe concept of hyperlattice was introduced by the Romanian algebraist Mihail

Benado in 1953 in the paper “Asupra unei generalizari a noţiunii de structura” (BENADO,1953). In this work, Benado presents two equivalent definitions of hyperlattice and alsosome examples.

Despite the introduction of the concept already appears in the work of Benado of1953, several authors uses the paper “Les ensembles partiellement ordonnés et le théorèmede raffinement de Schreier. II. Théorie des multistructures” (BENADO, 1955), publishedin 1955 as initial reference for the hyperlattice theory. In this work, Benado called théoriedes multistructures to what we call of hyperlattice theory. In this section our definitionwill be based on (BENADO, 1955).

Definition 1.2.1. (BENADO, 1955) A multistructure6 (now known as multilattice orhyperlattice) is any partially ordered set (poset)7 P that satisfies the following principles:4 Let �H, ·� be a hypergroup and a ∈ H, the regenerativity is the property: a.H = H.a = H.5 <http://aha.eled.duth.gr/>.6 Remember that in this work, we use the name multistructure/hyperstructure to denote the class of all

multialgebras such as hypergroups, hyperlattices, and so on.7 See Definition 0.1.11

M1) Let a, b ∈ P , if there is Ω ∈ P such that Ω ≥ a and Ω ≥ b, then there is also anM ∈ P such that M ≤ Ω, M ≥ a and M ≥ b and the conditions x ≤ M , x ≥ a andx ≥ b imply x = M .

M2) Let a, b ∈ P , if there is ω ∈ P such that ω ≤ a and ω ≤ b, then there is also a d ∈ P

such that d ≥ ω, d ≤ a and d ≤ b and the conditions y ≥ d, y ≤ a and y ≤ b implyy = d.

Benado uses the notation (a ∨ b)Ω to denote all M of (M1), that is (a ∨ b)Ω ={M : M is minimal in {x : x ≥ a, x ≥ b} and M ≤ Ω} and he uses the notation (a∧b)ω todenote all d of (M2), that is (a ∧ b)ω = {d : d is maximal in {y : y ≤ a, y ≤ b} and d ≥ ω}.

[. . .] I see that the meaning and the validity of several fundamental principlesof the structures theory do not depend on the fact that in the structure S,a ∨ b and a ∧ b are respectively the upper and lower bounds [12]8 of elementsa, b ∈ S, but it depends only on the fact that a ∨ b and a ∧ b are respectively aminimal element [12] between all x ∈ S, such that Ω ≥ x ≥ a, b and a maximalelement [12] between all y ∈ S, such that ω ≤ y ≤ a, b (here Ω, ω are arbitrary,but fixed) (BENADO, 1955, p. 309, our translation)9.

In this citation, the author makes clear the difference between lattices andhyperlattices. The main difference between Benado’s definition and the usual definitionof supremum (infimum, respectively) is that the minimal upper bounds (maximal lowerbounds, resp.) are considered instead of the minimum (the maximum, resp.)10

Still in the paper (BENADO, 1955), Benado presents also his definition ofsubmultistructure (or subhyperlattice as we call here).

Definition 1.2.2. (BENADO, 1955) Let M be any multistructure, a non-empty subsetR of M is a submultistructure of M if R satisfies the following conditions:

1�) If a, b ∈ R and there is at least an Ω ∈ R such that Ω ≥ a and Ω ≥ b, thenR ∩ (a ∨ b)Ω �= ∅.

2�) If a, b ∈ R and there is at least an ω ∈ R such that ω ≤ a and ω ≤ b, thenR ∩ (a ∧ b)ω �= ∅.

8 By [12], Benado refers to “N. Bourbaki, Théorie des ensembles (fasc. de résultats), Paris, Hermann,1939”.

9 [. . .] je me suis rendu compte de ce que le sens et la validité de plusieurs principes fondamentauxde la théorie des structures, ne dépendent nullement du fait que, dans la structure S, a ∨ b et a ∧ bsont respectivement les bornes supérieure et inférieure [12] des éléments a, b ∈ S; mais ils dépendentuniquement du fait que a ∨ b et a ∧ b y sont respectivement un élément minimal [12] parmi tous lesx ∈ S tels que Ω ≥ x ≥ a, b et un élément maximal [12] parmi tous les y ∈ S tels que ω ≤ y ≤ a, b (iciΩ, ω sont arbitraires mais fixes)(BENADO, 1955, p. 309).

10 For these concepts, see the Chapter 0.

Classically, a subset R� is a sublattice of a lattice R if R� is closed under theoperations ∨ and ∧, that is, for every a, b ∈ R�, (a∨b) ∈ R� and (a∧b) ∈ R�. Similarly, thedefinition of Benado provides that to R be a submultistructure of M, for every a, b ∈ R, Rmust have at least one element that satisfies the condition (M1) and at least one elementthat satisfies the condition (M2) (of multistructure definition).

And yet, Benado (BENADO, 1955) says that a submultistructure R of M isclosed if, for every a, b ∈ R of (1�) we have that (a ∨ b)Ω ⊆ R and for each a, b ∈ R of (2�)we have that (a ∧ b)ω ⊆ R.

Definition 1.2.3. (BENADO, 1955, p. 321) A multistructure is a non-empty set R withtwo operations ∧ and ∨11 satisfying the following axioms:

MI) Let a, b ∈ R, if a ∨ b �= ∅ (a ∧ b �= ∅) and b ∨ a �= ∅ (b ∧ a �= ∅). Then:

I �) a ∨ b = b ∨ a;

I ��) a ∧ b = b ∧ a.

MII) Let a, b, c ∈ R, if a ∨ b �= ∅ and (a ∨ b) ∨ c �= ∅ (a ∧ b �= ∅ and (a ∧ b) ∧ c �= ∅) and ifb ∨ c �= ∅ and a ∨ (b ∨ c) �= ∅ (b ∧ c �= ∅ and a ∧ (b ∧ c) �= ∅), then

II �) for each M ∈ (a ∨ b) ∨ c there is an M � ∈ a ∨ (b ∨ c) such that M ∨ M � �= ∅ andM ∨ M � = M ;

II ��) for each d ∈ (a ∧ b) ∧ c there is a d� ∈ a ∧ (b ∧ c) such that d ∧ d� �= ∅ andd ∧ d� = d.

MIII) Let a, b ∈ R, if a ∨ b �= ∅ (a ∧ b �= ∅) and if a ∧ (a ∨ b) �= ∅ (a ∨ (a ∧ b) �= ∅) then

III �) a ∧ (a ∨ b) = a;

III ��) a ∨ (a ∧ b) = a.

MIV ) For each a ∈ R we have a ∨ a �= ∅ and a ∧ a �= ∅.

MV ) Let a, b, c ∈ R such that a = b and if c ∨ a �= ∅ (c ∧ a �= ∅) and if c ∨ b �= ∅ (c ∧ b �= ∅)then

V �) c ∨ a = c ∨ b;

V ��) c ∧ a = c ∧ b.

MV I) Let a, b ∈ R such that a∨b �= ∅ (a∧b �= ∅) and let M, M � ∈ a∨b (d, d� ∈ a∧b) such thatM ∨ M � �= ∅ (d ∧ d� �= ∅) then if M �= M � (d �= d�) we have M �� �= M, M � (d�� �= d, d�)for each M �� ∈ M ∨ M � (d�� ∈ d ∧ d�).

11 According to Benado (BENADO, 1955) these operations are not necessarily universal and unambiguous,that is, there are a, b ∈ R, such that if a ∨ b �= ∅ (a ∧ b �= ∅), then the set a ∨ b (a ∧ b) may has at least

The axiom (MI) refers to commutativity, the axiom (MII) is a kind of partialassociativity. Marty called the axiom (MIII) of reduction (usually we call absorption).The axiom (MIV ) only says that a ∨ a and a ∧ a are not empty (for every a ∈ R), whilethe axiom (MV ) preserves equality between c ∨ a = c ∨ b and c ∧ a = c ∧ b in the case ofa = b.

Benado (BENADO, 1955) also showed that (MI)-(MV I) of Definition 1.2.3and (M1)-(M2) of Definition 1.2.1 are equivalent.

Hyperlattices have also been studied by other authors as D.J. Hansen (HANSEN,1981), that presents an alternative to the axiomatization given by Benado for characteriza-tion of a hyperlattice. The motivation of Hansen is to avoid partial associativity in Benadodefinition. The new axiomatic of Hansen only validates the axioms (MI) and (MIII)above and adds three new axioms.

Also in (MARTíNEZ et al., 2001), we found an alternative definition of hyper-lattice which aims to eliminate some disadvantages generated by generalized associativityin hyperlattice definitions of Benado and Hansen. Among the disadvantages cited by theauthors, we have a non natural generalization of the associative property and the fact thatsuch properties do not allow algebraic definition of submultilattice. So, (MARTíNEZ etal., 2001) introduce a new algebraic structure of hyperlattice with a weaker associativeproperty.

After Benado and before Hansen, the Brazilian mathematician Antonio AntunesMario Sette, motivated by algebraization of Cω

12, introduced the concept of hyperlatticeCω (SETTE, 1971) in his Master’s thesis (1971) supervised by Professor Newton da Costa.

To introduce the concept of hyperlattice Cω, Sette used the definition ofhyperlattice presented by José Morgado in the book “Introdução à Teoria dos Reticula-dos” (MORGADO, 1962). Morgado calls his hyperlattices of “reticuloides” and he usesthe concepts of “supremoide” and “infimoide” in his definition.

Definition 1.2.4. (MORGADO, 1962; SETTE, 1971) A hyperlattice (reticuloide) is asystem �R,�� consisting of a set R �= ∅ and a quasi-order ≤ (relation only reflexive andtransitive) such that, for all a, b ∈ R,

a � b �= ∅ �= a � b

where a�b (infimoide) is the set of all infimum of the pair (a, b) ∈ R2 and a�b (supremoide)is the set of all supremum of the pair (a, b) ∈ R2.

Remark 1.2.5. Note that in a partially ordered system, by antisymmetric property, wecan show the uniqueness of the supremum (infimum) (if it exists), but this result is not12 Cω is one of the paraconsistent logics wich is part of the systems Cn (1 � n � ω) of Newton da

obtained in the quasi-ordered systems13. Since to have more than one supremum (infimum)it is possible, it is legitimate to define the supremoide (infimoide) as the set of all suprema(infima).

Morgado also presents another reticuloide definition:

Definition 1.2.6. (MORGADO, 1962) A hyperlattice (reticuloide) is a system �R, �, ��consisting of a set R �= ∅ and two operations �, � : R × R −→ (P(R) − {∅}) such that,for all a, b, c ∈ R, the following conditions are satisfied:

h1) a � b = b � a;

h2) If x ∈ a � b and y ∈ b � c, then x � c = a � y;

h3) If x ∈ a � b, then a ∈ a � x.

h4) a � b = b � a;

h5) If x ∈ a � b and y ∈ b � c, then x � c = a � y;

h6) If x ∈ a � b, then a ∈ a � x;

The definitions of Morgado are more intuitive (more similar that usual latticedefinition) than those by Benado. In the Definition 1.2.4 what changes in relation tothe usual definition, is the loss of the uniqueness of the supremum and of the infimum.And in Definition 1.2.6, similar to the usual case, the items (h1) and (h4) correspond togeneralization of the commutativity, the items (h2) and (h5) are a kind of associativityand the items (h3) and (h6) are a kind of generalization of the absorption property.

In Morgado’s book and in the Sette’s dissertation we found only some consid-erations about the hyperlattices, a definition of an implicative hyperlattice and a definitionof the hyperlattice Cω. Sette, in the concluding remarks, notes that algebraization ofinconsistent formal systems can lead us to the consideration of hypersystems, this could bea generalization of reticuloides (hyperlattices), but he did not give any other informationabout the development of such hypersystems. The generalization of this idea of hypergroupand hyperlattice led to the emergence of hyperalgebras/multialgebras.

These more general structures will be the topic of the next section.

1.3 Multialgebras

13 That is the systems composed by a set and a quasi-order. For more information about these concepts,

In general, multialgebras (also known as hyperalgebras) are algebras such thatthe operations can return, for a given entry, a set of values instead of a single value. Theorigin of multialgebras is a little obscure, because of a very large number of papers camefrom Marty’s paper in 1934 (MARTY, 1934).

Some author, for example (WALICKI, 2005), consider as seminal work the twopapers: “Algebras with Operators. Part I” (JONSSON; TARSKI, 1951) and “Algebras withOperators” (JONSSON; TARSKI, 1952), both published by Bjarni Jonsson and AlfredTarski.

In these papers, Jonsson and Tarski introduce the concept of complex algebraand they prove the representation of Boolean algebras with operators by means of thesealgebras. However, the term “complex algebra” has several meanings in the literatureand is usually found in authors such as (CIRULIS, 2004) and (BOŠNJAK; MADARÁSZ,2003), with the name of full complex algebra.

Definition 1.3.1. (JONSSON; TARSKI, 1951, p. 933) A complex algebra of a relationalstructure14 U = �U, {Ri}i∈I� is defined by:

U+ = �P(U), {R+i }i∈I�

such that,

(i) U+ is a Boolean algebra with operators. A Boolean algebra with operators (BAO) isan algebra �A, {fi}i∈I� such that A is a Boolean algebra and fi are operators, thatis, operations over the Boolean algebra that are additive (distributive on the usualBoolean addition) on each of its arguments;

(ii) Let R ⊆ Un+1, then R+i : P(U)n −→ P(U) is an operation such that

R+(X0, X1, . . . , Xn−1) = {y ∈ U : (x0, x1, . . . , xn−1, y) ∈ R, for x0 ∈ X0, x1 ∈X1, . . . , xn−1 ∈ Xn−1}.

Following Marty and Benados’ line, that is, by defining hyperalgebras frommultioperations, we can say that the origin of multialgebras can be found in the paper“Bemerkungen zum Homomorphiebegriff” (Comments to the concept of homomorphism)of Günter Pickert, published in 1950 (PICKERT, 1950). The goal of the author in thispaper was to define homomorphism from structures, but what he calls “structure” is whatwe call multialgebra.

In 1958, and probably independently, the concept of multialgebra was introducedby P. Brunovský in “O zovšeobecnených algebraických systémoch” (A generalization ofalgebraic systems) (BRUNOVSKý, 1958). Brunovský’s definitions of multioperation andmultialgebra are the following:

Definition 1.3.2. (BRUNOVSKý, 1958) An n-ary generalized operation fα in the set A

is a function, such that for all sequences of elements n = n(α) in A, assigns any subset ofthe set A.

Definition 1.3.3. (BRUNOVSKý, 1958) Let be A a set and F any set of generalizedoperations in A. The set A with the set F will be designated multialgebra of A.

Brunovský cites the Benado multilattices as an example of multialgebras.

Note that the definition of Brunovský is quite similar to the definition of morerecent authors like Hansoul, Schweigert and Ameri and Rosenberg.

Definition 1.3.4. (HANSOUL, 1981; SCHWEIGERT, 1985; AMERI; ROSENBERG,2009) Let A be a non-empty set, a multioperation (or hyperoperation) (n-ary) σ on A is afunction σ : An → (P(A) − {∅}), such that n is a positive integer.

Definition 1.3.5. (HANSOUL, 1981; SCHWEIGERT, 1985; AMERI; ROSENBERG,2009) A multialgebra (or hyperalgebra) is the pair �A, {σi}i∈I�, such that A is a non-emptyset and {σi}i∈I is a family of multioperations on A.

The main difference between the Definitions 1.3.2, 1.3.3 and 1.3.4, 1.3.5 is thatthe value of an operation to be empty.

Multialgebras can be defined as relational structures with a composition ofrelations of arbitrary arity. Properties of multialgebras seen as relational systems can befound in (PICKETT, 1964). We present, now, some important concepts in the multialgebrastheory.

1.3.1 Homomorphisms of multialgebras and other conceptsThere are in the literature several generalizations of the notion of homomorphism

for multialgebras. We detected that Marty, in his paper “Rôle de la notion d’hypergroupedans l’étude des groupes non abéliens” (MARTY, 1935) already presented a concept ofhomomorphism for hypergroups: “[. . .] a representation of a hypergroup over (or in) otheris a homomorphism if the image of a determination of the product is the determination ofthe product of the images.”15 (MARTY, 1935, p. 636, our translation).

The definition of Marty means that given �G1, ·1� and �G2, ·2� two hypergroups16,the function h from G1 to G2 is a hypergroups homomorphism if, for all x,

x ∈ a.1b ⇒ h(x) ∈ (h(a).2h(b))15 The original in French: “[. . .] une représentation d’une hypergroupe sur (ou dans) un autre est une

homomorphie si l’image d’une détermination du produit est détermination du produit des images”.16 According to the definition of Marty (MARTY 1934).

.

Marty says, in the same paper, that a isomorphism between hypergroups is ahomomorphism such that the correspondence (in the definition above) is biunivocal. Theauthor also remarks the necessity to distinguish degrees of homomorphism and he presentshis definition of a quasi isomorphism, which is basically a kind of surjective homomorphismbetween hypergroups.

For the generalized notion of the multialgebras, again we can say that Brunovský,in (BRUNOVSKý, 1958), was the first to introduce a definition of homomorphism formultialgebras17.

Definition 1.3.6. (BRUNOVSKý, 1958) Let A = �A, F � and B = �B, G� be two mul-tialgebras of same type18 and let h be a function from A to B. Then, h is said to be ahomomorphism if, for all n-ary generalized operations fα ∈ F and for any sequence ofelements x1, . . . , xn in A, it is true:

h[fα(x1, . . . , xn)] = fα[h(x1), . . . , h(xn)].

In the literature, however, there are several definitions of homomorphismbetween multialgebras. In 1979, Francis Maurice Nolan, in his Ph.D. Thesis (NOLAN,1979), introduced five definitions of multialgebras homomorphisms and constructed acategory to each one. The definitions of Nolan are the following:

Definition 1.3.7. Let A = �A, F � and B = �B, F �� be two multialgebras of same typeand let h be a function from A to B,

• h is a full homomorphism between the multialgebras A and B if, for every f � ∈ F �,for every a ∈ A and for any sequence b1, . . . , bn ∈ B, h(a) ∈ f �(b1, . . . , bn) iff thereare a1, . . . , an ∈ A such that a ∈ f(a1, . . . , an) and h(ai) = bi for every i such that1 ≤ i ≤ n.

• h is a weak homomorphism between the multialgebras A and B if, for every f ∈ F

and for any sequence a1, . . . , an ∈ A, h(f(a1, . . . , an)) ⊆ f �(h(a1), . . . , h(an)).

• h is a strong homomorphism between the multialgebras A and B if, for every f ∈ F

and for any sequence a1, . . . , an ∈ A, h(f(a1, . . . , an)) = f �(h(a1), . . . , h(an)).17 Multialgebras in the sense of Definitions 1.3.2 and 1.3.3.18 That is, let A = �A, F � and B = �B, G� be two multialgebras such that F and G are their respective

sets of n-ary generalized operations. We say that the multialgebras A and B are of same type if itis possible to establish a biunivocal correspondence between the n-ary generalized operations of Aand the n-ary generalized operations of B, such that for each operation fα ∈ F , the operation fβ ∈ Gcorresponding to it, will be n-ary with the same n.

• h is a bimorphism between the multialgebras A and B if, for every f � ∈ F �, for everya ∈ A and for any sequence h(a1), . . . , h(an) ∈ h[A], h(a) ∈ f �(h(a1), . . . , h(an)) iffa ∈ f(a1, . . . , an).

• h is an absolute homomorphism between the multialgebras A and B if, for everyf � ∈ F �, for every b ∈ B and for any sequence b1, . . . , bn ∈ B, b ∈ f �(b1, . . . , bn) iffthere are a, a1, . . . , an ∈ A such that a ∈ f(a1, . . . , an), h(a) = b and h(ai) = bi forevery i such that 1 ≤ i ≤ n.

Similarly to concept of homomorphism, other concepts such as congruence,submultialgebra, direct product and so on, can also be defined in the context of themultialgebras. These concepts will be formally introduced in the next chapter and, therefore,we will talk briefly about the origin of some of them.

The concept of congruence was defined in the framework of multialgebras bySchweigert, in the paper called “Congruence relations of multialgebras”, published in1985 (SCHWEIGERT, 1985). The goal of Schweigert was to find a suitable concept ofvariety of multialgebras.

In this paper, Schweigert says that the Birkhoff theorem19 is valid for mul-tialgebras. However, the author simply notes that the demonstration is similar to thestatement of results for the usual algebras. The problem with this is that, as we saw before,in multialgebras we have many possibilities to define concepts such as homomorphism,submultialgebras, and others. Being so, this ambiguity is not a minor issue.

The concept of congruence of multialgebras is studied in detail in the paper“Congruences of multialgebras” of Reza Ameri and Ivo G. Rosenberg (AMERI; ROSEN-BERG, 2009). Other concepts such as identities and direct limit of multialgebras can befound on the Ph.D. Thesis by Cosmin Pelea (PELEA, 2003).

In the next section, we will briefly discuss an important result for the multial-gebras theory.

1.3.2 Representation theoremThe result known as representation theorem for multialgebras was studied firstly

in (GRÄTZER, 1962) and (HöFT; HOWARD, 1981). The representation theorem is oneof the most important results on multialgebras theory. Intuitively speaking, it proves thatthe study of multialgebras is a natural extension of the theory of universal algebra. Thistheorem was introduced by G. Grätzer, in the paper entitled “A representation theorem formulti-algebras” (GRÄTZER, 1962). But this theorem does not apply to multialgebras with19 All multialgebra can be represented as a product of directly irreducible multialgebras

multioperations such that the images can be an empty set, because these multialgebrasare closer to relational systems than to universal algebra.

The representation theorem of Grätzer uses the concept of concrete multialgebra.

Definition 1.3.8. (GRÄTZER, 1962, p. 453) Let A be an algebra with domain A and acollection of operations F , and let θ be an equivalence relation on A. A concrete multialgebrais a multialgebra A/θ which consists of the following elements:

(i) a set A/θ of the equivalence classes, such that if a ∈ A, then a/θ is the equivalenceclass represented by a; and

(ii) a set of multioperations of kind f(a1/θ, . . . , an/θ), such that if f ∈ F , then each mul-tioperation is defined by: f(a1/θ, . . . , an/θ) = {f(b1, . . . , bn)/θ : b1 ∈ a1/θ, . . . , bn ∈an/θ}.

Theorem 1.3.9. (GRÄTZER, 1962, p. 453) Every multialgebra is concrete.

This representation theorem states that every multialgebra is concrete, whatmeans that, for example, if A is a multialgebra with the only binary multioperation ·, thenthere is an algebra B with a binary operation · and with a equivalence relation θ such thatB/θ (defined as above) is isomorphic to A. So, every multialgebra has an algebra thatrepresents it.

Grätzer, also in his paper of 1962, lists some problems that arise naturallyfrom the representation theorem. For example, is the theorem equivalent to the axiom ofchoice? (Problem 1). This problem was solved by H. Höft and P. E. Howard in the paper“Representing multi-algebras by algebras, the axiom of choice, and the axiom of dependentchoice” (HöFT; HOWARD, 1981). In this paper, the authors showed that the axiom ofchoice is equivalent to the representation theorem for multialgebras.

Before Grätzer, Bjarni Jonsson and Alfred Tarski, in (JONSSON; TARSKI,1951, p. 933), also introduced a representation theorem for complex algebras (definedbriefly in Section 1.3). But, we present here the representation theorem of Grätzer insteadof the theorem of Jonsson and Tarski given that the focus of the work are the multialgebrasseen as algebras with multioperations (following the Marty’s line), instead of Jonsson andTarski perspective.

1.4 Hyperrings and hyperfields.In this section, we will briefly talk about the origin and definition of other

hyperstructures, namely as hyperrings and hyperfields.

A hyperring is a generalization of a ring where one of the operations is ahyperoperation and similarly, hyperfield is the structure that generalizes the usual conceptof field.

The concept of hyperfield was introduced by Marc Krasner in (KRASNER,1957) in connection with his work on valued fields.

Definition 1.4.1. (KRASNER, 1957; KRASNER, 1983) A hyperfield �C, +, ·� is a set C

with a operation (·) : C × C → C and with a multioperation (+) : C × C → P(C).

According to the established, the use of this multioperation can be extendedto subsets of C as following: A + B = �(a + b), for a ∈ A, b ∈ B, a + B = {a} + B andA + b = A + {b}. The structure �C, +, ·� satisfies the following properties:

(i) Properties of the operation (·):

a) C is a multiplicative semigroup20 with respect to operation (·) and has abilaterally absorbing element21 denoted by 0;

b) C − {0} is a group with respect to operation · and the identity element isdenoted by 1.

(ii) Properties of the multioperation (+):

a) For every a, b ∈ C, a + b = b + a (commutativity)

b) For every a, b, c ∈ C, (a + b) + c = a + (b + c) (associativity)

c) For every a ∈ C, there exists one and only one a� ∈ C, such that 0 ∈ a + a�22

(inverse element)

d) For every a, b, c ∈ C, if c ∈ a + b then b ∈ c + (−a) (quasi subtration)

(iii) Properties of distributivity:

a) For every a, b, c ∈ C, c · (a + b) = c · a + c · b

b) For every a, b, c ∈ C, (a + b) · c = a · c + b · c

The concept of hyperring was introduced in 1941, by Robert S. Pate in thepaper intitled “Rings with multiple-valued operations” (PATE, 1941). The main differencebetween the hyperrings and the usual rings is that in a hyperring, the addition is notnecessarily unique.20 See the Chapter 0 and (RASIOWA; SIKORSKI, 1963) for more about definitions and properties of

usual groups/fields.21 Let �S, ·� be a system composed by a set S and a binary operation ·, a bilaterally absorbing element on

�S, ·� is the element z such that, for every s in S, z · s = s · z = z.22 a� will be denoted by −a.

When modifying an axiom of the definition of Marc Krasner of hyperfield, wecan also get a hyperring definition. So, according to Marc Krasner (KRASNER, 1957;KRASNER, 1983), we have that a hyperring is a structure �C, +, ·� composed by a setC, an operation (·) : C × C → C and a multioperation (+) : C × C → P(C). Thestructure �C, +, ·� satisfy the properties of the items (ii) and (iii) and the property (i) (ofthe Definition 1.4.1) is replaced by the following property:

(i)’ C is a multiplicative semigroup with bilaterally absorbing element 0.

Marc Krasner define also a subhyperring as a subset C � of a hyperring C suchthat C � is closed for the multioperation (+), for the operation (·) and for the inverseelement, that is, if a, b ∈ C � then a + b ∈ C � and a · b ∈ C � and if a ∈ C �, then −a ∈ C �.

In the next section, we will introduce the definition of other kind of hyperstruc-tures, obtained by replacing some axioms (as the axiom of associativity and commutativity),by weaker versions.

1.5 Hv-structuresThe Hv-structure concept was introduced by Thomas Vougiouklis, in his paper

“The fundamental relation in hyperrings. The general hyperfield” (VOUGIOUKLIS, 1991).In the quote below, Vougiouklis reveals his motivation for the introduction of this concept:

This paper concludes with the definition of a new class of hyperstructures,more general than the known ones, introduced by the author at the FourthInternational Congress on Algebraic Hyperstructures and Applications (AHA).The motivation to introduce this class was the uniting elements procedure[2]23. (VOUGIOUKLIS, 1991, p. 210, our translation).

In the quote, the uniting elements procedure is a method that allows you to putin the same class two or more elements. Vougiouklis (VOUGIOUKLIS, 2014) claims that,by means of hyperstructures, this method leads to structures with additional properties.

The Hv-structures are weaker generalizations of some hyperstructures, forexample hypergroups, hyperrings, hyperlattices, and so on. In the Hv-structures, someaxioms are replaced by corresponding weaker versions. See, for example, the definition ofa Hv-group:

Definition 1.5.1. (VOUGIOUKLIS, 1991) A Hv-group �G, ·� is a set G equipped with amultioperation (·) : G × G → (P(G) − {∅}) that satisfies the following axioms:23 By [2] the author refers to his paper of 1989 written together with Piergiulio Corsini: “From groupoids

to groups through hypergroups” (CORSINI; VOUGIOUKLIS, 1989).

1. For every a, b, c ∈ G, ((a · b) · c) ∩ (a · (b · c)) �= ∅ [weak associativity]

2. For every a ∈ G, a · G = G · a = G.

In the definition above, the non-empty intersection, called by him weak asso-ciativity, substitutes the equality of the usual associativity.

In the same paper, Thomas Vougiouklis says that structures like the Hv-groups,Hv-rings and Hv-fields and so on, can be defined similarly, by replacing the associativelaw for its weaker version (1) and the commutative law by the following weaker version:

(a · b) ∩ (b · a) �= ∅ (for every a, b ∈ G).

The definition of the Hv-groups motivated that several papers related toHv-structures have appeared in the literature in recent years, as for example (VOU-GIOUKLIS; SPARTALIS; KESSOGLIDES, 1997), (VOUGIOUKLIS, 1999), (DAVVAZ,2000), (DAVVAZ, 2006), (VOUGIOUKLIS, 2008) and (VOUGIOUKLIS, 2014).

With relation to the application of hyperstructures, a lot of work have beendone in several areas of Mathematics (pure and applied) like in algebra, geometry, topology,graph theory, probability theory, theory of automata, fuzzy theory, and so on. However,this is not the only way to apply hyperstructures. In the next section, we will talk aboutother application of hyperstructures, strongly related to the non-determinism notion.

1.6 Non-deterministic matricesAt the beginning of this chapter, we already highlight the strong relationship

between non-deterministic matrices and hyperstructures. Now, we will talk about theorigin of the concept of non-deterministic matrix.

Non-deterministic matrix is a generalization of the usual concept of many-valued matrix24. So, before going to non-determinism, let us talk briefly about the originof the algebraic theory of logical matrices.

The idea of logical matrices was used by Pierce (PEIRCE, 1885) and bySchröder (SCHRÖDER, 1891). They applied truth-tables for manipulating logical problems,but both had as focus only the classical logic. According to (WÓJCICKI, 1984), thenotion of logical matrix was defined in a more rigorous and general way by Łukasiewiczand Tarski in (ŁUKASIEWICZ; TARSKI, 1930) and the foundations of the theory oflogical matrices were developed by Loś in (ŁOŚ, 1949) and in a series of papers of24 Additional information about many-valued matrices can be found at (ROSSER; TURQUETTE, 1952),

(BOLC; BOROWIK, 1992), (MALINOWSKI, 1993), (GOTTWALD, 2001) and (HAHNLE, 2001).

Kalicki (KALICKI, 1949), (KALICKI, 1950b), (KALICKI, 1950a) and (KALICKI, 1952).Wójcicki highlights the work of Wajsberg (WAJSBERG, 1935), Jaśkowski (JAŚKOWSKI,1936), Tarski (TARSKI, 1938) and Suszko (SUSZKO, 1957) as being of considerableimportance in the development of the theory of logical matrices. Formally:

Definition 1.6.1. A logical matrix for a propositional language L(Σ, Ξ) can be definedas a pair M = �A, D� such that A = �A, σA� is an algebra25 over Σ with domain A, andD is a subset of A. The elements of D are called designated elements.

Logical matrices are used to give a natural semantics for propositional logicsand they also play an important role in the general techniques of algebraization of logicsintroduced by W. Blok and D. Pigozzi (BLOK; PIGOZZI, 1986; BLOK; PIGOZZI, 1989).However, the logical matrices can be used not only like semantics for bivalent logics (as inthe case of classical propositional logic), but also for many-valued logics, as is the case ofsome modal logics and many-valued logics in general.

Although several propositional logics can be characterized semantically usinga many-valued matrix (ŁOŚ; SUSZKO, 1975), many of them have only infinite matricesand such matrices do not constitute a good decision procedure for these logics. So, analternative solution is the use of the non-deterministic matrices.

The aplication of non-deterministic matrices was quite studied by Avron andhis colaborators. At this point we cannot speak too much about the origin of this concept,but we discovered that in 1962, N. Rescher (RESCHER, 1962) was using non-deterministicmatrices with the name of quasi-truth-functional systems. This concept also were used byJ. Kearns in 1981 (KEARNS, 1981) and by Y. Ivlev in 1988 (IVLEV, 1988). We believethat all these authors gave their non-deterministic matrix definition independently.

The non-deterministic matrix was studied by Avron and Lev firstly in (AVRON;LEV, 2001). Although they were not the first to introduce this concept, this idea wasquite disseminated through his papers. So, in the next lines, we will give more attentionfor Avron and Lev’s definitions.

Avron and Zamansky (AVRON; ZAMANSKY, 2011) motivated by the con-flict between the truth-functionality principle and the non truth-functional characterof information present in the real world, use non-deterministic matrices to weaken thisprinciple.

See bellow, an example of application of non-determinism. This example wasbased on Example “linguistic ambiguity” in (AVRON; ZAMANSKY, 2011, p. 3).

Example 1.6.2. In the natural language, the word “or” can have two meanings: aninclusive and other exclusive. For example:

(1) My father is, right now, playing soccer in Brazil or in Japan.

(2) I’m going to buy the pair of shoes light blue or the pair of shoes dark blue.

In the item (1), the disjunction “or” is exclusive, because a person cannot be intwo places at the same time, but in the item (2) the disjunction “or” is inclusive, becauseif I’m with a doubt about which pair of shoes to buy, I can buy the two pair of shoes.

The problem associated with the use of “or” is because, in many cases, we can’tdistinguish if the “or” in question is inclusive or if it is exclusive, however, even in thesecases, we would like to be able to infer something from what was said, and this can be donethrough the non-deterministic matrices, see below:

∨1 1 {1,0}1 0 {1}0 1 {1}0 0 {0}

Now, we present the formal definition of non-deterministic matrix.

Definition 1.6.3. (AVRON; LEV, 2001, p. 536) A non-deterministic matrix (or in shortNmatrix) for a propositional language L(Σ, Ξ)26 is a ordered triple M = �V, D, O�, suchthat:

(i) V is a non-empty set of truth-values,

(ii) D is a proper and non-empty subset of V called set of designated truth values and

(iii) O assigns one corresponding function c : V n → (P(V ) − {∅}) for all n-ary connectivec ∈ Σn.

Note in the above definition that the function c is a multioperation and so thenon-deterministic matrices are multialgebras.

One of the main features of the non-deterministic matrices is that the truthvalue of a complex formula can be chosen non-deterministically from a non-empty set ofoptions. In (AVRON; LEV, 2001) we also found a definition of valuation and of semanticsconsequence in the non-deterministic matrices theory. These concepts will be used in thenext chapters and, therefore, they will be presented now.26 Recall the definition of L(Σ, Ξ) in the Chapter 0. Note that Avron denotes, in a simplified way, by L.

Definition 1.6.4. (AVRON; LEV, 2001, p. 536) A valuation in an Nmatrix M = �V, D, O�is a function v : L(Σ, Ξ) → V such that, for every c ∈ Σn, α1, ..., αn ∈ L(Σ, Ξ) and n ∈ Nthe following condition is satisfied:

v(c(α1, ..., αn)) ∈ c(v(α1), ..., v(αn)).

Definition 1.6.5. (AVRON; LEV, 2001, p. 536) Let be Δ ∪ {α} ⊆ L(Σ, Ξ), then Δ �M α

if, for all valuation v in an Nmatrix M = �V, D, O�,

v[Δ] ⊆ D ⇒ v(α) ∈ D.

In particular, if Δ = ∅, then �M α if, for all valuation v in M, v(α) ∈ D.

The non-deterministic matrices motivated the development of this work, because,even though much has already been done by Avron and his collaborators, the non-deterministic matrices have not been linked to multialgebras from the point of view ofuniversal algebra. So, the next chapter is dedicated to presenting a non-deterministicalgebraic theory based on multialgebras.

2 Some concepts in universal multialgebra

As has been said in the first chapter, multialgebras have been very much studiedin the literature, but the generalization from algebra universal to multialgebras of evenbasic conceps such as homomorphism, subalgebras and congruences is far from obvious,and several different alternatives were proposed in the literature.

In this chapter we begin the study of formal properties of multialgebras, withemphasis on universal algebra, through the development of complementary results to whathas already been developed in the theory of hyperstructures and already mentioned in theChapter 1 of this thesis.

This study will enable the appropriate choice of concepts in multialgebra theoryin order to obtain the results in subsequent chapters.

The definitions and results presented in this chapter were adapted to ourpurpose in accordance with results from literature of multialgebras. The reference for thischapter is the Chapter 1, whereas it is clear that the definitions have been adapted to ourcontext. Also worth to mention that the same definitions (or results) have been proposedby several authors using a different languages that one used in this Thesis. Therefore, it isdifficult to adopt and to reference definitions (or results) from a specific author.

2.1 Multialgebras and homomorphismsIn this section, we present several concepts in multialgebras theory that we will

adopt along this work as well as results in the category of multialgebras. At some placesthese algebras can be called non-deterministic algebras1.

Definition 2.1.1 (Multialgebra). Let Θ be a signature. A multialgebra (or hyperalgebra)over Θ is a pair A = �A, σA� such that A is a nonempty set (the universe or support ofA) and σA is a mapping assigning to each c ∈ Θn, a function (called multioperation orhyperoperation) cA : An → (P(A) − {∅}). In particular, ∅ �= cA ⊆ A if c ∈ Θ0.

Notation 2.1.2. If c ∈ Θ0 such that cA is a singleton, then we will denote by cA thesingle element of cA; that is cA = {cA}. On the other hand, the support of A will befrequently denoted by |A|.1 Because of the Nmatrices of Avron and his colaborators (AVRON; LEV, 2001; AVRON; LEV, 2005;

AVRON; KONIKOWSKA, 2005).

Now, we can present a similar definition of non-deterministic matrix (Definition1.6.3) using the concept of multialgebras:

Definition 2.1.3 (Non-deterministic matrix). Let Θ a signature. A non-deterministicmatrix (or Nmatrix) is a pair M = �A, D� such that A = �A, σA� is a multialgebra over Θwith support A, and D is a subset of A. The elements in D are called designated elements.

From the definition of non-deterministic matrices, recall that the semantics2

associated to non-deterministic matrices is given by:

Definition 2.1.4 (Valuation). Let M = �A, D� be a non-deterministic matrix over asignature Θ. A valuation3 over M is a function v : For(Θ) → |A| such that, for everyc ∈ Θn and every α1, . . . , αn ∈ For(Θ):

v(c(α1, . . . , αn)) ∈ cA(v(α1), . . . , v(αn)).

In particular, v(c) ∈ cA, for every c ∈ Θ0.

Definition 2.1.5 (Consequence relation). Let M = �A, D� be a non-deterministic matrixover a signature Θ, and let Γ ∪ {α} ⊆ For(Θ). We say that α is a consequence of Γ inthe non-deterministic matrix M, denoted by Γ |=M α, if the following holds: for everyvaluation v over M, if v[Γ] ⊆ D then v(α) ∈ D. In particular, α is valid in M, denotedby |=M α, if v(α) ∈ D for every valuation v over M.

Avron in (AVRON, 2005, p. 156 and p. 157) presents two non-deterministicmatrices MB

5 and MB3 , that semantically characterize the logical system B, which is

known in literature as mbC, one of the simplest Logics of Formal Inconsistency (LFIs) 4.These non-deterministic matrices will be presented in the following two examples, andsubsequently analyzed in the light of the concepts introduced, along with other non-deterministic matrices introduced in the literature.

Example 2.1.6. Let Σ = {∧, ∨, →, ¬, ◦} be a signature and M5 = �A5, D5� be thenon-deterministic matrix over Σ such that:

(i) |A5| = A5 = {t, tI , I, f, fI};

(ii) D5 = {t, tI , I};2 These definitions were adapted from the definitions of Avron (AVRON; LEV, 2001; AVRON; LEV,

2005; AVRON; KONIKOWSKA, 2005).3 It is worth noting that Avron and his collaborators (AVRON; LEV, 2001; AVRON, 2005; AVRON;

ZAMANSKY, 2011) use the term legal valuation to refer to valuations over an Nmatrix.4 Introduced by W. Carnielli and J. Marcos in (CARNIELLI; MARCOS, 2002) and after studied in

detail in (CARNIELLI; CONIGLIO; MARCOS, 2007) and (CARNIELLI; CONIGLIO, 2016).

(iii) For each connective c, the multioperation σA5(c) = cA5 is defined by the followingtables (here, F5 = {f, fI}).

∨A5 t tI I f fI

t D5 D5 D5 D5 D5

tI D5 D5 D5 D5 D5

I D5 D5 D5 D5 D5

f D5 D5 D5 F5 F5

fI D5 D5 D5 F5 F5

∧A5 t tI I f fI

t D5 D5 D5 F5 F5

tI D5 D5 D5 F5 F5

I D5 D5 D5 F5 F5

f F5 F5 F5 F5 F5

fI F5 F5 F5 F5 F5

→A5 t tI I f fI

t D5 D5 D5 F5 F5

tI D5 D5 D5 F5 F5

I D5 D5 D5 F5 F5

f D5 D5 D5 D5 D5

fI D5 D5 D5 D5 D5

¬A5

t F5

tI F5

I D5

f D5

fI D5

◦A5

t D5

tI F5

I F5

f D5

fI F5

Clearly, M5 induces a multialgebra A5 = �A5, σA5� over Σ.

Example 2.1.7. Let Σ = {∧, ∨, →, ¬, ◦} be a signature and let M3 = �A3, D3� be thenon-deterministic matrix over Σ such that:

(i) |A3| = A3 = {t�, I �, f �};

(ii) D3 = {t�, I �};

(iii) For each connective c, the multioperation σA3(c) = cA3 is defined by the followingtables:

∨A3 t� I � f �

t� D3 D3 D3

I � D3 D3 D3

f � D3 D3 {f �}

∧A3 t� I � f �

t� D3 D3 {f �}I � D3 D3 {f �}f � {f �} {f �} {f �}

→A3 t� I � f �

t� D3 D3 {f �}I � D3 D3 {f �}f � D3 D3 D3

¬A3

t� {f �}I � D3

f � D3

◦A3

t� A3

I � {f �}f � A3

Let A �A � So A is multialgebr Σ

Remark 2.1.8. The example below will be denoted by M�3 although it is a 5-valued

matrix because M�3 is an extension of the 3-valued matrix M3.

Example 2.1.9. Let Σ = {∧, ∨, →, ¬, ◦} be a signature and M�3 = �A�

3, D�3� be the

non-deterministic matrix over Σ such that:

(i) |A�3| = A�

3 = {t�, t�I , I �, f �, f �

I};

(ii) D�3 = {t�, I �};

(iii) For each connective c, the multioperation σA�3(c) = cA�3 is defined by the followingtables (here, F �

3 = {f �}).

∨A�3 t� t�I I � f � f �

I

t� D�3 D�

3 D�3 D�

3 D�3

t�I D�

3 D�3 D�

3 D�3 D�

3

I � D�3 D�

3 D�3 D�

3 D�3

f � D�3 D�

3 D�3 F �

3 F �3

f �I D�

3 D�3 D�

3 F �3 F �

3

∧A�3 t� t�I I � f � f �

I

t� D�3 D�

3 D�3 F �

3 F �3

t�I D�

3 D�3 D�

3 F �3 F �

3

I � D�3 D�

3 D�3 F �

3 F �3

f � F �3 F �

3 F �3 F �

3 F �3

f �I F �

3 F �3 F �

3 F �3 F �

3

→A�3 t� t�I I � f � f �

I

t� D�3 D�

3 D�3 F �

3 F �3

t�I D�

3 D�3 D�

3 F �3 F �

3

I � D�3 D�

3 D�3 F �

3 F �3

f � D�3 D�

3 D�3 D�

3 D�3

f �I D�

3 D�3 D�

3 D�3 D�

3

¬A�3

t� F �3

t�I F �

3

I � D�3

f � D�3

f �I D�

3

◦A�3

t� {t�, I �, f �}t�I {t�, I �, f �}

I � F �3

f � {t�, I �, f �}f �

I {t�, I �, f �}

Clearly, M�3 induces a multialgebra A�

3 = �A�3, σA�3� over Σ.

Definition 2.1.10 (Homomorphisms of multialgebras). Let A = �A, σA� and B = �B, σB�be two multialgebras over a signature Θ, and let h : A → B be a function.(i) h is said to be a homomorphism5 from A to B, denoted by h : A → B, if

h[cA(�a)] ⊆ cB(h(a1), . . . , h(an))

for every c ∈ Θn and �a ∈ An. In particular, h[cA] ⊆ cB for every c ∈ Θ0.(ii) h is said to be a full homomorphism6 from A to B, which is denoted by h : A →s B, if

h[cA(�a)] = cB(h(a1), . . . , h(an))

for every c ∈ Θn and �a ∈ An. In particular, h[cA] = cB for every c ∈ Θ0.5 Nolan called weak homomorphism, see Definition 1.3.7.

Notation 2.1.11. When there is no risk of confusion, a full homomorphism will be simplydenoted by h : A → B instead of h : A →s B.

Example 2.1.12. Let A5 = �A5, σA5� and A3 = �A3, σA3� be the multialgebras introducedin the Examples 2.1.6 and 2.1.7. Let h : A5 → A3 be a function such that h(t) = h(tI) = t�,h(I) = I � and h(f) = h(fI) = f �. Clearly,

h[D5] = D3 and h(F5) = {f �}.

From here, and by definitions of σA5 and σA3, is immediate that h[cA5 (a, b)] ⊆ cA3(h(a), h(b))for every a, b ∈ A5 and c ∈ {∨, ∧, →} (in fact, the equality holds). In the same wayh[¬A5 a] ⊆ ¬A3 h(a) for every a ∈ A5 (again, the equality holds).

Finally, h[◦A5 a] = h[D5] = D3 ⊂ A3 = ◦A3 h(a), for a ∈ {t, f}, whileh[◦A5 I] = h[F5] = {f �} = ◦A3 I � = ◦A3 h(I) and h[◦A5 a] = h[F5] = {f �} ⊂ A3 = ◦A3 h(a),for a ∈ {tI , fI}.

For example, h[◦A5 t] = h[D5] = D3 ⊂ A3 = ◦A3 h(t) = ◦A3 t�, and h[◦A5 fI ] =h[F5] = {f �} ⊂ A3 = ◦A3 f � = ◦A3 h(fI).

So, h[cA5 a] ⊆ cA3 h(a) for every a ∈ A5 and c ∈ {¬, ◦} and therefore h definesa homomorphism h : A5 → A3.

Example 2.1.13. Let A5 = �A5, σA5� and A3 = �A3, σA3� be the multialgebras introducedin the Examples 2.1.6 and 2.1.7. Let h : A3 → A5 the function such that h(t�) = I,h(I �) = fI and h(f �) = tI .

We have that h[◦A3 I �] = h[f �] = {tI} � F5 = ◦A5 fI = ◦A5 h(I �). It shows thatthere is b ∈ A3, such that h[◦A3 b] � ◦A5 h(b) and so, h does not define a homomorphismh : A3 → A5.

However, we still note that h[◦A3 a] = h[A3] = {fI , tI , I} � F5 = ◦A5 h(a) fora ∈ {t�, f �} that is h[◦A3 a] � ◦A5 h(a) for all a ∈ A3.

When there is no risk of confusion, we will assume that multialgebras aredefined over a fixed signature Θ.

Just as Benado (BENADO, 1955) introduced the concept of submultilattice,other authors such as Pickett (PICKETT, 1967), Hansoul (HANSOUL, 1981) and Pe-lea (PELEA, 2003) also studied the concepts of submultialgebra. In the next section, wewill introduced some new results related to submutialgebras.

2.2 SubmultialgebrasFrom the definitions of homomorphisms, we will now analyze the notion of

Definition 2.2.1 (Submultialgebra). Let B = �B, σB� and A = �A, σA� be two multial-gebras over Θ. Then B is said to be a submultialgebra of A, denoted by B ⊆ A, if thefollowing conditions hold:

(i) B ⊆ A,

(ii) if c ∈ Θn and �b ∈ Bn, then cB(�b) ⊆ cA(�b); in particular, cB ⊆ cA if c ∈ Θ0.

In Proposition 2.5.1 the meaning of submultialgebras will be clarified.

Example 2.2.2. Let A3 = �A3, σA3� and A�3 = �A�

3, σA�3� be the multialgebras introducedin the Examples 2.1.7 and 2.1.9.

We can see that A3 ⊆ A�3 and by definitions of σA3 and σA�3 we have, for every

c ∈ Θn and �a ∈ An3 ,

cA3(�a) ⊆ cA�3(�a).

Therefore, A3 is submultialgebra of A�3, that is, A3 ⊆ A�

3.

In fact, we have more, we have cA3(�a) = cA�3(�a).

In a similar way to universal algebra, we can apply the concept of subuniverseto multialgebras.

Definition 2.2.3 (Submultiuniverse). Let A = �A, σA� a multialgebra. A submultiuniverseof A is a non-empty subset B of A which is closed under the multioperations of A, that is,for every n ≥ 0, c ∈ Θn and �b ∈ Bn, cA(�b) ⊆ B.

Note that if B = �B, σB� is a submultialgebra of A = �A, σA�, then B is asubmultiuniverse of A. So:

Example 2.2.4. Let A3 = �A3, σA3� and A�3 = �A�

3, σA�3� be the multialgebras introducedin the Examples 2.1.7 and 2.1.9. We have that A3 is submultiuniverse of A�

3.

Definition 2.2.5 (Generated submultiuniverse). Let A = �A, σA� be a multialgebra and∅ �= X ⊆ A. The submultiuniverse of A generated by X, denoted by sgA(X) (in shortsg(X)) is defined by:

sg(X) =�

{B : B is submultiuniverse of A and X ⊆ B}.

Remark 2.2.6. Note that A is submultiuniverse of A that contains X. So, sg(X) is welldefined, since the set {B : B is submultiuniverse of A and X ⊆ B} is non-empty.

Proposition 2.2.7. If A = �A, σA� is a multialgebra and ∅ �= X ⊆ A, then sg(X) is, in

Proof. Note that sg(X) ⊆ A and sg(X) �= ∅. Let n ≥ 0, c ∈ Θn and b1, . . . , bn ∈ sg(X). IfB is a subuniverse of A such that X ⊆ B, then cA(b1, . . . , bn) ⊆ B, thus cA(b1, . . . , bn) ⊆sg(X) and so sg(X) is a submultiuniverse of A.

As in the case of usual algebras, we can give a constructive definition of the setsg(X).

Proposition 2.2.8. Let A = �A, σA� be a multialgebra, ∅ �= X ⊆ A and let {En(X) :n ≥ 0} be a family of subsets of A, defined by recursion as follows:

E0(X) = X

En+1(X) = En(X) ∪ �{cA(a1, . . . , ak) : k ≥ 0, c ∈ Θk and a1, . . . , ak ∈ En(X)}.

Then, sg(X) = �{En(X) : n ≥ 0}.

Proof. In (AMERI; ROSENBERG, 2009, Theorem 3.17, p. 11).

Lemma 2.2.9. Let A = �A, σA� and B = �B, σB� be two multialgebras, ∅ �= X ⊆ A andlet h : A → B be a homomorphism. If En(X) and En(h[X]) are defined recursively as inProposition 2.2.8 then h[En(X)] ⊆ En(h[X]).

Proof. The proof is by induction over n, n ≥ 0. If n = 0, h[E0(X)] = h[X] = E0(h[X]).Suppose that h[En(X)] ⊆ En(h[X]); then

h[En+1(X)] = h[En(X) ∪ �{cA(a1, . . . , ak) : k ≥ 0, c ∈ Θk and a1, . . . , ak ∈ En(X)}] =h[En(X)] ∪ h[�{cA(a1, . . . , ak) : k ≥ 0, c ∈ Θk and a1, . . . , ak ∈ En(X)}] ⊆En(h[X]) ∪ �{h[cA(a1, . . . , ak)] : k ≥ 0, c ∈ Θk and a1, . . . , ak ∈ En(X)}) ⊆En(h[X]) ∪ �{cB(h(a1), . . . , h(ak)) : k ≥ 0, c ∈ Θk and h(a1), . . . , h(ak) ∈ En(h(X))} =En+1(h[X]).

Remark 2.2.10. In the previous lemma, if h is a full homomorphism, then h[En(X)] =En(h[X]).

Theorem 2.2.11. Let A = �A, σA� and B = �B, σB� be two multialgebras, ∅ �= X ⊆ A

and let h : A → B be a homomorphism. Then,

h[sg(X)] ⊆ sg(h[X]).

Proof. By Proposition 2.2.8, we have

h[sg(X)] = h� �

{En(X) : n ≥ 0}�

=�

{h[En(X)] : n ≥ 0}.

Using the previous lemma and Proposition 2.2.8, we have�

{h[En(X)] : n ≥ 0} ⊆�

{En(h[X]) : n ≥ 0} = sg(h[X]).

Definition 2.2.12 (Direct image). Let A = �A, σA� and B = �B, σB� be two multialgebras,and let h : A → B be a homomorphism. The direct image of h is the multialgebrah(A) = �h[A], σh(A)�, such that, for every c ∈ Θn and �b ∈ h[A], ch(A)(�b) = � �

h[cA(�a)] :�a ∈ h−1(�b)

�. In particular, ch(A) = h[cA] for every c ∈ Θ0.

Proposition 2.2.13. Let A = �A, σA� and B = �B, σB� be two multialgebras, and let h :A → B be a homomorphism. The direct image of h, h(A) = �h[A], σh(A)� is submultialgebraof B.

Proof. If �b ∈ h[A] and �a ∈ h−1(�b) then h[cA(�a)] ⊆ cB(h(�a)) = cB(�b). Hence ch(A)(�b) ⊆ cB(�b).So, h(A) ⊆ B.

Although some of the results that were presented, already exist in the literatureof multialgebras, the notion of interpretation of formulas has not yet been properly studiedand it will be the topic of the next section.

2.3 Interpretation of formulasIn the book “Fuzzy algebraic hyperstructures: an introduction”, the authors

defined a semihypergroup from the definition of hypergroupoid using a generalized associa-tivity law, that is:

Definition 2.3.1. (DAVVAZ; CRISTEA, 2015, Definition 1.3.2, p. 17) A semihypergroupis a hypergroupoid �S, ·� such that for all a, b, c ∈ S,

�

u∈a·bu · c =

�

v∈b·ca · v

From generalized associativity law and by definition of legal valuation (Defini-tion 2.1.4), we can think about define a notion of interpretation of formulas in multialgebrastheory. For instance, suppose that if c ∈ Θ0, then cA is a singleton7 (that is, cA = {cA})and consider the two definitions below:7

Definition 2.3.2 (Assignment). Let A = �A, σA� be a multialgebra over a signature Θand let Ξ be the set of variables. An assignment in A is a function ρ : Ξ → A.

Definition 2.3.3 (Interpretation of formulas). Let A = �A, σA� be a multialgebra andlet ρ be an assignment in A. The multifunction (·)Aρ : L(Θ, Ξ) →M A is an interpretationαAρ of the formula α in A by ρ, and it is defined, by induction on the complexity of theformula α, as being the non-empty subset of A such that:

(i) ξAρ = {ρ(ξ)}, if ξ ∈ Ξ;

(ii) cAρ = {cA}, if c ∈ Θ0;

(iii) c(α1, . . . , αn)Aρ = �{cA(�a) : ai ∈ αAρi , for 1 ≤ i ≤ n}, if n > 0, c ∈ Θn and

αi ∈ L(Θ, Ξ) (for 1 ≤ i ≤ n).

After looking quickly through the example below we can think that the last isa good definition of interpretation of formulas in a multialgebra:

Example 2.3.4. Let A3 = �A3, σA3� be the multialgebra introduced in the Example 2.1.7and let ρ be an assignment in A3.

The interpretation of the formula ¬ ◦ ξ → (ξ ∧ ¬ξ) in A3, for ρ(ξ) = t� isobtained as follows:

(¬ ◦ ξ → (ξ ∧ ¬ξ))A3ρ = �{→A3(a1, a2) : a1 ∈ (¬ ◦ ξ)A3ρ and a2 ∈ (ξ ∧ ¬ξ)A3ρ}. But,(¬ ◦ ξ)A3ρ = �{¬A3(a) : a ∈ (◦ξ)A3ρ} and (◦ξ)A3ρ = �{◦A3(a) : a ∈ ξA3ρ}. On the otherhand (ξ∧¬ξ)A3ρ = �{∧A3(a1, a2) : a1 ∈ ξA3ρ and a2 ∈ (¬ξ)A3ρ}. But (¬ξ)A3ρ = �{¬A3(a) :a ∈ ξA3ρ} and ξA3ρ = {ρ(ξ)}. Since ρ(ξ) = t� we have (¬ξ)A3ρ = ¬A3(t�) = {f �}, (◦ξ)A3ρ =◦A3(t�) = {t�, I �, f �}. So, (¬ ◦ ξ)A3ρ = {t�, I �, f �}. Thus, (ξ ∧ ¬ξ)A3ρ = ∧A3(t�, f �) = {f �}.Therefore (¬ ◦ ξ → (ξ ∧ ¬ξ))A3ρ = {f �} ∪ {f �} ∪ {t�, I �, f �} = {t�, I �, f �}.

But see in the following example that this notion of interpretation does notalways works well:

Example 2.3.5. Let A5 = �A5, σA5� be the multialgebra introduced in the Example 2.1.6,let α the formula schema given by α = ¬(ξ1 ∨ ξ2) and let ρ be an assignment in A5

such that ρ(ξ1) = {I} and ρ(ξ2) = {f}. Then, αA5ρ = A5 and (¬α)A5ρ = A5 and thus,(α ∨ ¬α)A5ρ = A5. But, since this formula schema is an axiom of mbC, its interpretationshould belong to D5.

So, the question of defining the notion of interpretation of formulas in multial-gebras is still an open problem.

We already mentioned in Chapter 1 that the congruence concept was definedin the environment of multialgebras by Schweigert (SCHWEIGERT, 1985) and it wasextensively studied by Reza Ameri and Ivo G. Rosenberg (AMERI; ROSENBERG, 2009).In the next section, we will present this concept.

2.4 Multicongruences and quotient multialgebrasThe concepts of congruence and quotient algebra are fundamental tools in

the algebraization theory. Thus the generalization of these concepts to multialgebrastheory will be fundamental tools to obtain the results in the next chapters with respect toalgebraization of logical systems.

The natural generalization of the notion of congruence to multialgebras is asfollows:

Definition 2.4.1 (Multicongruence). Let A = �A, σA� be a multialgebra, and let θ ⊆ A×A.Then θ is said to be a multicongruence over A if the following properties hold:

(i) θ is an equivalence relation;

(ii) for every n > 0, c ∈ Θn and �a,�b ∈ An: if (ai, bi) ∈ θ for every 1 ≤ i ≤ n then, forevery a ∈ cA(�a) there is b ∈ cA(�b) such that (a, b) ∈ θ;

(ii) for every c ∈ Θ0 and every a, b ∈ A: if a, b ∈ cA then (a, b) ∈ θ.

Remark 2.4.2. Observe that, since θ is an equivalence relation, from Definition 2.4.1(ii), we can obtain the symmetric condition:

(ii’) for every n > 0, c ∈ Θn and �a,�b ∈ An: if (ai, bi) ∈ θ for every 1 ≤ i ≤ n then, forevery b ∈ cA(�b) there is a ∈ cA(�a) such that (a, b) ∈ θ.

Example 2.4.3. Let A�3 be the multialgebra introduced in the Example 2.1.9 and θ =

{(t�I , t�), (t�, t�

I), (f �I , f �), (f �, f �

I)} ∪ {(a, a) : a ∈ A�3} ⊆ A�

3 × A�3.

It’s easy to see that θ is an equivalence relation.

Let a1, b1 ∈ A�3 and c ∈ {¬, ◦}, if (a1, b1) ∈ θ then, for every a ∈ cA�3(a1), there

is b ∈ cA�3(b1) such that (a, b) ∈ θ. See, for example, if c = ◦:

(a1, b1) ∈ θ ◦A�3(a1) ◦A�3(b1)(t�, t�

I) {t�, I �, f �} {t�, I �, f �}(t�

I , t�) {t�, I �, f �} {t�, I �, f �}(f �, f �

I) {t�, I �, f �} {t�, I �, f �}(f �

I , f �) {t�, I �, f �} {t�, I �, f �}(I �, I �) {f �} {f �}

(a�, a�) : a� ∈ A�3 − {I �} {t�, I �, f �} {t�, I �, f �}

Note that, in all lines of the table we have ◦A�3(a1) = ◦A�3(b1).

We also have that if (a1, b1) ∈ θ and (a2, b2) ∈ θ, for every a1, b1, a2, b2 ∈ A�3 and

c ∈ {∧, ∨, →} then, for every a ∈ cA�3(a1, a2) there is b ∈ cA�3(b1, b2) such that (a, b) ∈ θ.See, for example, if c = ∨:

(a1, b1) ∈ θ (t�, t�I) (t�

I , t�) (f �, f �I) (f �

I , f �) (a�, a�) : a ∈{t�, t�

I , I �}(a�, a�) : a ∈{f �, f �

I}(t�, t�

I) D�3 D�

3 D�3 D�

3 D�3 D�

3

(t�I , t�) D�

3 D�3 D�

3 D�3 D�

3 D�3

(f �, f �I) D�

3 D�3 F � F � D�

3 F �

(f �I , f �) D�

3 D�3 F � F � D�

3 F �

(a�, a�) : a ∈ {t�, t�I , I �} D�

3 D�3 D�

3 D�3 D�

3 D�3

(a�, a�) : a ∈ {f �, f �I} D�

3 D�3 F � F � D�

3 F �

Note that, in all lines of the table we have a ∈ cA�3(a1, a2) = cA�3(b1, b2).

Therefore, θ is a multicongruence over A�3.

Example 2.4.4. Let A�3 be the multialgebra introduced in the Example 2.1.9 and θ� =

{(t�I , I �), (I �, t�

I), (f �I , f �), (f �, f �

I)} ∪ {(a, a) : a ∈ A�3} ⊆ A�

3 × A�3.

It’s easy to see that θ� is an equivalence relation.

If c = ¬, since (I �, t�I) ∈ θ�, then ¬A�3(I �) = {t�, I �}. But, ¬A�3(t�

I) = {f �}and is not the case that (t�, f �) ∈ θ�. So, there isn’t b ∈ {f �}, such that, for every a ∈{t�, I �}, (a, b) ∈ θ�.

Therefore, θ� is not a multicongruence over A�3.

Definition 2.4.5 (Quotient multialgebra). Let A = �A, σA� be a multialgebra, and letθ be a multicongruence over A. The quotient multialgebra (or factor multialgebra) ofA modulo θ is the multialgebra A/θ = �A/θ, σA/θ

� such that, for every c ∈ Θn andevery (a1/θ, . . . , an/θ) ∈ (A/θ)n, cA/θ(a1/θ, . . . , an/θ) =

�a/θ : a ∈ cA(�a)

�. In particular,� �

Proposition 2.4.6. If A = �A, σA� is a multialgebra and θ is a multicongruence over A,then A/θ = �A/θ, σA/θ

� (as in previous definition) is, in fact, a multialgebra with the sametype.

Proof. Let θ be a multicongruence such that (ai, bi) ∈ θ (for every 1 ≤ i ≤ n) and letc ∈ Θn (for n > 0). If a/θ ∈ {a/θ : a ∈ cA(�a)} then, since a ∈ cA(�a), there is b ∈ cA(�b)such that (a, b) ∈ θ. That is, a/θ = b/θ and a/θ ∈ {b/θ : b ∈ cA(�b)}. Similarly, we canshow that {b/θ : b ∈ cA(�b)} ⊆ {a/θ : a ∈ cA(�a)}.

Definition 2.4.7 (Canonical map). Let A = �A, σA� be a multialgebra, and let θ be amulticongruence over A. The canonical map p : A → A/θ is given by p(a) = a/θ for everya ∈ A.

Proposition 2.4.8. Let A be a multialgebra and let θ be a multicongruence over A. Thenthe canonical map p : A → A/θ is a (full) homomorphism of multialgebras such thatp(A) = A/θ.

Proof. p[cA(a1, . . . , an)] = cA(a1, . . . , an)/θ = cA/θ(a1/θ, . . . , an/θ) = cA/θ(p(a1), . . . , p(an)).In particular, if c ∈ Θ0, then p[cA] = cA/θ = cA/θ.

Definition 2.4.9 (Compatible homomorphism). Let A = �A, σA� and B = �B, σB� be twomultialgebras, let θ be a multicongruence over A and let h : A → B be a homomorphism.We say that h is a homomorphism compatible with θ, if for every a, b ∈ A, such that(a, b) ∈ θ, then h(a) = h(b).

Theorem 2.4.10 (Homomorphism theorem). Let A = �A, σA� and B = �B, σB� be twomultialgebras, let θ be a multicongruence over A, let h : A → B be a homomorphismcompatible with θ and let p : A → A/θ be the homomorphism of the Proposition 2.4.8.Then, there is a unique homomorphism h from A/θ to B, such that h(a/θ) = h(a), that ish ◦ p = h.

Proof. For every a, b ∈ A, if (a, b) ∈ θ then h(a) = h(b). Thus, h is well defined.

h[cA/θ(a1/θ, . . . , an/θ)] = h[cA(a1, . . . , an)/θ] = h[cA(a1, . . . , an)] ⊆

cB(h(a1), . . . , h(an)) = cB(h(a1/θ), . . . , h(an/θ)).

Therefore, h is a homomorphism and if h1 is another homomorphism from A/θ to B suchthat h1(a/θ) = h(a), then h1(a/θ) = h(a) = h(a/θ). So, h1 and h are the same.

Example 2.4.11. Let A�3 be the multialgebra introduced in the Example 2.1.9 and let θ =

{(t�I , t�), (t�, t�

I), (f �I , f �), (f �, f �

I)} ∪ {(a, a) : a ∈ A�3} be the congruence over A�

3 introducedin the Example 2.4.3.

If A�3/θ = �A�

3/θ, σA�3/θ�, such that A�3/θ = {t�/θ, I �/θ, f �/θ}, D�

3/θ = {t�/θ, I �/θ}and for every c ∈ Σ, the multioperation σA�3/θ(c) = cA�3/θ is defined by the following tables:

∨A�3/θ t�/θ I �/θ f �/θ

t�/θ D3/θ D3/θ D3/θ

I �/θ D3/θ D3/θ D3/θ

f �/θ D3/θ D3/θ {f �/θ}

∧A�3/θ t�/θ I �/θ f �/θ

t�/θ D3/θ D3/θ {f �/θ}I �/θ D3/θ D3/θ {f �/θ}f �/θ {f �/θ} {f �/θ} {f �/θ}

→A�3/θ t�/θ I �/θ f �/θ

t�/θ D3/θ D3/θ {f �/θ}I �/θ D3/θ D3/θ {f �/θ}f �/θ D3/θ D3/θ D3/θ

¬A�3/θ

t�/θ {f �/θ}I �/θ D3/θ

f �/θ D3/θ

◦A�3/θ

t�/θ A�3/θ

I �/θ {f �/θ}f �/θ A�

3/θ

Then, A�3/θ is a quotient multialgebra of A�

3 modulo θ over Σ.

Example 2.4.12. Let A�3/θ = �A�

3/θ, σA�3/θ� and A3 = �A3, σA3� be the multialgebrasintroduced in the Examples 2.4.11 and 2.1.7, and let h1 : A�

3/θ → A3 be the function suchthat h1(t�/θ) = t�, h1(I �/θ) = I � and h1(f �/θ) = f �. So,

h1[D�3/θ] = D3 and h1[{f �/θ}] = {f �}.

And, therefore h1 define a (full) homomorphism h1 : A�3/θ → A3.

2.5 The category of multialgebrasSeveral authors, some of them already mentioned in the Chapter 1, studied the

categories in which the objects are multialgebras. In particular, Nolan (NOLAN, 1979)defines five different homomorphisms between multialgebras and studies the categoryobtained with each one.

Note that due to the large number of possibilities for each definition in mul-tialgebra theory, the study of categories on multialgebras does not always coincide. Inthis section we will introduce the basic notions and results concerning the category ofmultialgebras that will be used mainly for what will be done in the Chapter 4. This resultsare based on the pre-print (CONIGLIO; ORELLANO; GOLZIO, 2016).

Proposition 2.5.1. If B and A are two multialgebras over Θ such that |B| ⊆ |A| then:B ⊆ A iff the inclusion map i : |B| → |A| is a homomorphism from B to A.

Proof. For every c ∈ Θn and �b ∈ |B|n,(⇒) i[ B(�b)] B(�b) ⊆ A(�b) A(i(b ) i(b )) In particular, if ∈ Θ i[ B] B ⊆ A

(⇐) cB(�b) = i(cB(�b)) ⊆ cA(i(b1), . . . , i(bn)) = cA(�b). In particular, if c ∈ Θ0, cB = i[cB] ⊆cA.

Proposition 2.5.2. Let Θ a signature. There is a category of multiagebras over Θ togetherwith their (full) homomorphisms, that will be denoted by MAlg(Θ).

Proof. Observe that, if f : A → B and g : B → C are homomorphisms of multialgebras,then g ◦ f : A → C is also a homomorphism of multialgebras: g ◦ f [cA(�a)] = g[f [cA(�a)]] ⊆g[cB(f(a1), . . . , f(an))] ⊆ cC(g(f(a1)), . . . , g(f(an))) = cC(g ◦ f(a1), . . . , g ◦ f(an)). Theresult still holds when we consider full homomorphisms. And, if f : A → B, g : B → C andh : C → D are homomorphisms of multialgebras, then because f, g and h are functions,clearly f ◦ (g ◦ h) = (f ◦ g) ◦ h.

On the other hand, the identity mapping 1A : A → A is a (full) homo-morphism from A to A, for every multialgebra A = �A, σA�: 1A[cA(�a)] = cA(�a) =cA(1A(a1), . . . , 1A(an)). In particular, 1A[cA] ⊆ cA, if c ∈ Θ0. And, if f � : B → Aand g� : A → C are homomorphisms of multialgebras, then clearly 1A ◦ f � = f � andg� ◦ 1A = g�.

The following results will be useful in the Chapter 4.

Proposition 2.5.3. Let A = �A, σA� and B = �B, σB� be two multialgebras over Θ, andlet f : A → B be a function. Then, f is an isomorphism f : A → B in the categoryMAlg(Θ) iff f is a full homomorphism f : A →s B which is a bijective function.

Proof. (⇒) Since f : A → B is an isomorphism in the category MAlg(Θ), then f isa homomorphism and so f [cA(a1, . . . , an)] ⊆ cB(f(a1), . . . , f(an)). On the other hand,since f is an isomorphism, then there is a homomorphism f−1 : B → A, such thatf ◦ f−1 = 1B. So, cB(f(a1), . . . , f(an)) = f ◦ f−1(cB(f(a1), . . . , f(an))). Since, f−1 isa homomorphism, then f−1[cB(f(a1), . . . , f(an))] ⊆ cA(f−1(f(a1)), . . . , f−1(f(an))) andso, f(f−1[cB(f(a1), . . . , f(an))]) ⊆ f(cA(f−1(f(a1)), . . . , f−1(f(an)))) = f [cA(a1, . . . , an)]because f−1 ◦f = 1A. Therefore, f is a full homomorphism. Suppose that f(a1) = f(a2) forsome a1, a2 ∈ A. Since f is an isomorphism, there is f−1 : B → A such that f−1 ◦ f = 1A,where 1A : A → A is the identity mapping. So, a1 = 1A(a1) = f−1 ◦ f(a1) = f−1(f(a1)) =f−1(f(a2)) = f−1 ◦f(a2) = 1A(a2) = a2. Therefore f is one-to-one. Let b ∈ B. Again, sincef is an isomorphism, there is f−1 : B → A such that f ◦ f−1 = 1B, where 1B : B → B isthe identity mapping. So, b = 1B(b) = f ◦ f−1(b) = f(f−1(b)) = f(a) for some a ∈ A and,therefore f is onto.

(⇐) Suppose that f is a bijection; then, there is f−1 : B → A such that f−1 is theinverse function of f . Since f is a full homomorphism which is a surjection function, for

b ∈ B with 1 ≤ i ≤ there is ∈ A h that f( ) b then f−1[ B(�b)]

f−1[cB(f(a1), . . . , f(an))] = f−1[f([cA(a1, . . . , an)])] = f−1 ◦ f([cA(a1, . . . , an)]) = cA(�a) =cA(f−1 ◦ f(a1), . . . , f−1 ◦ f(an)) = cA(f−1(b1), . . . , f−1(bn)). Therefore, f−1 : B → A is afull homomorphism. So, for every a ∈ A, f−1 ◦ f(a) = f−1(f(a)) = a = 1A(a) and forevery b ∈ B, f ◦ f−1(b) = f(f−1(b)) = b = 1B(b). Therefore f is an isomorphism.

Proposition 2.5.4. Let A = �A, σA� and B = �B, σB� be two multialgebras over Θ, andlet f : A → B be a homomorphism. If f : A → B is an injective (one-to-one) functionthen f is a monomorphism in the category MAlg(Θ).

Proof. Let f : A → B be a homomorphism in MAlg(Θ), such that f : A → B is aninjection and let g, h : C → A be two homomorphisms in MAlg(Θ) such that f ◦ g = f ◦ h.If c ∈ |C|, then f(g(c)) = f ◦ g(c) = f ◦ h(c) = f(h(c)) and since f is an injection,g(c) = h(c). Therefore, f is a monomorphism.

Proposition 2.5.5. Let A = �A, σA� and B = �B, σB� be two multialgebras over Θ, andlet f : A → B be a function. Then, f is an epimorphism f : A → B in the categoryMAlg(Θ) iff f is a homomorphism in MAlg(Θ) such that f is a surjective (onto) function.

Proof. (⇐) Let f : A → B be a surjective (onto) homomorphism in MAlg(Θ) and letg, h : B → C be two homomorphisms in MAlg(Θ) such that g ◦ f = h ◦ f . Since f isonto, for every b ∈ B, there is a ∈ A, such that f(a) = b. So, g(b) = g(f(a)) = g ◦ f(a) =h ◦ f(a) = h(f(a)) = h(b). Therefore g = h and so, f is an epimorphism.

(⇒) Conversely, suppose that f : A → B is an epimorphism in MAlg(Θ) and let A� bea multialgebra over Θ with domain {0, 1} such that cA�(�a) = {0, 1} for every c ∈ Θn

and �a ∈ {0, 1}n. In particular, cA� = {0, 1} for every c ∈ Θ0. Consider g, h : B → {0, 1}such that g(x) = 1 if there exists y ∈ A such that x = f(y), g(x) = 0 otherwiseand h(x) = 1 for every x ∈ B. Clearly, g and h are homomorphisms from B to A�

in MAlg(Θ): g[cB(�b)] ⊆ {0, 1} = cA�(g(b1), . . . , g(bn)), for every c ∈ Θn and �b ∈ Bn.In particular, if c ∈ Θ0, then g[cB] ⊆ {0, 1} = cA� . The same is true for h. If a ∈ A,g ◦ f(a) = g(f(a)) = 1 = g(h(a)) = h ◦ f(a). So, g ◦ f = h ◦ f and since f is epimorphismin MAlg(Θ) then g = h. So, if b ∈ B, then g(b) = h(b) = 1 and by definition of g thismeans that f is a surjective (onto) function.

In the next, we will present a definition of direct product in multialgebras.

Definition 2.5.6 (Product in multialgebras). Let {Ai}i∈I be a family of multialgebrasover Θ and let A = �

i∈I Ai be the standard cartesian product of the family of sets {Ai}i∈I

with canonical projections πi : A → Ai for every i ∈ I. The (direct) product A = �i∈I Ai

of the family {Ai}i∈I is the multialgebra A = �A, σA� over Θ such that, for every c ∈ Θn

and every �a ∈ An, cA(�a) = �i∈I cAi(πi(a1), . . . , πi(an)). In particular, cA = �

i∈I cAi for

In particular, the category MAlg(Θ) has a terminal object that correspondsto the product of the empty family of multialgebras over Θ. See the following proposition:

Proposition 2.5.7 (Terminal object). The terminal object in MAlg(Θ) is the multialgebra1 = �{∗}, σ1�, such that c1(∗, . . . , ∗) = {∗}, for every c ∈ Θn (with n > 0) and c1 = {∗},for every c ∈ Θ0.

Proof. Let A be a multialgebra over Θ, then !A : A → {∗}, such that for every a ∈ A,!A(a) = ∗ is the only (full) homomorphism from A to 1.

Proposition 2.5.8. The category MAlg(Θ) has arbitrary products.

Proof. Let {Ai}i∈I be a family of multialgebras over Θ. If I = ∅, since the multialgebra1 = �{∗}, σ1� is the terminal object in MAlg(Θ), then the result is obvious. Now, assumethat I �= ∅.

Consider the product A = �i∈I Ai as in Definition 2.5.6. Then, πi[cA(�a)] =

πi[�

i∈I(cAi(πi(a1), . . . , πi(an)))] = cAi(πi(a1), . . . , πi(an)). In particular, we have πi[cA] =πi[

�i∈I cAi ] = cAi . Therefore, πi is a (full) homomorphism from A to Ai.

Let B be an object in MAlg(Θ), let µi : B → Ai be a morphism in MAlg(Θ),for every i ∈ I and let δ : B → A such that δ : |B| → |A| be the function defined byδ(b) = �

i∈I µi(b) for every b ∈ |B|, then:

(i) If b ∈ |B|, πi ◦ δ(b) = πi(δ(b)) = πi(�

i∈I µi(b)) = µi(b), so πi ◦ δ = µi.(ii) δ[cB(�b)] = �

i∈I µi(cB(�b)) ⊆ �i∈I cAi(µi(b1), . . . , µi(bn)) = �

i∈I cAi(πi ◦ δ(b1), . . . , πi ◦δ(bn)) = �

i∈I cAi(πi(δ(b1)), . . . , πi(δ(bn))) = cA(δ(b1), . . . , δ(bn)). In particular, if c ∈ Θ0,δ[cB] = �

i∈I µi[cB] ⊆ �i∈I cAi = cA, so δ is a homomorphism in MAlg(Θ).

(iii) Supose that δ� : B → A is a morphism in MAlg(Θ) such that πi ◦ δ� = µi. If b ∈ |B|,δ(b) = �

i∈I µi(b) = �i∈I(πi ◦ δ�(b)) = �

i∈I πi(δ�(b)) = �i∈I(δ�(b))(i) = δ�(b), so δ is unique.

By (i), (ii) and (iii), we have that �A, {πi}i∈I� is the product in MAlg(Θ) of the family{Ai}i∈I .

Moreover, the following useful result holds in MAlg(Θ):

Proposition 2.5.9 (Epi-mono factorization). Let A = �A, σA� and B = �B, σB� betwo multialgebras over Θ, and let f : A → B be a homomorphism in MAlg(Θ). Letf : A → f [A] be the mapping given by f(x) = f(x) for every x ∈ A, and let g : f [A] → B

be the inclusion map. Then f and g are homomorphisms f : A → f(A) and g : f(A) → Bsuch that f is an epimorphism in MAlg(Θ), g is a monomorphism in MAlg(Θ), and

f = g ◦ f .A f ��

f ��

B

f(A)��

g

��

Moreover, if f is (one-to-one) injective (as a function) then f is an isomorphism inMAlg(Θ).

Proof. (i) If �a ∈ An(n > 0), f [cA(�a)] = f [cA(�a)]. By Proposition 2.2.13, f [cA(�a)] ⊆�{f [cA(�a)] : �a ∈ f−1(f(a1), . . . , f (an))} = cf(A)(f(a1), . . . , f (an)) = cf(A)(f(a1), . . . , f(an)).In particular, f [cA] = f [cA] = cf(A). Therefore, f is a homomorphism from A to f(A).(ii) By Proposition 2.2.13 and by Proposition 2.5.1, we have that g is a homomorphismfrom f(A) to B.(iii) Note that f is onto: If b ∈ f [A] = {f(a) : a ∈ A} = {f(a) : a ∈ A}, there is a ∈ A suchthat f(a) = b and by Proposition 2.5.5 we have that f is an epimorphism in MAlg(Θ).(iv) Note that g is one-to-one: If b1, b2 ∈ f [A] and g(b1) = g(b2), then b1 = g(b1) = g(b2) =b2. So, by Proposition 2.5.4 we have that g is a monomorphism in MAlg(Θ).(v) If a ∈ A, g ◦ f(a) = g(f(a)) = f(a) = f(a), so f = g ◦ f .(vi) Note that f is one-to-one: if a1, a2 ∈ A, f(a1) = f(a1) = f(a2) = f(a2). Since byhypothesis f is one-to-one, then a1 = a2. By (iii) we have also that f is onto. By (i), If�a ∈ An(n > 0), f [cA(�a)] ⊆ cf(A)(f(a1), . . . , f(an)), on other hand cf(A)(f(a1), . . . , f(an)) =cf(A)(f(a1), . . . , f(an)) = �{f [cA(�a)] : �a ∈ f−1(f(a1), . . . , f(an))} = f [cA(�a)] = f [cA(�a)].Therefore, f is a full homomorphism and by Proposition 2.5.3 f is an isomorphism inMAlg(Θ).

In the next chapters, we will show the use of the multialgebras in the alge-braization of logical systems.

3 Non-deterministic semantics for non-normal modal logics

In this chapter we will introduce swap structures for some non-normal modalsystems and we will use the Lindenbaum-Tarski swap structures to obtain completenesstheorems w.r.t Hilbert-style version of these systems. To begin with, however, we willintroduce, formally, these modal logics.

3.1 The Systems Tm, T4m, T45m, TBm and DmJohn T. Kearns in the paper “Modal semantics without possible worlds” pub-

lished in 1981 (KEARNS, 1981) proposes a semantics of four values for modal logicswithout using the concept of possible worlds. That is, different of the well-known Kripkerelational semantics. In this semantics, non-deterministic matrices are used to characterizethe logical operators. It is interesting to observe that this is one of the first relevantapplications of non-deterministic matrices in the literature.

The truth-values consider by Kearns are:

T : Necessary truth

t: Contingent truth

f : Contingent falsity

F : Necessary falsity

In 1988, in the paper called “A semantics for modal calculi” (IVLEV, 1988), J.V. Ivlev presented a semantics of non-deterministic matrices of four values, in order tosemantically characterize a family of weaker modal logics in which the necessitation rule1

is not valid.

Ivlev, to define the non-deterministic matrices of four values, considers thefollowing truth-values:

tn: Necessary truth1 For more information about modal logic and about the necessitation rule, see for instance (CHELLAS,

1980).

tc: Possible truth

f c: Possible falsity

f i: Necessary falsity

The set of designated truth-values is {tn, tc} and the set of non-designatedtruth-values is {f i, f c}.

If we consider the signature Σ� = {¬, →,�} and tn = T , tc = t, f c = f andf i = F , then the Ivlev2 and Kearns3 (four-valued) non-deterministic matrix semanticscoincide. They are given by:

→ tn tc f c f i

tn tn tc f c f i

tc tn tn/tc f c f c

f c tn tn/tc tn/tc tc

f i tn tn tn tn

α ¬α

tn f i

tc f c

f c tc

f i tn

α �α

tn tn/tc

tc f c/f i

f c f c/f i

f i f c/f i

The operators ∨ and ♦ can be defined, as in the classical case, respectively byα ∨ β = ¬α → β and ♦α = ¬�¬α, so they produce the following matrices:

∨ tn tc f c f i

tn tn tn tn tn

tc tn tn/tc tn/tc tc

f c tn tn/tc f c f c

f i tn tc f c f i

α ♦α

tn tn/tc

tc tn/tc

f c tn/tc

f i f c/f i

A Hilbert-style deductive system for Sa+ and of T was introduced in (CONIGLIO;CERRO; PERON, 2015) with the notation Tm. A slightly different version will be intro-duced here, following the modifications suggested in (OMORI; SKURT, 2016; CONIGLIO;CERRO; PERON, 2016).

Definition 3.1.1. The system Tm over Σ� = {¬, →,�} is composed of the followingaxiom schemes:

α → (β → α) (Ax1)

(α → (β → σ)) → ((α → β) → (α → σ)) (Ax2)2 Semantics matrices for the system Sa+, see (IVLEV, 1988).3 Semantics matrices for the system T , see (KEARNS, 1981).

(¬β → ¬α) → ((¬β → α) → β) (Ax3)

�(α → β) → (�α → �β) (K)

�(α → β) → (�¬β → �¬α) (K1)

¬�¬(α → β) → (�α → ¬�¬β) (K2)

�¬α → �(α → β) (M1)

�β → �(α → β) (M2)

�¬(α → β) → �¬β (M3)

�¬(α → β) → �α (M4)

�α → α (T )

�α → �¬¬α (DN1)

�¬¬α → �α (DN2)

And by the rule of inference:

α, α → β

β(MP )

The systems T4m, T45m, TBm and Dm were introduced in (CONIGLIO;CERRO; PERON, 2015). Now, to define them, we consider the following axioms:

�α → ��α (4)

¬�¬�α → �α (5)

¬�¬�α → α (B)

�α → ¬�¬α (D)

The versions of Hilbert-style deductive systems for T4m, T45m and TBmare given by:

T4m = Tm ∪ {(4)}

T45m = T4m ∪ {(5)}

TBm = Tm ∪ {(B)}

The non-deterministic matrix semantics for the systems T4m, T45m andTBm are given by the same multioperation → and by the same operation ¬ of Tm only

α �T4mα

tn tn

tc f c/f i

f c f c/f i

f i f c/f i

α �T45mα

tn tn

tc f i

f c f i

f i f i

α �TBmα

tn tn/tc

tc f c/f i

f c f i

f i f i

The operation �T45m is the same operation � proposed by Kearns in (KEARNS,1981) for the system S5 and by Ivlev for the system Sb+ in (IVLEV, 1988). And, bytaking �(f i) = {f i} instead of �(f i) = {f c, f i}, then the multioperation � proposedby Kearns (KEARNS, 1981) in order to interpret the system S4 and the multioperation�T4m of T4m will be the same too.

Deontic logics are known to interpret, intuitively, the operators � and ♦,respectively, by it is obligatory that and by it is permitted that. In (CONIGLIO; CERRO;PERON, 2015) it was proposed that, under this interpretation, to accept a weaker principle�α → ♦α instead of principles like �α → α and α → ♦α would be more convenient. Andso a new (weakly) deontic system characterized by six-valued non-deterministic matrixsemantics was obtained. This new system was called Dm.

In Dm a proposition p is said to be:

• fulfil led whenever either it is obligatory and it is the case, or it is forbidden and it isnot the case (that is, it is the case of its negation). In symbols: (�p∧p)∨ (�¬p∧¬p).

• infringed whenever either it is obligatory and it is not the case, or it is forbiddenand it is the case. In symbols: (�p ∧ ¬p) ∨ (�¬p ∧ p).

• optional if it is neither obligatory nor forbidden. In symbols: ¬�p ∧ ¬�¬p.

It generates six truth-values as follows:

T +: p is fulfilled;

C+: p is optional;

F +: p is infringed;

T −: ¬p is infringed;

C−: ¬p is optional and

F −: ¬p is fulfilled.

The versions of Hilbert-style deductive system for Dm is given by:

Dm = Tm ∪ {(D)} − {(T )}

And the semantics matrices for the system Dm is given by the multioperations→Dm and �Dm and by the operation ¬Dm:

→Dm T + T − C+ C− F + F −

T + T + T − C+ C− F + F −

T − T + T + C+ C+ F + F +

C+ T + T − {T +, C+} {T −, C−} C+ C−

C− T + T + {T +, C+} {T +, C+} C+ C+

F + T + T − T + T − T + T −

F − T + T + T + T + T + T +

α ¬Dmα

T + F −

T − F +

C+ C−

C− C+

F + T −

F − T +

α �Dmα

T + {T +, C+, F +}T − {T +, C+, F +}C+ {T −, C−, F −}C− {T −, C−, F −}F + {T −, C−, F −}F − {T −, C−, F −}

Let

• {T +, T −} is the set interpreted as obligatory;

• {C+, C−} is the set interpreted as optional;

• {F +, F −} is the set interpreted as forbidden;

• + = {T +, C+, F +} is the set of designated truth-values and

• − = {T −, C−, F −} is the set of non-designated truth-values.

In the next section, we will introduce swap structures for the systems Tm,T4m, T45m, TBm and Dm.

3.2 Swap structures for Tm, T4m, T45m, TBm and DmIn this chapter we will introduce an original class of swap structures as semantics

for a family of non-normal modal systems. Through swap structures we are proposinga class of multialgebras, whose elements are triples in a particular Boolean algebra andthe operations change of place (swap) some components of the triples. The elements of agiven swap structure are called snapshots4. In the swap structures, we ‘swap’ in a certainmanner the components of the snapshots and this justifies the name adopted for thesemultialgebras. The consequence relation over swap structures will be presented by meansof logical matrices, but the swap structures for the modal systems presented here are moregeneral than the corresponding non-deterministic matrices. Intuitively, in this Chapter,the triples for any formula α are (α,�α,�¬α).

After we define swap structures for these modal logics, we will prove soundnessand completeness theorems of the Hilbert calculi defining these non-normal modal systemswith respect to such non-deterministic matrix semantics. In order to obtain the complete-ness theorem, we will to apply the method of Lindenbaum-Tarski swap structures, thatgeneralizes in a quite natural way the classical Lindenbaum-Tarski method, allowing todeal with logics which are not algebraizable in the usual sense as is the case of the modallogics presented in the previous sections of this chapter.

Definition 3.2.1 (Swap structures for Tm). Let

A = �A, ∨, ∧, →, 0, 1�

be a Boolean algebra and let

BTmA = {(a1, a2, a3) ∈ A3 : a2 ≤ a1 and a1 ∧ a3 = 0}.

A swap structure for Tm over A is any multialgebra

B = �B, →, ¬,��

over Σ� = {→, ¬,�} such that B ⊆ BTmA and the multioperations satisfy the following, for

every (a1, a2, a3) and (b1, b2, b3) in B:

(i) (a1, a2, a3) → (b1, b2, b3) = {(c1, c2, c3) ∈ B : c1 = a1 → b1, c3 = a2∧b3 and a3∨b2 ≤c2 ≤ (a1 → b1) ∧ (a2 → b2) ∧ (b3 → a3)}

(ii) ¬(a1, a2, a3) = {(¬a1, a3, a2)}

(iii) �(a1, a2, a3) = {(c1, c2, c3) ∈ B : c1 = a2}

The unique swap structure for Tm, with domain BTmA , will be denoted by BTm

A .

The next proposition shows that the implication in Tm is a well-definedmultioperation: (a1, a2, a3) → (b1, b2, b3) �= ∅ for (a1, a2, a3), (b1, b2, b3) ∈ BTm

A .

Proposition 3.2.2. : If (a1, a2, a3), (b1, b2, b3) ∈ BTmA , then

a3 ∨ b2 ≤ (a1 → b1) ∧ (a2 → b2) ∧ (b3 → a3).

Proof. Let be (a1, a2, a3), (b1, b2, b3) ∈ BTmA ,

As a1 ∧ a3 = 0, then a3 ≤ ¬a1. But, as a2 ≤ a1, then ¬a1 ≤ ¬a2 and, therefore,a3 ≤ ¬a2. So,

a3 ∨ b2 ≤ ¬a2 ∨ b2 = a2 → b2 (3.1)

It also holds that (a3 ∨ b2) ∧ a1 = (a3 ∧ a1) ∨ (b2 ∧ a1) = 0 ∨ (b2 ∧ a1) = b2 ∧ a1 ≤ b2 ≤b1 ≤ a1 → b1. So,

a3 ∨ b2 ≤ a1 → b1 (3.2)

On the other hand, (a3 ∨b2)∧b3 = (a3 ∧b3)∨(b2 ∧b3) ≤ (a3 ∧b3)∨(b1 ∧b3) = (a3 ∧b3)∨0 =a3 ∧ b3 ≤ a3. So,

a3 ∨ b2 ≤ b3 → a3 (3.3)

From (3.1), (3.2) and (3.3), we have a3 ∨ b2 ≤ (a1 → b1) ∧ (a2 → b2) ∧ (b3 → a3).

Proposition 3.2.3. : If (a1, a2, a3), (b1, b2, b3) ∈ BTmA , c1 = a1 → b1 and c3 = a2 ∧ b3, then

c3 ∧ c1 = 0.

Proof. (a2 ∧ b3) ∧ (a1 → b1) = (a2 ∧ b3) ∧ (¬a1 ∨ b1) = (b3 ∧ (a2 ∧ ¬a1)) ∨ ((b3 ∧ b1) ∧ a2).Since, a2 ∧ ¬a1 = (a2 ∧ a1) ∧ ¬a1 = a2 ∧ 0 = 0, then (b3 ∧ (a2 ∧ ¬a1)) ∨ ((b3 ∧ b1) ∧ a2) =(b3 ∧ 0) ∨ (0 ∧ a2) = 0.

In the particular case of the two-element Boolean algebra A2 with domainA2 = {0, 1}, eight triples are possible: (1, 1, 1), (1, 0, 1), (0, 1, 0), (0, 1, 1), (1, 1, 0), (1, 0, 0),(0, 0, 0) and (0, 0, 1). However, a triple (a1, a2, a3) in A3

2 belongs to BTmA2 , whenever satisfies

the conditions a2 ≤ a1 and a1 ∧ a3 = 0. By the first condition, (0, 1, 0) and (0, 1, 1) arediscarded and by the second condition, the triples (1, 1, 1) and (1, 0, 1) are discarded too.So, BTm

A2 = {(1, 1, 0), (1, 0, 0), (0, 0, 0), (0, 0, 1)} and if we denote the values (1, 1, 0), (1, 0, 0),(0, 0, 0) and (0, 0, 1), respectively by tn, tc, f c and f i then the truth tables of Ivlev for Sa+are a particular case of our definition of swap structure for Tm.

Definition 3.2.4 (Swap structures for T4m). A swap structure for T4m over A is a swapstructure for Tm such that the multioperation � is given by �(a1, a2, a3) = {(c1, c2, c3) ∈B : c1 = a2 and a2 ≤ c2} for every (a1, a2, a3) in B.

Definition 3.2.5 (Swap structures for T45m). A swap structure for T45m over A isa swap structure for Tm such that the multioperation � is given by �(a1, a2, a3) ={(c1, c2, c3) ∈ B : c1 = a2, a2 ≤ c2 and c3 ∨ c1 = 1} for every (a1, a2, a3) in B.

Definition 3.2.6 (Swap structures for TBm). A swap structure for TBm over A isa swap structure for Tm such that the multioperation � is given by �(a1, a2, a3) ={(c1, c2, c3) ∈ B : c1 = a2 and a1 ∨ c3 = 1} for every (a1, a2, a3) in B.

Definition 3.2.7 (Swap structures for Dm). Let

A = �A, ∨, ∧, →, 0, 1�

be a Boolean algebra and let

BDmA = {(a1, a2, a3) ∈ A3 : a2 ∧ a3 = 0}.

A swap structure for Dm over A is any multialgebra

B = �B, →, ¬,��

over Σ� = {→, ¬,�} such that B ⊆ BDmA and the multioperations satisfy the following,

for every (a1, a2, a3) and (b1, b2, b3) in B:

(i) (a1, a2, a3) → (b1, b2, b3) = {(c1, c2, c3) ∈ B : c1 = a1 → b1, c3 = a2∧b3 and a3∨b2 ≤c2 ≤ (a2 → b2) ∧ (b3 → a3)}

(ii) ¬(a1, a2, a3) = {(¬a1, a3, a2)}

(iii) �(a1, a2, a3) = {(c1, c2, c3) ∈ B : c1 = a2}

The unique swap structure for Dm, with domain BDmA , will be denoted by

BDmA .

As in the case of Tm, it will be shown that the implication (a1, a2, a3) →(b1, b2, b3) in Dm returns a non-empty set, for (a1, a2, a3), (b1, b2, b3) ∈ BDm

A :

Proposition 3.2.8. : If (a1, a2, a3), (b1, b2, b3) ∈ BDmA , then

a3 ∨ b2 ≤ (a2 → b2) ∧ (b3 → a3).

Proof. Let be (a1, a2, a3), (b1, b2, b3) ∈ BDmA . We have

a3 ∧ ¬a2 = a3 ∧ ¬a2

a3 ∧ ¬a2 = (a3 ∧ a2) ∨ (a3 ∧ ¬a2)

a3 ∧ ¬a2 = a3 ∧ (a2 ∨ ¬a2)

a3 ∧ ¬a2 = a3 ∧ 1

a3 ∧ ¬a2 = a3

.

But, a3 ∧ ¬a2 = a3 iff a3 ≤ ¬a2. Then

a3 ∨ b2 ≤ ¬a2 ∨ b2 = a2 → b2 (3.4)

On the other hand,

b2 ∧ ¬b3 = b2 ∧ ¬b3

b2 ∧ ¬b3 = 0 ∨ (b2 ∧ ¬b3)

b2 ∧ ¬b3 = (b2 ∧ b3) ∨ (b2 ∧ ¬b3)

b2 ∧ ¬b3 = b2 ∧ (b3 ∨ ¬b3)

b2 ∧ ¬b3 = b2 ∧ 1

b2 ∧ ¬b3 = b2

.

But, b2 ∧ ¬b3 = b2 iff b2 ≤ ¬b3. Then

a3 ∨ b2 ≤ a3 ∨ ¬b3 = b3 → a3 (3.5)

From (3.4) and (3.5), we have a3 ∨ b2 ≤ (a2 → b2) ∧ (b3 → a3).

Proposition 3.2.9. : If (a1, a2, a3), (b1, b2, b3) ∈ BDmA and c3 = a2 ∧ b3, then there is c2

such that c2 ∧ c3 = 0.

Proof. Let be c2 = a3 ∨ b2, then (a3 ∨ b2) ∧ (a2 ∧ b3) = (a2 ∧ b3 ∧ a3) ∨ (a2 ∧ b3 ∧ b2) =(0 ∧ b3) ∨ (a2 ∧ 0) = 0 ∨ 0 = 0.

Notation 3.2.10. If L ∈ {Tm, T4m, T45m, TBm, Dm}, then we denote by KL theclass of swap structures for the system L .

Definition 3.2.11. Let L ∈ {Tm, T4m, T45m, TBm, Dm} and DB = {(z1, z2, z3) ∈|B| : z1 = 1} for each B ∈ KL. The non-deterministic matrix associated to B isM(B) = �B, DB�.

Notation 3.2.12. If L ∈ {Tm, T4m, T45m, TBm, Dm}, then we denote by Mat(KL) ={M(B) : B ∈ KL} the class of non-deterministic matrix associated to swap structure tothe system L.

Recall the semantics associated to non-deterministic matrices where the relationΔ |=M α is the consequence relation defined over a non-deterministic matrix M (seeDefinitions 2.1.4, 2.1.5):

Definition 3.2.13. For L ∈ {Tm, T4m, T45m, TBm, Dm}, let Δ ∪ {α} ⊆ For(Σ�)be a set of formulas of L. We say that α is a consequence of Δ in the class Mat(KL)of non-deterministic matrices, and we denote it by Δ |=Mat(KL) α, if Δ |=M α for everyM ∈ Mat(KL). In particular, α is valid in Mat(KL), denoted by |=Mat(KL) α, if it is validin every M ∈ Mat(KL).

Theorem 3.2.14. (Deduction theorem) For L ∈ {Tm, T4m, T45m, TBm, Dm}, letΔ ∪ {α, β} be a set of formulas in L. Then Δ ∪ {α} �L β iff Δ �L α → β.

Proof. Since in the systems Tm, T4m, T45m, TBm and Dm the necessitation rule isnot valid, then this classical result follows from the fact that these logics are axiomaticextensions of classical logic. A proof can be found in (MENDELSON, 1987).

Lemma 3.2.15. Let πi : A3 → A be the function projection on the ith coordinate of A3,such that πi((a1, a2, a3)) = ai for i = 1, 2, 3. Let α, β ∈ For(Σ�) and let v be any valuationfor a swap structure for L ∈ {Tm, T4m, T45m, TBm, Dm}. Then,

i) π1(v(α → β)) = π1(v(α)) → π1(v(β));

ii) π1(v(¬α)) = ¬π1(v(α)).

Proof.

i) v(α → β) ∈ v(α) → v(β) = {(c1, c2, c3) ∈ B : c1 = π1(v(α)) → π1(v(β)), c3 =π2(v(α)) ∧ π3(v(β)) and π3(v(α)) ∨ π2(v(β)) ≤ c2 ≤ (π1(v(α)) → π1(v(β))) ∧(π2(v(α)) → π2(v(β))) ∧ (π3(v(β)) → π3(v(α)))}. So, π1(v(α → β)) = π1(v(α)) →π1(v(β)).

ii) v(¬α) ∈ ¬v(α) = {(¬π1(v(α)), π3(v(α)), π2(v(α)))}. So, π1(v(¬α)) = ¬π1(v(α)).

Lemma 3.2.16. Let πi : A3 → A be the function projection on the ith coordinate ofA3, such that πi((a1, a2, a3)) = ai for i = 1, 2, 3. Let α, β ∈ For(Σ�) and let v be anyvaluation for a swap structure for L ∈ {Tm, T4m, T45m, TBm, Dm}. So, the following

i) v(α → β) ∈ DB;

ii) π1(v(α)) ≤ π1(v(β));

iii) π1(v(α)) → π1(v(β)) = 1.

Proof. (i) ⇔ (iii) : v(α → β) ∈ DB iff π1(v(α → β)) = 1. But, by Lemma 3.2.15, we haveπ1(v(α → β)) = π1(v(α)) → π1(v(β)) and, therefore π1(v(α)) → π1(v(β)) = 1.

On the other hand, it is well-known that a ≤ b iff a → b = 1 in any Booleanalgebra. This completes the proof.

Theorem 3.2.17. (Soundness) If L ∈ {Tm, T4m, T45m, TBm, Dm} and α ∈ For(Σ�),then

�L α ⇒ �Mat(KL) α.

Proof. Let v be a valuation for a swap structure B for L. We will prove by induction overthe length of the deduction of α in L, that v(α) ∈ DB.

If there is only one formula in the deduction of α, this formula is α itself. Thenα can only be an axiom of L. So, we must verify that the result is true for every axiom ofL:

To simplify the proof, we will use Lemma 3.2.16 and the following notation:

If α is a formula and v is a valuation, we’ll write |α|1, |α|2 and |α|3 instead of,respectively, π1(v(α)), π2(v(α)) and π3(v(α)).

The task of checking the axioms will be divided into three parts:

Part 1: If α is an axiom in {(Ax1), (Ax2), (Ax3), (K2), (M1), (M2), (M3), (M4), (DN1),(DN2)}, then the proof is the same for any system L. See below:

• If α is of the form δ → (β → δ):

Note that |β → δ|1 = |β|1 → |δ|1 = ¬|β|1 ∨ |δ|1, so |δ|1 ≤ ¬|β|1 ∨ |δ|1 = |β → δ|1.

• If α is of the form (δ → (β → σ)) → ((δ → β) → (δ → σ)):

Note that |(δ → β) → (δ → σ)|1 = |δ → β|1 → |δ → σ|1 = (|δ|1 → |β|1) →(|δ|1 → |σ|1) = ¬(|δ|1 → |β|1) ∨ (|δ|1 → |σ|1) = ¬(¬|δ|1 ∨ |β|1) ∨ (¬|δ|1 ∨ |σ|1) =(|δ|1 ∧ ¬|β|1) ∨ (¬|δ|1 ∨ |σ|1) = (|δ|1 ∨ (¬|δ|1 ∨ |σ|1)) ∧ (¬|β|1 ∨ (¬|δ|1 ∨ |σ|1)) =(1 ∨ |σ|1) ∧ (¬|δ|1 ∨ (¬|β|1 ∨ |σ|1)) = 1 ∧ (|δ|1 → (|β|1 → |σ|1)) = |δ|1 → (|β|1 →|σ|1) = |δ → (β → σ)|1. In particular, |δ → (β → σ)|1 ≤ |(δ → β) → (δ → σ)|1.

• If α is of the form (¬β → ¬δ) → ((¬β → δ) → β):

Note that |(¬β → δ) → β)|1 = (¬|β|1 → |δ|1) → |β|1 = ¬(¬|β|1 → |δ|1) ∨ |β|1 =

1 ∧ (¬|δ|1 ∨ |β|1) = |δ|1 → |β|1 = ¬|β|1 → ¬|δ|1 = |¬β → ¬δ|1. In particular,|¬β → ¬δ|1 ≤ |(¬β → δ) → β)|1.

• If α is of the form ¬�¬(δ → β) → (�δ → ¬�¬β):

Note that |¬�¬(δ → β)|1 = ¬|�¬(δ → β)|1 = ¬|¬(δ → β)|2 = ¬|(δ → β)|3 =¬(|δ|2 ∧ |β|3) = ¬|δ|2 ∨ ¬|β|3 = |δ|2 → ¬|β|3. And |�δ → ¬�¬β|1 = |�δ|1 →|¬�¬β|1 = |δ|2 → ¬|�¬β|1 = |δ|2 → ¬|¬β|2 = |δ|2 → ¬|β|3. In particular,|¬�¬(δ → β)|1 ≤ |�δ → ¬�¬β|1.

• If α is of the form �¬δ → �(δ → β):

Note that |�¬δ|1 = ¬|δ|2 = |δ|3 and |�(δ → β)|1 = |δ → β|2. But, |δ|3 ≤ |δ|3∨|β|2 ≤|δ → β|2. So, |�¬δ|1 ≤ |�(δ → β)|1.

• If α is of the form �β → �(δ → β):

Note that |�β|1 = |β|2 and |�(δ → β)|1 = |δ → β|2. But, |β|2 ≤ |δ|3∨|β|2 ≤ |δ → β|2.So, |�β|1 ≤ |�(δ → β)|1.

• If α is of the form �¬(δ → β) → �¬β:

Note that |�¬β|1 = |¬β|2 = |β|3 and |�¬(δ → β)|1 = |¬(δ → β)|2 = |δ → β|3 =|δ|2 ∧ |β|3. So, |�¬(δ → β)|1 = |δ|2 ∧ |β|3 ≤ |β|3 = |�¬β|1.

• If α is of the form �¬(δ → β) → �δ:

Note that |�δ|1 = |δ|2 and |�¬(δ → β)|1 = |¬(δ → β)|2 = |δ → β|3 = |δ|2 ∧ |β|3. So,|�¬(δ → β)|1 = |δ|2 ∧ |β|3 ≤ |δ|2 = |�δ|1.

• If α is of the form �δ → �¬¬δ or α is of the form �¬¬δ → �δ:

Note that |�δ|1 = |δ|2 and |�¬¬δ|1 = |¬¬δ|2 = |¬δ|3 = |δ|2. In particular, |�δ|1 ≤|�¬¬δ|1 and |�¬¬δ|1 ≤ |�δ|1.

Part 2: If α is an axiom in {(K), (K1)}, then the proof is the same for any systemL ∈ {Tm, T4m, T45m, TBm}. So, we will be divided the proof of each one into twoparts.

• If α is of the form �(δ → β) → (�δ → �β):

If L ∈ {Tm, T4m, T45m, TBm}:

From |�(δ → β)|1 = |δ → β|2 and we have |δ|3 ∨ |β|2 ≤ |δ → β|2 ≤ (|δ|1 →|β|1) ∧ (|δ|2 → |β|2) ∧ (|β|3 → |δ|3).On the other hand, |�δ → �β|1 = |�δ|1 → |�β|1 = |δ|2 → |β|2. So, |�(δ → β)|1 ≤(|δ|1 → |β|1) ∧ |�δ → �β|1 ∧ (|β|3 → |δ|3). Therefore, |�(δ → β)|1 ≤ |�δ → �β|1.

If L = Dm:

From |�(δ → β)|1 = |δ → β|2 and we have |δ|3 ∨ |β|2 ≤ |δ → β|2 ≤ (|δ|2 →|β|2) ∧ (|β|3 → |δ|3).On the other hand, |�δ → �β|1 = |�δ|1 → |�β|1 = |δ|2 → |β|2. So, |�(δ → β)|1 ≤|�δ → �β|1 ∧ (|β|3 → |δ|3). Therefore, |�(δ → β)|1 ≤ |�δ → �β|1.

• If α is of the form �(δ → β) → (�¬β → �¬δ):

If L ∈ {Tm, T4m, T45m, TBm}:

From |�(δ → β)|1 = |δ → β|2 and we have |δ|3 ∨ |β|2 ≤ |δ → β|2 ≤ (|δ|1 →|β|1) ∧ (|δ|2 → |β|2) ∧ (|β|3 → |δ|3).On the other hand, |�¬β → �¬δ|1 = |�¬β|1 → |�¬δ|1 = |¬β|2 → |¬δ|2 = |β|3 →|δ|3. So, |�(δ → β)|1 ≤ (|δ|1 → |β|1) ∧ (|δ|2 → |β|2) ∧ |�¬β → �¬δ|1. Therefore,|�(δ → β)|1 ≤ |�¬β → �¬δ|1.If L = Dm:

From |�(δ → β)|1 = |δ → β|2 and we have |δ|3 ∨ |β|2 ≤ |δ → β|2 ≤ (|δ|2 →|β|2) ∧ (|β|3 → |δ|3).On the other hand, |�¬β → �¬δ|1 = |�¬β|1 → |�¬δ|1 = |¬β|2 → |¬δ|2 = |β|3 →|δ|3. So, |�(δ → β)|1 ≤ (|δ|2 → |β|2) ∧ |�¬β → �¬δ|1. Therefore, |�(δ → β)|1 ≤|�¬β → �¬δ|1.

Part 3: In this part, we will check the specific axioms of each system. See below:

• If α is of the form �δ → δ and L ∈ {Tm, T4m, T45m, TBm}:

As |�δ|1 = |δ|2 then, by definition of BTmA , we have |δ|2 ≤ |δ|1. So, |�δ|1 ≤ |δ|1.

• If α is of the form �δ → ¬�¬δ and L = Dm:

Note that |�δ|1 = |δ|2 and |¬�¬δ|1 = ¬|�¬δ|1 = ¬|¬δ|2 = ¬|δ|3 and by definitionof BDm

A , we have |δ|2 ∧ |δ|3 = 0. From this |δ|2 ≤ ¬|δ|3. Therefore, |�δ|1 ≤ |¬�¬δ|1.

• If α is of the form ¬�¬�δ → δ and L = TBm:

Note that |¬�¬�δ|1 = ¬|�¬�δ|1 = ¬|¬�δ|2 = ¬|�δ|3, such that |δ|1 ∨ |�δ|3 = 1(by Definition 3.2.6). But, then ¬|�δ|3 → |δ|1 = 1. And so, ¬|�δ|3 ≤ |δ|1. Therefore,|¬�¬�δ|1 ≤ |δ|1.

• If α is of the form ¬�¬�δ → �δ and L = T45m:

Note that |¬�¬�δ|1 = ¬|�¬�δ|1 = ¬|¬�δ|2 = ¬|�δ|3, such that |�δ|1 ∨ |�δ|3 = 1(by Definition 3.2.5). But, then ¬|�δ|3 → |�δ|1 = 1. And so, ¬|�δ|3 ≤ |�δ|1.

• If α is of the form �δ → ��δ and L ∈ {T4m, T45m}:

As |�δ|1 = |δ|2 such that (by Definitions 3.2.4 and 3.2.5) |δ|2 ≤ |�δ|2 and |��δ|1 =|�δ|2. So, |�δ|1 ≤ |��δ|1.

Suppose the result is true for every formula that has a shorter deduction lengththan the deduction of α.

• If α is an axiom, we have already proved that v(α) ∈ DB, for all valuation v forL ∈ {Tm, T4m, T45m, TBm, Dm}.

• If α follows from two previous formulas by (MP ), so these formulas must be theform β and β → α. By hypothesis, we have �L β and �L β → α. Then, byhypothesis of induction, we have �Mat(KL) β and �Mat(KL) β → α. Let v be avaluation. By definition, we have �Mat(KL) β iff v(β) ∈ DB iff π1(v(β)) = 1 and�Mat(KL) β → α iff v(β → α) ∈ DB iff π1(v(β → α)) = 1 and by Lemma 3.2.15 wehave π1(v(β → α)) = π1(v(β)) → π1(v(α)). So, π1(v(β)) → π1(v(α)) = 1 and asπ1(v(β)) = 1, then π1(v(α)) = 1. Therefore, v(α) ∈ DB for every valuation and so�Mat(KL) α.

Theorem 3.2.18. (Strong soundness) If L ∈ {Tm, T4m, T45m, TBm, Dm} and Δ ∪{α} ⊆ For(Σ�), then

Δ �L α ⇒ Δ �Mat(KL) α.

Proof. If Δ = ∅, the result has already been proved in the Theorem 3.2.17. If Δ �= ∅. Letv be a valuation such that v(δ) ∈ DB for every δ ∈ Δ. Let β1, . . . , βn be all the elementsof Δ that appear in a deduction of α from Δ. As hypothesis so, we have β1, . . . , βn �L α

and, by successive applications of the Deduction Theorem, we have �L β1 → (. . . →(βn → α) . . .). However, by the Theorem 3.2.17, we have �Mat(KL) β1 → (. . . → (βn →α) . . .). But, �Mat(KL) β1 → (. . . → (βn → α) . . .) ⇔ (β1 → (. . . → (βn → α) . . .)) ∈DB ⇒ π1(v(β1 → (. . . → (βn → α) . . .))) = 1. On the other hand, by Lemma 3.2.15,π1(v(β1 → (. . . → (βn → α) . . .))) = π1(v(β1)) → (. . . → ((π1(v(βn)) → π1(v(α))) . . .). So,π1(v(β1)) → (. . . → ((π1(v(βn)) → π1(v(α))) . . .) = 1. But, by hypothesis v(βi) ∈ DB, forall βi with (1 ≤ i ≤ n) and consequently π1(v(βi)) = 1 for all βi with (1 ≤ i ≤ n). So,π1(v(α)) = 1, but π1(v(α)) = 1 ⇔ v(α) ∈ DB. Then, Δ �Mat(KL) α.

Definition 3.2.19. If L ∈ {Tm, T4m, T45m, TBm, Dm} and Δ is a theory in L.Then, ≡Δ is a relation between formulas of L, such that:

In particular, if Δ = ∅ we have:

α ≡ β iff �L α → β and �L β → α.

Proposition 3.2.20. If L ∈ {Tm, T4m, T45m, TBm, Dm}. Then, the relation ≡Δ isa congruence w.r.t. connectives into Σ�� = {→, ¬}.

Proof. It follows from the fact that L contains classical logic over the signature Σ�. Seefor instance (RASIOWA; SIKORSKI, 1963).

Notation 3.2.21. If L ∈ {Tm, T4m, T45m, TBm, Dm} and ≡Δ is the congruencedefined above. Then, [α]Δ = α/≡Δ = {β ∈ For(Σ�) : α ≡Δ β} is the equivalence class ofα ∈ For(Σ�) and For(Σ�)/≡Δ = {α/≡Δ : α ∈ For(Σ�)} is the set of all equivalence classes.

Note that the congruence ≡Δ is well-defined in relation to operations intoΣ�� = {→, ¬} but, the connective � is not congruential. See below:

Proposition 3.2.22. If L ∈ {Tm, T4m, T45m, TBm, Dm}. Then, the connective �is not congruential, that is α ≡Δ β � �α ≡Δ �β.

Proof. The proof will be divided into two cases, the first to L ∈ {Tm, T4m, T45m, TBm}and the second to L = Dm.

So, if L ∈ {Tm, T4m, T45m, TBm}, then we have p ≡Δ ¬p → p but,�p �≡Δ �(¬p → p), because there exist a valuation v : For(Σ�) → {tn, tc, f c, f i} for L,such that:

• If L ∈ {Tm, TBm}, then v(p) = tc and for v(¬p → p) = tn, we have v(�p) ∈{f i, f c}, but v(�(¬p → p)) ∈ {tn, tc};

• If L = T4m, then v(p) = tc and for v(¬p → p) = tn, we have v(�p) ∈ {f i, f c}, butv(�(¬p → p)) ∈ {tn};

• If L = T45m, then v(p) = tc and for v(¬p → p) = tn, we have v(�p) ∈ {f i}, butv(�(¬p → p)) ∈ {tn}.

And if L = Dm, we have p ≡Δ ¬p → p but, �p �≡Δ �(¬p → p) too.Because there exist a valuation v : For(Σ�) → {T +, T −, C+, C−, F +, F −} for Dm, suchthat if v(p) = C+ and for v(¬p → p) = T +, we have v(�p) ∈ {T −, F −, C−}, butv(�(¬p → p)) ∈ {T +, F +, C+}.

Through the example, we see that the operator � is not well defined becauseits representative can change. As Carnielli and Coniglio did in (CARNIELLI; CONIGLIO,2016), to circumvent a similar problem with the operators ◦ and ¬ of the system mbC,we will use the concept of multioperation to dribble the problem, that is, we will de-fine a structure where the range of operations is a set that contains all their possiblerepresentatives.

Proposition 3.2.23. If L ∈ {Tm, T4m, T45m, TBm, Dm} and let

AΔ = �For(Σ�)/≡Δ , ∨, ∧, →, 0Δ, 1Δ�

be the structure such that

[α]Δ ∨ [β]Δ = [¬α → β]Δ

[α]Δ ∧ [β]Δ = [¬(α → ¬β)]Δ

0Δ = [¬(α → α)]Δ,

1Δ = [α → α]Δ,

Then, AΔ is a Boolean algebra.

Proof. The operations ∧ and ∨ are well defined by Proposition 3.2.20. Since L containsclassical logic, it is easy to see that AΔ = �For(Σ�)/≡Δ , ∨, ∧, →, 0Δ, 1Δ� is a Heytingalgebra. For any elements [α]Δ, [β]Δ ∈ For(Σ�)/≡Δ , we have [α ∨ (α → β)]Δ = [α]Δ ∨([α]Δ → [β]Δ) = [α]Δ ∨ (¬[α]Δ ∨ [β]Δ) = [α]Δ ∨ ([¬α]Δ ∨ [β]Δ) = ([α]Δ ∨ [¬α]Δ) ∨ [β]Δ =[α ∨ ¬α]Δ ∨ [β]Δ = [1]Δ ∨ [β]Δ = [1]Δ. Then, by Proposition 0.2.42, AΔ is a Booleanalgebra.

Definition 3.2.24 (Lindenbaum-Tarski swap structure for Tm). Given the Booleanalgebra

AΔ = �For(Σ�)/≡Δ , ∨, ∧, →, 0Δ, 1Δ�,

the Lindenbaum-Tarski swap structure for Tm over AΔ is the unique swap structure

BTmAΔ

= �BTmAΔ

, →, ¬,��,

such that

BTmAΔ

= {([α1]Δ, [α2]Δ, [α3]Δ) ∈ (For(Σ�)/≡Δ)3 : [α2 → α1]Δ = 1Δ and [α1 ∧ α3]Δ = 0Δ}.

Being so, for every ([α1]Δ, [α2]Δ, [α3]Δ) and ([β1]Δ, [β2]Δ, [β3]Δ) in BTmAΔ

theultioperations defined follo

(i) ([α1]Δ, [α2]Δ, [α3]Δ) → ([β1]Δ, [β2]Δ, [β3]Δ) = {([δ1]Δ, [δ2]Δ, [δ3]Δ) ∈ BTmAΔ

: [δ1]Δ =[α1]Δ → [β1]Δ, [α3]Δ ∨ [β2]Δ ≤ [δ2]Δ ≤ ([α1]Δ → [β1]Δ) ∧ ([α2]Δ → [β2]Δ) ∧ ([β3]Δ →[α3]Δ) and [δ3]Δ = [α2]Δ ∧ [β3]Δ};

(ii) ¬([α1]Δ, [α2]Δ, [α3]Δ) = {([¬α1]Δ, [α3]Δ, [α2]Δ)};

(iii) �([α1]Δ, [α2]Δ, [α3]Δ) = {([δ1]Δ, [δ2]Δ, [δ3]Δ) ∈ BTmAΔ

: [δ1]Δ = [α2]Δ}.

Definition 3.2.25 (Lindenbaum-Tarski swap structure for T4m). The Lindenbaum-Tarski swap structure for T4m over AΔ is the unique swap structure for T4m withdomain BTm

AΔ.

Note that, for every ([α1]Δ, [α2]Δ, [α3]Δ) in BTmAΔ

= BT4mAΔ

, the multioper-ation � is given by: �([α1]Δ, [α2]Δ, [α3]Δ) = {([δ1]Δ, [δ2]Δ, [δ3]Δ) ∈ BTm

AΔ: [δ1]Δ =

[α2]Δ and [α2]Δ ≤ [δ2]Δ}.

Definition 3.2.26 (Lindenbaum-Tarski swap structure for T45m). The Lindenbaum-Tarski swap structure for T45m over AΔ is the unique swap structure for T45m withdomain BTm

AΔ.

Note that, for every ([α1]Δ, [α2]Δ, [α3]Δ) in BTmAΔ

= BT45mAΔ

, the multioperation� is given by: �([α1]Δ, [α2]Δ, [α3]Δ) = {([δ1]Δ, [δ2]Δ, [δ3]Δ) ∈ B Tm

AΔ: [δ1]Δ = [α2]Δ, [α2]Δ ≤

[δ2]Δ and [δ3]Δ ∨ [δ1]Δ = 1Δ}.

Definition 3.2.27 (Lindenbaum-Tarski swap structure for TBm). The Lindenbaum-Tarski swap structure for TBm over AΔ is is the unique swap structure for TBm withdomain BTm

AΔ.

Note that, for every ([α1]Δ, [α2]Δ, [α3]Δ) in BTmAΔ

= BTBmAΔ

, the multioperation �is given by: �([α1]Δ, [α2]Δ, [α3]Δ) = {([δ1]Δ, [δ2]Δ, [δ3]Δ) ∈ BTm

AΔ: [δ1]Δ = [α2]Δ and [α1]Δ ∨

[δ3]Δ = 1Δ}.

Definition 3.2.28 (Lindenbaum-Tarski swap structure for Dm). Given the Booleanalgebra

AΔ = �For(Σ�)/≡Δ , ∨, ∧, →, 0Δ, 1Δ�,the Lindenbaum-Tarski swap structure for Dm over AΔ is the unique swap structure

BDmAΔ

= �BDmAΔ

, →, ¬,��,

such that

BDmAΔ

= {([α1]Δ, [α2]Δ, [α3]Δ) ∈ (For(Σ�)/≡Δ)3 : [α2 ∧ α3]Δ = 0Δ}.

Being so, for every ([α1]Δ, [α2]Δ, [α3]Δ) and ([β1]Δ, [β2]Δ, [β3]Δ) in BDmAΔ

the

(i) ([α1]Δ, [α2]Δ, [α3]Δ) → ([β1]Δ, [β2]Δ, [β3]Δ) = {([δ1]Δ, [δ2]Δ, [δ3]Δ) ∈ BDmAΔ

: [δ1]Δ =[α1]Δ → [β1]Δ, [α3]Δ ∨ [β2]Δ ≤ [δ2]Δ ≤ ([α2]Δ → [β2]Δ)∧ ([β3]Δ → [α3]Δ) and [δ3]Δ =[α2]Δ ∧ [β3]Δ};

(ii) ¬([α1]Δ, [α2]Δ, [α3]Δ) = {([¬α1]Δ, [α3]Δ, [α2]Δ)};

(iii) �([α1]Δ, [α2]Δ, [α3]Δ) = {([δ1]Δ, [δ2]Δ, [δ3]Δ) ∈ BDmAΔ

: [δ1]Δ = [α2]Δ}.

Definition 3.2.29 (Nmatrix associated to Lindenbaum-Tarski swap structure).

Let L ∈ {Tm, T4m, T45m, TBm, Dm} and AΔ be the Boolean algebradefined as above and let BL

AΔbe the Lindenbaum-Tarski swap structure over AΔ with

domain BLAΔ

.

If DBLAΔ

= {([α1]Δ, [α2]Δ, [α3]Δ) ∈ BLAΔ

: [α1]Δ = 1Δ}, then the non-deterministicmatrix associated to BL

AΔis:

M(BLAΔ

) = �BLAΔ

, DBLAΔ

�.

In particular, if Δ = ∅, M(BLA∅) = �BL

A∅ , DBLA∅

�.

Definition 3.2.30. If L ∈ {Tm, T4m, T45m, TBm, Dm} and AΔ is defined as in theProposition 3.2.23, then the relation �M(BL

AΔ) is the consequence relation defined over the

non-deterministic matrix M(BLAΔ

) (see Definitions 2.1.4, 2.1.5).

Lemma 3.2.31. If L ∈ {Tm, T4m, T45m, TBm, Dm} and AΔ is defined as in theProposition 3.2.23, then [α]Δ = 1Δ iff Δ �L α.

Proof. (⇒) By Definition 3.2.29, we have [α]Δ = 1Δ = [p1 ∨ ¬p1]Δ. So:

1. Δ �L p1 ∨ ¬p1 → α Definition of ≡Δ

2. Δ �L p1 ∨ ¬p1 CPC3. Δ �L α MP in 1 and 2

(⇐) Suppose Δ �L α:

1. Δ �L α Premise2. Δ �L p1 ∨ ¬p1 CPC3. Δ �L α → ((p1 ∨ ¬p1) → α) Axiom (Ax1)4. Δ �L (p1 ∨ ¬p1) → α MP in 1 e 35. Δ �L (p1 ∨ ¬p1) → (α → (p1 ∨ ¬p1)) Axiom (Ax1)6. Δ �L α → (p1 ∨ ¬p1) MP in 2 e 5

It shows that Δ �L (p1∨¬p1) → α and Δ �L α → (p1∨¬p1). So, [α]Δ = [p1∨¬p1]Δ =1Δ.

Lemma 3.2.32. If L ∈ {Tm, T4m, T45m, TBm, Dm} and AΔ is defined as in theProposition 3.2.23, then the following conditions are equivalent:

(i) [α]Δ ≤ [β]Δ

(ii) [α]Δ → [β]Δ = 1Δ

(iii) [α → β]Δ = 1Δ

(iv) Δ �L α → β

Proof. (i) ⇔ (ii) it is a well known classical result in a Boolean algebra.(ii) ⇔ (iii) it follows from the Proposition 3.2.20.(iii) ⇔ (iv) it follows from the Lemma 3.2.31.

Lemma 3.2.33. If AΔ is defined as in the Proposition 3.2.23. The following results holdin the indicated Lindenbaum-Tarski swap structures:

(i) [�¬¬α]Δ = [�α]Δ in BLAΔ

, for L ∈ {Tm, T4m, T45m, TBm, Dm};

(ii) [�α]Δ ≤ [��α]Δ in BLAΔ

, for L ∈ {T4m, T45m};

(iii) [�¬�α]Δ ∨ [�α]Δ = 1Δ in BT45mAΔ

;

(iv) [α]Δ ∨ [�¬�α]Δ = 1Δin BTBmAΔ

;

(v) [�¬(α → β)]Δ = [�α]Δ∧[�¬β]Δ in BLAΔ

, for L ∈ {Tm, T4m, T45m, TBm, Dm};

(vi) [�¬α]Δ ∨ [�β]Δ ≤ [�(α → β)]Δ in BLAΔ

, for L ∈ {Tm, T4m, T45m, TBm, Dm};

(vii) [�(α → β)]Δ ≤ [α]Δ → [β]Δ in BLAΔ

, for L ∈ {Tm, T4m, T45m, TBm};

(viii) [�(α → β)]Δ ≤ [�α]Δ → [�β]Δ in BLAΔ

, for L ∈ {Tm, T4m, T45m, TBm, Dm};

(ix) [�(α → β)]Δ ≤ [�¬β]Δ → [�¬α]Δ in BLAΔ

, for L ∈ {Tm, T4m, T45m, TBm, Dm}

Proof. In this proof we will use the Lemma 3.2.32 to simplify the proof.

(i) [�¬¬α]Δ = [�α]Δ iff Δ �L �¬¬α → �α and Δ �L �α → �¬¬α. We can get thesetwo conditions by applying the axioms (DN1) and (DN2).

(ii) [�α]Δ ≤ [��α]Δ iff Δ �L �α → ��α. But this holds by Axiom (4).

(iii) [�¬�α]Δ ∨ [�α]Δ = 1Δ iff [�¬�α ∨ �α]Δ = 1Δ iff [¬�¬�α → �α]Δ = 1Δ iffΔ �T45m ¬�¬�α → �α. But this holds by Axiom (5).

(iv) [α]Δ ∨ [�¬�α]Δ = 1Δ iff [α ∨ �¬�α]Δ = 1Δ iff [¬�¬�α → α]Δ = 1Δ iff Δ �TBm

¬�¬�α → α. But this holds by Axiom (B).

(v) [�¬(α → β)]Δ = [�α]Δ ∧ [�¬β]Δ iff [�¬(α → β)]Δ = [�α ∧ �¬β]Δ iff Δ �L

�¬(α → β) → (�α ∧ �¬β) and Δ �L (�α ∧ �¬β) → �¬(α → β). See thededuction below:1. Δ �L ¬�¬(α → β) → (�α → ¬�¬β) Axiom (k2)2. Δ �L ¬(�α → ¬�¬β) → ¬¬�¬(α → β) CPC in 13. Δ �L ¬(¬�α ∨ ¬�¬β) → �¬(α → β) CPC in 24. Δ �L (¬¬�α ∧ ¬¬�¬β) → �¬(α → β) CPC in 35. Δ �L (�α ∧ �¬β) → �¬(α → β) CPC in 4

On the other hand:1. Δ �L �¬(α → β) → �¬β Axiom (M3)2. Δ �L �¬(α → β) → �α Axiom (M4)3. Δ �L �¬(α → β) → (�α ∧ �¬β) CPC in 1 and 2

(vi) [�¬α]Δ ∨ [�β]Δ ≤ [�(α → β)]Δ iff [�¬α ∨ �β]Δ ≤ [�(α → β)]Δ iff Δ �L (�¬α ∨�β) → �(α → β). See the deduction below:

1. Δ �L �¬α → �(α → β) Axiom (M1)2. Δ �L �β → �(α → β) Axiom (M2)3. Δ �L (�¬α ∨ �β) → �(α → β) CPC in 1 and 2

(vii) [�(α → β)]Δ ≤ [α]Δ → [β]Δ iff [�(α → β)]Δ ≤ [α → β]Δ iff Δ �L �(α → β) →(α → β). But this is a consequence of Axiom (T).

(viii) [�(α → β)]Δ ≤ [�α]Δ → [�β]Δ iff [�(α → β)]Δ ≤ [�α → �β]Δ iff Δ �L �(α →β) → (�α → �β). But this follows by Axiom (K).

(ix) [�(α → β)]Δ ≤ [�¬β]Δ → [�¬α]Δ iff [�(α → β)]Δ ≤ [�¬β → �¬α]Δ iff Δ �L

�(α → β) → (�¬β → �¬α). But this is a consequence of Axiom (K1).

Proposition 3.2.34. If L ∈ {Tm, T4m, T45m, TBm, Dm}, then the function vΔ :For(Σ�) → BL

AΔ, such that vΔ(α) = ([α]Δ, [�α]Δ, [�¬α]Δ) is a valuation in M(BL

AΔ).

That is, for every α, β ∈ For(Σ�):

(i) vΔ(¬α) ∈ ¬vΔ(α);

(ii) vΔ(�α) ∈ �vΔ(α);

(iii) vΔ(α → β) ∈ vΔ(α) → vΔ(β).

Proof. (i) vΔ(¬α) = ([¬α]Δ, [�¬α]Δ, [�¬¬α]Δ).

On the other hand, for L ∈ {Tm, T4m, T45m, TBm, Dm}, we have by Proposi-tion 3.2.20:

¬vΔ(α) = ¬([α]Δ, [�α]Δ, [�¬α]Δ) = {([¬α]Δ, [�¬α]Δ, [�α]Δ)} and by Lemma 3.2.33(i), [�¬¬α]Δ = [�α]Δ. Thus vΔ(¬α) ∈ ¬vΔ(α);

(ii) vΔ(�α) = ([�α]Δ, [��α]Δ, [�¬�α]Δ). On the other hand:

• For L ∈ {Tm, Dm}, we have:�vΔ(α) = �([α]Δ, [�α]Δ, [�¬α]Δ) = {([β1]Δ, [β2]Δ, [β3]Δ) ∈ BL

AΔ: [β1]Δ =

[�α]Δ}. Therefore vΔ(�α) ∈ �vΔ(α);

• For L = T4m, we have:�vΔ(α) = �([α]Δ, [�α]Δ, [�¬α]Δ) = {([β1]Δ, [β2]Δ, [β3]Δ) ∈ BL

AΔ: [β1]Δ =

[�α]Δ and [�α]Δ ≤ [β2]Δ} and by Lemma 3.2.33 (ii) [�α]Δ ≤ [��α]Δ. There-fore vΔ(�α) ∈ �vΔ(α);

• For L = T45m, we have:�vΔ(α) = �([α]Δ, [�α]Δ, [�¬α]Δ) = {([β1]Δ, [β2]Δ, [β3]Δ) ∈ BL

AΔ: [β1]Δ =

[�α]Δ, [�α]Δ ≤ [β2]Δ and [β3]Δ ∨ [β1]Δ = 1Δ} and by Lemma 3.2.33 (ii) and(iii) [�α]Δ ≤ [��α]Δ and [�¬�α]Δ∨[�α]Δ = 1Δ. Therefore vΔ(�α) ∈ �vΔ(α);

• For L = TBm, we have:�vΔ(α) = �([α]Δ, [�α]Δ, [�¬α]Δ) = {([β1]Δ, [β2]Δ, [β3]Δ) ∈ BL

AΔ: [β1]Δ =

[�α]Δ and [α]Δ ∨ [β3]Δ = 1Δ} and by Lemma 3.2.33 (iv) [α]Δ ∨ [�¬�α]Δ = 1Δ.Therefore vΔ(�α) ∈ �vΔ(α);

(iii) vΔ(α → β) = ([α → β]Δ, [�(α → β)]Δ, [�¬(α → β)]Δ). On the other hand:

• For L ∈ {Tm, T4m, T45m, TBm}, we have:vΔ(α) → vΔ(β) =([α]Δ, [�α]Δ, [�¬α]Δ) → ([β]Δ, [�β]Δ, [�¬β]Δ) = {([γ1]Δ, [γ2]Δ, [γ3]Δ) ∈ BL

AΔ:

[γ1]Δ = [α]Δ → [β]Δ, [γ3]Δ = [�α]Δ ∧ [�¬β]Δ and [�¬α]Δ ∨ [�β]Δ ≤ [γ2]Δ ≤([α]Δ → [β]Δ) ∧ ([�α]Δ → [�β]Δ) ∧ ([�¬β]Δ → [�¬α]Δ)}.By Lemma 3.2.33 (v) and (vi), we have [�¬(α → β)]Δ = [�α]Δ ∧ [�¬β]Δ and[�¬α]Δ ∨ [�β]Δ ≤ [�(α → β)]Δ. And by Lemma 3.2.33 (vii), (viii) and (ix), wehave [�(α → β)]Δ ≤ ([α]Δ → [β]Δ) ∧ ([�α]Δ → [�β]Δ) ∧ ([�¬β]Δ → [�¬α]Δ).Therefore vΔ(α → β) ∈ vΔ(α) → vΔ(β).

• For L = Dm, we have:vΔ(α) → vΔ(β) =

([α]Δ, [�α]Δ, [�¬α]Δ) → ([β]Δ, [�β]Δ, [�¬β]Δ) = {([γ1]Δ, [γ2]Δ, [γ3]Δ) ∈ BLAΔ

:[γ1]Δ = [α]Δ → [β]Δ, [γ3]Δ = [�α]Δ ∧ [�¬β]Δ and [�¬α]Δ ∨ [�β]Δ ≤ [γ2]Δ ≤([�α]Δ → [�β]Δ) ∧ ([�¬β]Δ → [�¬α]Δ)}.By Lemma 3.2.33 (v) and (vi), we have [�¬(α → β)]Δ = [�α]Δ ∧ [�¬β]Δ and[�¬α]Δ ∨ [�β]Δ ≤ [�(α → β)]Δ. And by Lemma 3.2.33 (viii) and (ix), we have[�(α → β)]Δ ≤ ([�α]Δ → [�β]Δ) ∧ ([�¬β]Δ → [�¬α]Δ).Therefore vΔ(α → β) ∈ vΔ(α) → vΔ(β).

Proposition 3.2.35. If L ∈ {Tm, T4m, T45m, TBm, Dm}, then the function vΔ :For(Σ�) → BL

AΔ, such that vΔ(α) = ([α]Δ, [�α]Δ, [�¬α]Δ) is a canonical valuation in

M(BLAΔ

), that is vΔ(α) ∈ DBLAΔ

iff Δ �L α.

Proof. Suppose vΔ(α) = ([α]Δ, [�α]Δ, [�¬α]Δ) ∈ DBLAΔ

, by Definition 3.2.29, we have[α]Δ = 1Δ. So, by Proposition 3.2.34, vΔ(α) is a valuation in M(BL

AΔ) and by Lemma 3.2.31,

vΔ(α) ∈ DBLAΔ

iff Δ �L α.

Corollary 3.2.36. For L ∈ {Tm, T4m, T45m, TBm, Dm}, �L α iff �L (β∨¬β) → α.

The canonical valuation allows to prove immediately the completeness ofL ∈ {Tm, T4m, T45m, TBm, Dm} w.r.t. swap structures:

Theorem 3.2.37. (Completeness) For L ∈ {Tm, T4m, T45m, TBm, Dm} and forevery Δ ∪ {α} ⊆ For(Σ�),

Δ |=Mat(KL) α ⇒ Δ �L α.

Proof. Suppose that Δ ��L α. Then by Proposition 3.2.35, there exist a valuation vΔ overa Lindenbaum-Tarski swap structure BL

AΔsuch that vΔ(α) �∈ DBL

AΔ.

If β ∈ Δ, then Δ �L β and applying again the Proposition 3.2.35, we havevΔ(β) ∈ DBL

AΔ. Thus, vΔ[Δ] ⊆ DBL

AΔbut vΔ(α) �∈ DBL

AΔ. From this, Δ �|=Mat(KL) α.

Corollary 3.2.38. For L ∈ {Tm, T4m, T45m, TBm, Dm}, �M(BLAΔ

) α iff �M(BLAΔ

)

(β ∨ ¬β) → α.

Proposition 3.2.39. For L ∈ {Tm, T4m, T45m, TBm, Dm} and for every Δ ∪{α} ∪ {β} ⊆ For(Σ�), we have Δ, α �M(BL

AΔ) β iff Δ �M(BL

AΔ) α → β.

Proof. Δ, α �M(BLAΔ

) β ⇔ Δ, α �L β ⇔ Δ �L α → β ⇔ Δ �M(BLAΔ

) α → β.

Proposition 3.2.40. For L ∈ {Tm, T4m, T45m, TBm, Dm} and for every Δ ∪{α} ⊆ For(Σ�), we have Δ �M(BL

AΔ) α iff there exist Δ0 finite ⊆ Δ, such that Δ0 �M(BL

AΔ)

α.

Proof. Δ �M(BLAΔ

) α ⇔ Δ �L α ⇔ there exist Δ0 finite ⊆ Δ, such that Δ0 �L α ⇔ thereexist Δ0 finite ⊆ Δ, such that Δ �M(BL

AΔ) α.

Remark 3.2.41. In the proof of the next proposition we are assuming that the infimumof the empty set is a tautology.

Proposition 3.2.42. For L ∈ {Tm, T4m, T45m, TBm, Dm}, let Δ0 be finite. Forevery Δ0 ∪ {α} ⊆ For(Σ�), we have Δ0 �M(BL

AΔ) α iff �M(BL

AΔ)

� Δ0 → α.

Proof. • If Δ0 = ∅:

�M(BLAΔ

) α ⇔(3.2.38)�M(BLAΔ

) (β ∨ ¬β) → α ⇔ �M(BLAΔ

)� Δ0 → α;

• If Δ0 �= ∅, let Δ0 = {γ1, γ2, . . . , γn−1, γn} :

Δ0 �M(BLAΔ

) α ⇔γ1, γ2, . . . , γn−1, γn �M(BL

AΔ) α ⇔(3.2.39)

γ1, γ2, . . . , γn−1 �M(BLAΔ

) γn → α ⇔(3.2.39)

γ1, γ2, . . . , γn−2 �M(BLAΔ

) γn−1 → (γn → α) ⇔(3.2.39)

�M(BLAΔ

) γ1 → (γ2 → (. . . → (γn−1 → (γn → α)) . . .)) ⇔�M(BL

AΔ) (γ1 ∧ γ2 ∧ . . . ∧ γn−1 ∧ γn) → α ⇔

�M(BLAΔ

)� Δ0 → α.

Theorem 3.2.43. For L ∈ {Tm, T4m, T45m, TBm, Dm} and for every Δ ∪ {α} ⊆For(Σ�), are equivalent:

1. Δ �L α;

2. Δ �Mat(KL) α;

3. Δ �M(BLAΔ

) α;

4. Δ �M(BLA2

) α;

5. Δ �M(BLA∅)

α.

Proof.

(1) ⇔ (2) By Theorems 3.2.18 and 3.2.37

(1) ⇔ (3) By Theorem 3.2.18 and the proof of Theorem 3.2.37;

(1) ⇔ (4) By the soundness and completeness theorems proved by (CONIGLIO;CERRO; PERON, 2016) and by observing that BL

A2 are the finite-valued non-deterministic matrices presented there;

(1) ⇔ (5)

Δ �M(BLA∅)

α ⇔(3.2.40)

There exist Δ0 finite ⊆ Δ, such that Δ0 �M(BLA∅)

α ⇔(3.2.42)

There exist Δ0 finite ⊆ Δ, such that �M(BLA∅)

� Δ0 → α ⇔There exist Δ0 finite ⊆ Δ, such that �L

� Δ0 → α ⇔5

There exist Δ0 finite ⊆ Δ, such that Δ0 �L α ⇔6 Δ �L α.

Corollary 3.2.44. For L ∈ {Tm, T4m, T45m, TBm, Dm} and for every α ∈ For(Σ�),are equivalent:

1. �L α;

2. �Mat(KL) α;

3. �M(BLAΔ

) α;

4. �M(BLA2

) α.

5 By Deduction Theorem and properties.

4 An algebraic study of LFIs by means ofswap structures

In this chapter, we will present an algebraic study of LFIs theory using swapstructures and the properties and definitions of multialgebras introduced in previouschapters. For this, we will use definitions and properties of multialgebras and the develop-ment in the category theory started in the Chapter 2, and we will apply the method ofLindenbaum-Tarski swap structures, introduced in the Chapter 3 to obtain completenesstheorems w.r.t Hilbert-style version of these logical systems. This results are based on thepre-print (CONIGLIO; ORELLANO; GOLZIO, 2016).

The C-systems of da Costa (COSTA, 1963; COSTA, 1974) motivated theemergence of several paraconsistent logics in the last years. That was the case of Logicsof Formal Inconsistency (LFIs, for short) introduced by W. Carnielli and J. Marcosin (CARNIELLI; MARCOS, 2002), as a generalization of da Costa systems Cn.

In its simplest form, the LFIs have a non-explosive negation ¬, as well as a(primitive or derived) consistency connective ◦ which allows to recover the explosion lawin a controlled way.

Definition 4.0.1. Let L = �Θ, �� be a Tarskian, finitary and structural logic defined overa propositional signature Θ, which contains a negation ¬, and let ◦ be a (primitive ordefined) unary connective. Then, L is said to be a Logic of Formal Inconsistency withrespect to ¬ and ◦ if the following holds:

(i) α, ¬α � β for some α and β;

(ii) there are two formulas ϕ and ψ such that

(ii.a) ◦ϕ, ϕ � ψ;

(ii.b) ◦ϕ, ¬ϕ � ψ;

(iii) ◦α, α, ¬α � β for every α and β.

Condition (ii) of the definition of LFIs is required in order to satisfy condi-tion (iii) in a non-trivial way. The hierarchy of LFIs studied in (CARNIELLI; CONIGLIO;MARCOS, 2007) and (CARNIELLI; CONIGLIO, 2016) starts from a logic called mbC,which extends classical positive logic CPL+ by adding a negation ¬ and an unary consis-

As it is well-known, several logics in the hierarchy of the LFIs cannot besemantically characterized by a single finite matrix. Moreover, they lie outside the scopeof the usual techniques of algebraization of logics such as Blok and Pigozzi’s method.Several alternative semantical tools were introduced in the literature in order to deal withsuch systems: (non-truth-functional) bivaluations, possible-translations semantics, andnon-deterministic matrices (or Nmatrices), obtaining so, decision procedures for theselogics. However, the problem of finding an algebraic counterpart for this kind of logic, in asense to be determined, remains open.

So, we use the concepts of multialgebra, swap structure and non-deterministicmatrix to get some results towards the possibility of defining an algebraic theory of swapstructures semantics. As a first step, we will concentrate our efforts on the algebraictheory of KmbC, the class of swap structures for the logic mbC and we will prove that theclass KmbC is closed under sub-swap-structures and products, but it is not closed underhomomorphic images, hence it is not a variety in the usual sense.

We will also show that is possible to give a representation theorem for KmbC

which is analogous to the Birkhoff’s theorem in traditional algebraic logics (see Theo-rem 4.2.15). As a consequence of this, the class KmbC is generated by the structure withfive elements, which is constructed over the 2-element Boolean algebra. Such structure isprecisely Avron’s 5-valued characteristic non-deterministic matrix for mbC.

And finally, we will prove that, under the present approach, the classes of swapstructures for the axiomatic extensions of mbC found in (CARNIELLI; CONIGLIO, 2016)are subclasses of KmbC. They are obtained by requiring that its elements satisfy preciselythe additional axioms which define the corresponding logic. Analogous representationtheorems for each class of multialgebras will be found and this allow a modular treatmentof the algebraic theory of swap structures, as happens in the traditional algebraic setting.

From now on, the following signatures will be considered:

Σ+ = {∧, ∨, →};

ΣBA = {∧, ∨, →, 0, 1} and

Σ = {∧, ∨, →, ¬, ◦}.

Remark 4.0.2. Although we use the same symbols ¬ and → in Σ = {∧, ∨, →, ¬, ◦}and Σ� = {¬, →,�} (recall Chapter 3), in each signature these symbols have a differentinterpretation. So the approach given to operators ¬ and → in the previous chapter isdifferent from that is done in this chapter.

Definition 4.0.3 (Classical Positive Logic). The classical positive logic CPL+ is definedover the language For(Σ+) by the following Hilbert-style deductive system:

Axiom schemas:

α → (β → α) (Ax1)

(α → (β → γ)) → ((α → β) → (α → γ)) (Ax2)

α → (β → (α ∧ β)) (Ax3)

(α ∧ β) → α (Ax4)

(α ∧ β) → β (Ax5)

α → (α ∨ β) (Ax6)

β → (α ∨ β) (Ax7)

(α → γ) → ((β → γ) → ((α ∨ β) → γ)) (Ax8)

(α → β) ∨ α (Ax9)

Inference rule modus ponens:

α, α → β

β(MP )

Definition 4.0.4. The logic mbC, defined over signature Σ, is obtained from CPL+ byadding the following axiom schemas:

α ∨ ¬α (Ax10)

◦α → (α → (¬α → β)) (bc1)

For convenience, a (purely linguistic) expansion of CPL+ over signature Σ willbe considered from now on, besides CPL+ itself. This logic, denoted by CPL+

e , is nothingmore than CPL+ defined over Σ, that is, by adding ¬ and ◦ without any axioms or rulesfor them.

The simplest semantical characterization of mbC is given in terms of bivalua-tions or mbC-valuations (see (CARNIELLI; CONIGLIO; MARCOS, 2007)).

Definition 4.0.5 (Valuations for mbC). A function v : For(Σ) → {0, 1} is an mbC-valuation, if it satisfies the following conditions:

v(α ∧ β) = 1 ⇔ v(α) = 1 and v(β) = 1 (vAnd)

v(α ∨ β) = 1 ⇔ v(α) = 1 or v(β) = 1 (vOr)

v(α → β) = 1 ⇔ v(α) = 0 or v(β) = 1 (vImp)

v(α) = 0 ⇒ v(¬α) = 1 (vNeg)

v(◦α) = 1 ⇒ v(α) = 0 or v(¬α) = 0 (vCon)

Definition 4.0.6 (Semantical consequence relation). For every Γ∪{α} ⊆ For(Σ), Γ �mbC

α iff, for every mbC-valuation v, if v(γ) = 1 for every γ ∈ Γ, then v(α) = 1.

Proposition 4.0.7 (Soundness and completeness w.r.t. valuations). For every Γ ∪ {α} ⊆For(Σ), Γ �mbC α iff Γ �mbC α.

Proof. In (CARNIELLI; CONIGLIO; MARCOS, 2007, p. 38 and 40).

4.1 Swap structures for CPL+e

In (CARNIELLI; CONIGLIO, 2016) it was introduced the notion of swapstructures for mbC, as well as for some axiomatic extensions of it. In this section, thesestructures will be reintroduced in a slightly more general form, in order to define a hierarchyof classes of multialgebras associated to the corresponding hierarchy of logics. This is inline with the traditional approach of algebraic logic, in which hierarchies of classes ofalgebraic models are associated to hierachies of logics.

Since mbC is an axiomatic extension of CPL+e , it is natural to begin with

swap structures for the latter logic.

The algebraic semantics for CPL+ is given by classical implicative lattices(recall Definition 0.2.45). Namely, Γ �CPL+ α iff, for every classical implicative lattice Aand for every homomorphism v : For(Σ+) → A, if v(γ) = 1 for every γ ∈ Γ then v(α) = 1.

Now a semantics of swap structures over a given classical implicative latticeA will be introduced for CPL+

e . The idea is that a triple (c1, c2, c3) in such structurerepresents a (composite) truth-value in which c1 represents the truth-value of a formula α,while c2 and c3 represent a possible truth-value for ¬α and ◦α, respectively. Observe theanalogy with the semantics approach to modal logics given in the previous chapter.

Let A = �A, ∧, ∨, →� be classical implicative lattice and let πj : A3 → A bethe canonical projections, for 1 ≤ j ≤ 3

Definition 4.1.1. Let A be a classical implicative lattice with domain A. The universeof swap structures for CPL+

e over A is the set BCPL+e

A = A3.

Definition 4.1.2. Let A = �A, ∧, ∨, →� be a classical implicative lattice and let B ⊆BCPL+

eA . A swap structure for CPL+

e over A is any multialgebra B = �B, ∧, ∨, →, ¬, ◦� overΣ such that the multioperations satisfy the following, for every (a1, a2, a3) and (b1, b2, b3)in B:

(i) ∅ �= (a1, a2, a3)#(b1, b2, b3) ⊆ {(c1, c2, c3) ∈ B : c1 = a1#b1}, for # ∈ {∧, ∨, →};

(ii) ∅ �= ¬(a1, a2, a3) ⊆ {(c1, c2, c3) ∈ B : c1 = a2};

(iii) ∅ �= ◦(a1, a2, a3) ⊆ {(c1, c2, c3) ∈ B : c1 = a3}.

For every classical implicative lattice A, there is a unique swap structure BCPL+e

Awith domain BCPL+

eA = A3 such that, for every (a1, a2, a3) and (b1, b2, b3) in A3:

(i) (a1, a2, a3)#(b1, b2, b3) = {(c1, c2, c3) ∈ A3 : c1 = a1#b1}, for # ∈ {∧, ∨, →};

(ii) ¬(a1, a2, a3) = {(c1, c2, c3) ∈ A3 : c1 = a2};

(iii) ◦(a1, a2, a3) = {(c1, c2, c3) ∈ A3 : c1 = a3}.

Remark 4.1.3. Observe that, if B is a swap structure for CPL+e over A, then B ⊆ BCPL+

eA

in the sense of Definition 2.2.1 (as submultialgebra).

Proposition 4.1.4. Let A be a classical implicative lattice with domain A and let B aswap structure for CPL+

e over A with domain B, then π1[B] (with the operations naturallyinduced by composing π1 with the multioperations of B) is a classical implicative latticeincluded in A as subalgebra.

Proof. By definition of multioperations in B, π1[(c1, c2, c3)#(d1, d2, d3)] = {c1#d1} for# ∈ {∧, ∨, →}. So, π1[B] is closed under the operations ∧, ∨ and → of A: if a1, b1 ∈ π1[B],then a1#b1 ∈ π1[B] for # ∈ {∧, ∨, →}. Now, let c = (c1, c2, c3) ∈ B. By definitionof multioperations in B, π1[c → c] = {c1 → c1} = {1} and so 1 ∈ π1[B]. Now, sinceπ1[(a1, a2, a3) → (b1, b2, b3)] = {a1 → b1} and A is a classical implicative lattice, then a1 →b1 = �{c1 ∈ A : a1 ∧c1 ≤ b1}, for every a1, b1 ∈ A. Additionally, let (a1, a2, a3), (b1, b2, b3) ∈B, π1[(a1, a2, a3) ∨ ((a1, a2, a3) → (b1, b2, b3))] = {a1 ∨ (a1 → b1)} = {1}. Therefore, π1[B]is a classical implicative lattice.

As in the previous chapter, elements of a swap structure for CPL+e are called

snapshots for CPL+e (or simply snapshots). Since no axioms or rules are given in CPL+

e

for the new connectives ¬ and ◦, the multioperations associated to it in a swap structurejust put in evidence (or ‘swap’) its value on the first coordinate, leaving free the value ofthe other cordinates, obtaining so a (non-empty) set of snapshots. As we shall see in thenext sections, when axioms are given for ¬ and ◦, the multioperations (and the domain ofthe swap structures themselves) must be restricted.

Now, we will use the notions of semantics associated to non-deterministicmatrices introduced by Avron and Lev (see Definitions 2.1.4 and 2.1.5).

Let K+CPL+

ebe the class of swap structures for CPL+

e .

As it was done in (CARNIELLI; CONIGLIO, 2016, Chapter 6) with severalLFIs, and as it was done in the previous chapter, it is easy to see that each B ∈ K+

CPL+e

induces naturally a non-deterministic matrix such that the class of such non-deterministicmatrices semantically characterizes CPL+

e . More precisely:

Definition 4.1.5. For each B ∈ K+CPL+

e, let DB = {(c1, c2, c3) ∈ |B| : c1 = 1}.

1. The non-deterministic matrix associated to B is M(B) = (B, DB);

2. Mat(K+CPL+

e) =

�M(B) : B ∈ K+

CPL+e

�is the class of the non-deterministic matrices

associated to B.

In this particular case, Definition 2.1.5 assumes the following form:

Definition 4.1.6. Let B ∈ K+CPL+

eand M(B) as above. A valuation over M(B) is a

function v : For(Σ) → |B| such that, for every α, β ∈ For(Σ):

1. v(α#β) ∈ v(α)#v(β), for every # ∈ {∧, ∨, →};

2. v(¬α) ∈ ¬v(α);

3. v(◦α) ∈ ◦v(α).

Definition 4.1.7. Let Γ ∪ {α} ⊆ For(Σ) be a set of formulas of CPL+e . We say that α

is a consequence of Γ in the class Mat(K+CPL+

e) of non-deterministic matrices, denoted

by Γ |=Mat(K+CPL+

e) α, if Γ |=M α for every M ∈ Mat(K+

CPL+e). In particular, α is valid in

Mat(K+CPL+

e), denoted by |=Mat(K+

CPL+e

) α, if it is valid in every M ∈ Mat(K+CPL+

e).

In the following theorem, in order to prove completeness, we will apply, onceagain the method of the Lindenbaum-Tarski swap structure already used in the previous

Theorem 4.1.8 (Adequacy of CPL+e w.r.t. swap structures). Let Γ ∪ {α} ⊆ For(Σ) be

a set of formulas of CPL+e . Then: Γ �CPL+

eα iff Γ |=Mat(K+

CPL+e

) α.

Proof. ‘Only if’ part (Soundness): It follows by the definition of semantics of swap struc-tures, and by the fact that CPL+ is sound w.r.t. classical implicative lattices.

Indeed, if v is a valuation over B ⊆ BCPL+e

A then h = π1 ◦ v : For(Σ) → A is aΣ+-homomorphism such that h(γ) = 1 iff v(γ) ∈ DB: If # ∈ {∧, ∨, →}, v(ϕ) # v(ψ) ={(c1, c2, c3) ∈ |B| : c1 = π1 ◦ v(ϕ) # π1 ◦ v(ψ)}, so π1[v(ϕ) # v(ψ)] = {π1 ◦ v(ϕ) # π1 ◦v(ψ)}. By Definition 4.1.6, we have v(ϕ # ψ) ∈ v(ϕ) # v(ψ), thus π1 ◦ v(ϕ # ψ) =π1(v(ϕ) # v(ψ)) = π1 ◦ v(ϕ) # π1 ◦ v(ψ) and therefore h(ϕ # ψ) = h(ϕ) # h(ψ). Finally,by Definition 4.1.5 we have h(γ) = 1 iff π1 ◦ v(γ) = 1 iff v(γ) ∈ DB.

If α is an (instance of an) axiom of CPL+e then h(α) = 1 and so v(α) ∈ DB:

See, for instance, if α is the (instance of the) Axioma (Ax1): h(γ → (β → γ)) = π1 ◦v(γ →(β → γ)) = π1 ◦ v(γ) → (π1 ◦ v(β) → π1 ◦ v(γ)) = 1. Since h(ϕ # ψ) = h(ϕ) # h(ψ) for# ∈ {∧, ∨, →}, the proof for the remaining axioms of CPL+

e is quite similar.

On the other hand, if α follows from two previous formulas by (MP ), that is:if v(γ) ∈ DB and v(γ → α) ∈ DB then v(α) ∈ DB, because by hypothesis π1 ◦ v(γ) = 1and π1 ◦ v(γ → α) = π1 ◦ v(γ) → π1 ◦ v(α) = 1, so, clearly π1 ◦ v(α) = 1.

Thus, by induction on the length of a derivation of α from Γ, Γ �CPL+e

α impliesΓ |=Mat(K+

CPL+e

) α.

‘If’ part (Completeness): Suppose that Γ �CPL+e

α. Define in For(Σ) thefollowing relation: ϕ ≡Γ ψ iff Γ �CPL+

eϕ → ψ and Γ �CPL+

eψ → ϕ. It is clearly an

equivalence relation. Let AΓdef= For(Σ)/≡Γ be the quotient set, and define over AΓ the

following operations: [ϕ]Γ # [ψ]Γ def= [ϕ # ψ]Γ, for # ∈ {∧, ∨, →} (here, [ϕ]Γ denotes theequivalence class of ϕ w.r.t. ≡Γ). These operations are clearly1 well-defined, and so theyinduce a structure of classical implicative lattice over the set AΓ, that is [ϕ]Γ ≤ [ψ]Γ seeΓ �CPL+

eϕ → ψ is a partial ordering and [ϕ]Γ ∧ [ψ]Γ ≤ [θ]Γ see [ψ]Γ ≤ [ϕ → θ]Γ. Let AΓ

be the obtained classical implicative lattice, and let BCPL+e

AΓbe the corresponding swap

structure. Let MCPL+e

Γdef= M(BCPL+

eAΓ

), and consider the mapping vΓ : For(Σ) → (AΓ)3

given by vΓ(ϕ) = ([ϕ]Γ, [¬ϕ]Γ, [◦ϕ]Γ). Then vΓ is a valuation over the non-deterministicmatrix MCPL+

eΓ such that, for every ϕ, vΓ(ϕ) ∈ D

BCPL+e

AΓ

iff Γ �CPL+e

ϕ, see the proof below:

For every ϕ, ψ ∈ For(Σ), we have:

• vΓ(ϕ # ψ) = ([ϕ # ψ]Γ, [¬(ϕ # ψ)]Γ, [◦(ϕ # ψ)]Γ) and1 Using the axioms (Ax1) − (Ax8) of the Definition 4.0.3.

vΓ(ϕ) # vΓ(ψ) = ([ϕ]Γ, [¬ϕ]Γ, [◦ϕ]Γ) # ([ψ]Γ, [¬ψ]Γ, [◦ψ]Γ) = {(c1, c2, c3) ∈ (AΓ)3 :c1 = [ϕ]Γ # [ψ]Γ}, for # ∈ {∧, ∨, →}. Since [ϕ]Γ # [ψ]Γ def= [ϕ # ψ]Γ, for # ∈{∧, ∨, →}, then vΓ(ϕ # ψ) ∈ vΓ(ϕ) # vΓ(ψ).

• vΓ(¬ϕ) = ([¬ϕ]Γ, [¬¬ϕ]Γ, [◦¬ϕ]Γ) and ¬(vΓ(ϕ)) = ¬([ϕ]Γ, [¬ϕ]Γ, [◦ϕ]Γ) = {(c1, c2, c3) ∈(AΓ)3 : c1 = [¬ϕ]Γ}. So vΓ(¬ϕ) ∈ ¬(vΓ(ϕ)).

• vΓ(◦ϕ) = ([◦ϕ]Γ, [¬◦ϕ]Γ, [◦◦ϕ]Γ) and ◦(vΓ(ϕ)) = ◦([ϕ]Γ, [¬ϕ]Γ, [◦ϕ]Γ) = {(c1, c2, c3) ∈(AΓ)3 : c1 = [◦ϕ]Γ}. So vΓ(◦ϕ) ∈ ◦(vΓ(ϕ)).

Thus vΓ is a valuation over MCPL+e

Γ .

Fact: vΓ(ϕ) ∈ DBCPL+

eAΓ

iff Γ �CPL+e

ϕ. Indeed, suppose that vΓ(ϕ) = ([ϕ]Γ, [¬ϕ]Γ, [◦ϕ]Γ) ∈D

BCPL+e

AΓ

. By definition [ϕ]Γ = 1Γ = [p1 ∨ ¬p1]Γ. So:

1. Γ �CPL+e

(p1 ∨ ¬p1) → ϕ Definition of ≡Γ

2. Γ �CPL+e

p1 ∨ ¬p1 Ax103. Γ �CPL+

eϕ MP in 1 and 2

Conversely, suppose that Γ �CPL+e

ϕ:

1. Γ �CPL+e

ϕ hypothesis2. Γ �CPL+

ep1 ∨ ¬p1 Ax10

3. Γ �CPL+e

ϕ → ((p1 ∨ ¬p1) → ϕ) Ax14. Γ �CPL+

e(p1 ∨ ¬p1) → ϕ MP in 1 and 3

5. Γ �CPL+e

(p1 ∨ ¬p1) → (ϕ → (p1 ∨ ¬p1)) Ax16. Γ �CPL+

eϕ → (p1 ∨ ¬p1) MP in 2 and 5

So Γ �CPL+e

(p1 ∨ ¬p1) → ϕ and Γ �CPL+e

ϕ → (p1 ∨ ¬p1). Thus, [ϕ]Γ =[p1 ∨ ¬p1]Γ = 1Γ and therefore, vΓ(ϕ) = ([ϕ]Γ, [¬ϕ]Γ, [◦ϕ]Γ) ∈ D

BCPL+e

Γ. This proves the

fact.

From the fact above, for every γ ∈ Γ we have that Γ �CPL+e

γ and thereforevΓ(γ) ∈ D

BCPL+e

AΓ

. But since Γ �CPL+e

α, then vΓ(α) �∈ DBCPL+

eAΓ

. Hence, Γ �|=Mat(K+CPL+

e) α.

The previous result can be improved a bit, by considering swap structuresdefined over Boolean algebras instead of classical implicative lattices. The reason to do thisis that, when considering LFIs, a bottom element arises, and so each classical implicativelattice obtained as in the proof of Theorem 4.1.8 becomes a Boolean algebra.

In the sequel, it will be proven that a given classical implicative lattice A canbe extended to a Boolean algebra A∗ such that the non-deterministic matrices induced byboth algebras validate the sames formulas.

Definition 4.1.9. Let A = �A, ∧, ∨, →� be a classical implicative lattice, and let A∗ def= A×{0, 1}. Consider the operations ∧, ∨ and → defined over A∗ as follows, for every a, b ∈ A:

(a, 1)#(b, 1) = (a#b, 1), for # ∈ {∧, ∨, →};

(a, 1) ∧ (b, 0) = (b, 0) ∧ (a, 1) = (a → b, 0);

(a, 0) ∧ (b, 0) = (a ∨ b, 0);

(a, 1) ∨ (b, 0) = (b, 0) ∨ (a, 1) = (b → a, 1);

(a, 0) ∨ (b, 0) = (a ∧ b, 0);

(a, 1) → (b, 0) = (a ∧ b, 0);

(a, 0) → (b, 1) = (a ∨ b, 1);

(a, 0) → (b, 0) = (b → a, 1).

Remark 4.1.10. The intuitive reading of (a, 1) is “a”, moreover (a, 0) denotes its Booleancomplement ∼a.2

Proposition 4.1.11. The structure A∗ = �A∗, ∧, ∨, →, 0∗, 1∗� where the binary operators{∧, ∨, →} are defined as in Definition 4.1.9, is a non-trivial Boolean algebra (that is,1∗ �= 0∗) such that 0∗ def= (1, 0) and 1∗ def= (1, 1).

Proof. Taking into account that, for any a ∈ A, the pairs (a, 1) and (a, 0) can be consideredin A∗ as representing uniquely a and its Boolean complement ∼a, respectively, then we needto prove that �A∗, ∧, ∨� is a distributive lattice. See the proof of one side of distributivity:((a, 1) ∧ (b, 1)) ∨ (c, 1) = (a ∧ b, 1) ∨ (c, 1) = ((a ∧ b) ∨ c, 1). Since A is a classicalimplicative lattice we have ((a ∧ b) ∨ c, 1) = ((a ∨ b) ∧ (a ∨ c), 1) = (a ∨ b, 1) ∧ (a ∨ c, 1) =((a, 1)∨(b, 1))∧((a, 1)∨(c, 1)). And see the proof that ((a → b) → c, 1) = (a → b, 0)∨(c, 1):((a → b) → c, 1) = (∼(∼a ∨ b) ∨ c, 1) = ((a ∧ ∼b) ∨ c, 1) = ((a ∨ c) ∧ (∼b ∨ c), 1) =((a, 1) ∨ (c, 1), 1) ∧ ((b, 0) ∨ (c, 1), 1) = ((a, 1) ∧ (b, 0)) ∨ (c, 1) = (a → b, 0) ∨ (c, 1). Theproof of the other items is quite similar. Now, we have (a, 1) ∧ (1, 0) = (a → 1, 0) = (1, 0)(then 0∗ is the bottom element), (a, 1) ∨ (1, 1) = (a ∨ 1, 1) = (1, 1) (then 1∗ is the topelement), (a, 1) ∨ (a, 0) = (a → a, 1) = (1, 1) and (a, 1) ∧ (a, 0) = (a → a, 0) = (1, 0) (then(a, 0) is the Boolean complement of (a, 1)).

Proposition 4.1.12. Given a classical implicative lattice A, let A∗ as in Proposition 4.1.11.(1) Let i∗ : A → A∗ be the mapping given by i∗(a) = (a, 1), for every a ∈ A. Then i∗ is amonomorphism of classical implicative lattices.2 In this chapter we will use the symbol ∼ instead of ¬ to denote the Boolean complement, different

from what we did in the previous chapter, because the operator ¬ in the LFIs is not classic as in

(2) The pair (A∗, i∗) has the following universal property: if A� is a non-trivial Booleanalgebra and h : A → A� is a homomorphism of classical implicative lattices then thereexists a unique homomorphism of Boolean algebras h∗ : A∗ → A� such that h = h∗ ◦ i∗.That is, the diagram below commutes.

A � � i∗��

h��

A∗

h∗��

A�

Proof. (1) It is immediate from the definition of A∗ that i∗ is a homomorphism of classicalimplicative lattices. Suppose that there are two homomorphisms of classical implicativelattices f : A�� → A and g : A�� → A such that i∗ ◦ f = i∗ ◦ g. By definition, for everya ∈ |A��|, i∗ ◦ f(a) = i∗(f(a)) = (f(a), 1) and i∗ ◦ g(a) = i∗(g(a)) = (g(a), 1). But,(f(a), 1) = (g(a), 1) iff f(a) = g(a), for every a ∈ |A��|. Therefore, f = g.

(2) Let h∗(a, 1) = h(a) and h∗(a, 0) = ∼h(a) for every a ∈ A, where ∼ denotesthe Boolean complement in A�. Clearly, h∗ is a homomorphism of Boolean algebras.Moreover, for every a ∈ A, h∗ ◦ i∗(a) = h∗(i∗(a)) = h∗(a, 1) = h(a), thus h = h∗ ◦ i∗.To prove the uniqueness, suppose that there exist another �h∗ such that h = �h∗ ◦ i∗.We have, for every a ∈ A, h∗(a, 1) = h(a) = �h∗ ◦ i∗(a) = �h∗(i∗(a)) = �h∗(a, 1) andh∗(a, 0) = ∼h(a) = ∼(�h∗◦i∗(a)) = ∼(�h∗(i∗(a))) = ∼(�h∗(a, 1)). Since �h∗ is a homomorphismof Boolean algebras, ∼(�h∗(a, 1)) = �h∗(∼(a, 1)) = �h∗(a, 0), so �h∗ = h∗.

Now, let KCPL+e

be the subclass of K+CPL+

eformed by swap structures B for

CPL+e over a classical implicative lattice A such that A is in fact a Boolean algebra

(that is, A has a bottom element 0). Let Mat(KCPL+e) be the corresponding class of

non-deterministic matrices.

Theorem 4.1.13 (Adequacy of CPL+e w.r.t. swap structures). Let Γ ∪ {α} ⊆ For(Σ) be

a set of formulas of CPL+e . Then: Γ �CPL+

eα iff Γ |=Mat(KCPL+

e) α.

Proof. ‘Only if’ part (Soundness): It follows from Theorem 4.1.8, since KCPL+e

⊆ K+CPL+

e.

‘If’ part (Completeness): Suppose that Γ �CPL+e

ϕ, and consider the classical implicativelattice AΓ defined as in the proof of Theorem 4.1.8. Let (AΓ)∗ be the Boolean algebrainduced by AΓ as in Definition 4.1.9 and Proposition 4.1.11. Let BCPL+

e(AΓ)∗ be the corresponding

swap structure in KCPL+e. Let

�MCPL+

eΓ

�∗ def= M(BCPL+e

(AΓ)∗ ), and consider now a mappingv∗

Γ : For(Σ) → ((AΓ)∗)3 given by v∗Γ(ϕ) = (([ϕ]Γ, 1), ([¬ϕ]Γ, 1), ([◦ϕ]Γ, 1)). Then, it is

easy to see that v∗Γ is a valuation over the non-deterministic matrix (MCPL+

eΓ )∗ such that

v∗Γ(ϕ) ∈ D

BCPL+e

(AΓ)∗iff Γ �CPL+

eϕ, for every ϕ. See the proof below:

For every ϕ, ψ ∈ For(Σ), we have:

• v∗Γ(ϕ # ψ) = (([ϕ # ψ]Γ, 1), ([¬(ϕ # ψ)]Γ, 1), ([◦(ϕ # ψ)]Γ, 1)) and

v∗Γ(ϕ) # v∗

Γ(ψ) = (([ϕ]Γ, 1), ([¬ϕ]Γ, 1), ([◦ϕ]Γ, 1)) # (([ψ]Γ, 1), ([¬ψ]Γ, 1), ([◦ψ]Γ, 1))= {(c1, c2, c3) ∈ ((AΓ)∗)3 : c1 = ([ϕ]Γ, 1) # ([ψ]Γ, 1)}, for # ∈ {∧, ∨, →}. Since([ϕ]Γ, 1) # ([ψ]Γ, 1) def= ([ϕ]Γ # [ψ]Γ, 1) and [ϕ # ψ]Γ def= [ϕ]Γ # [ψ]Γ , for # ∈{∧, ∨, →}, then v∗

Γ(ϕ # ψ) ∈ v∗Γ(ϕ) # v∗

Γ(ψ).

• v∗Γ(¬ϕ) = (([¬ϕ]Γ, 1), ([¬¬ϕ]Γ, 1), ([◦¬ϕ]Γ, 1)) and

¬(v∗Γ(ϕ)) = ¬(([ϕ]Γ, 1), ([¬ϕ]Γ, 1), ([◦ϕ]Γ, 1)) = {(c1, c2, c3) ∈ ((AΓ)∗)3 : c1 =

([¬ϕ]Γ, 1)}. So v∗Γ(¬ϕ) ∈ ¬(v∗

Γ(ϕ)).

• v∗Γ(◦ϕ) = (([◦ϕ]Γ, 1), ([¬ ◦ ϕ]Γ, 1), ([◦ ◦ ϕ]Γ, 1)) and

◦(v∗Γ(ϕ)) = ◦(([ϕ]Γ, 1), ([¬ϕ]Γ, 1), ([◦ϕ]Γ, 1)) = {(c1, c2, c3) ∈ ((AΓ)∗)3 : c1 =

([◦ϕ]Γ, 1)}. So v∗Γ(◦ϕ) ∈ ◦(v∗

Γ(ϕ)).

Thus v∗Γ is a valuation over (MCPL+

eΓ )∗.

Now, suppose that v∗Γ(ϕ) = (([ϕ]Γ, 1), ([¬ϕ]Γ, 1), ([◦ϕ]Γ, 1)) ∈ D

BCPL+e

(AΓ)∗. Then,

([ϕ]Γ, 1) = (1Γ)∗ = (1Γ, 1) and 1Γ = [p1 ∨ ¬p1]Γ. So, by a similar proof to that for the Factincluded in the proof of Theorem 4.1.8, we have Γ �CPL+

eϕ.

If we suppose that Γ �CPL+e

ϕ, again by a similar proof for the Fact mentionedabove, we have [ϕ]Γ = [p1∨¬p1]Γ = 1Γ and therefore, v∗

Γ(ϕ) = (([ϕ]Γ, 1), ([¬ϕ]Γ, 1), ([◦ϕ]Γ, 1))∈ D

BCPL+e

(AΓ)∗.

Hence, v∗Γ(γ) ∈ D

BCPL+e

(AΓ)∗for every γ ∈ Γ, but v∗

Γ(α) �∈ DBCPL+

e(AΓ)∗

. From this,

Γ �|=Mat(KCPL+e

) α.

The full subcategory in MAlg(Σ) of swap structures for CPL+e will be denoted

by SWCPL+e

. That is, the class of objects of SWCPL+e

is KCPL+e

, and the morphisms betweentwo given swap structures are just the homomorphisms between them as multialgebras.

4.2 Swap structures for mbCA special subclass of KCPL+

eis formed by the swap structures for mbC, defined

as follows:

Definition 4.2.1. The universe of swap structures for mbC over a Boolean algebra Awith domain A is the set BmbC

A = {(c1, c2, c3) ∈ A3 : c1 ∨ c2 = 1 and c1 ∧ c2 ∧ c3 = 0}.

Definition 4.2.2. Let A be a Boolean algebra. A swap structure for CPL+e over A is

Let KmbC = {B ∈ KCPL+e

: B is a swap structure for mbC} be the class ofswap structures for mbC. The following is immediate:

Proposition 4.2.3. KmbC = {B ∈ KCPL+e

: |=M(B) (Ax10) ∧ (bc1)}.

Proof. Suppose that KmbC � {B ∈ KCPL+e

: |=M(B) (Ax10) ∧ (bc1)} and let B bethe domain of B. Then there exist a swap structure B ∈ KmbC and a valuation v overM(B) such that v((Ax10) ∧ (bc1)) �∈ DB, that is π1(v((Ax10) ∧ (bc1))) �= 1. But, bydefinition of BmbC

A = {(c1, c2, c3) ∈ A3 : c1 ∨ c2 = 1 and c1 ∧ c2 ∧ c3 = 0} we have thatπ1(v((Ax10) ∧ (bc1))) = 1, see below:

Since v((Ax10) ∧ (bc1)) ∈ v(Ax10) ∧ v(bc1), then π1(v((Ax10) ∧ (bc1))) =π1(v(Ax10))∧π1(v(bc1)). But, π1(v(bc1)) = π1(v(◦α → (α → (¬α → β)))) = π1(v(◦α)) →(π1(v(α)) → (π1(v(¬α)) → π1(v(β)))) = ∼(π1(v(◦α))∧(π1(v(α))∧(π1(v(¬α)))))∨π1(v(β)).However, v(◦α) ∈ ◦v(α) and v(¬α) ∈ ¬v(α), then π1(v(◦α)) = π3(v(α)) and π1(v(¬α)) =π2(v(α)). So, ∼(π1(v(◦α))∧(π1(v(α))∧(π1(v(¬α)))))∨π1(v(β)) = ∼(π3(v(α))∧π1(v(α))∧π2(v(α))) ∨ π1(v(β)) = ∼0 ∨ π1(v(β)) = 1 ∨ π1(v(β)) = 1. Similarly we show thatπ1(v(Ax10)) = 1.

Conversely, let B ∈ KCPL+e

such that |=M(B) (Ax10) ∧ (bc1) and let p, q twodifferent propositional variables. Let c = (c1, c2, c3) ∈ B and d ∈ π1[B]. Consider avaluation v over M(B) such that v(p) = c and π1(v(q)) = d. Then π1(v(p ∨ ¬p)) = 1 andπ1(v(◦p → (p → (¬p → q)))) = 1. That is, c1 ∨ c2 = 1 and (c3 → (c1 → (c2 → d))) = 1.From this, (c1 ∧ c2 ∧ c3) ≤ d, for every d ∈ π1[B]. Since by Proposition 4.1.4, π1[B] is aclassical implicative lattice included in A as subalgebra, then c1 ∧ c2 ∧ c3 = 0. From thisB ∈ KmbC.

Given a Boolean algebra A, let BmbCA be the unique swap structure for mbC

with domain BmbCA such that, for every (a1, a2, a3) and (b1, b2, b3) in BmbC

A :

(i) (a1, a2, a3)#(b1, b2, b3) = {(c1, c2, c3) ∈ BmbCA : c1 = a1#b1}, for # ∈ {∧, ∨, →};

(ii) ¬(a1, a2, a3) = {(c1, c2, c3) ∈ BmbCA : c1 = a2};

(iii) ◦(a1, a2, a3) = {(c1, c2, c3) ∈ BmbCA : c1 = a3}.

The full subcategory in SWCPL+e

of swap structures for mbC will be denotedby SWmbC. Clearly, SWmbC is a full subcategory in MAlg(Σ). Thus, the class of objectsof SWmbC is KmbC, and the morphisms between two given swap structures for mbC arethe homomorphisms between them, seeing as multialgebras over Σ

Let BAlg be the category of Boolean algebras defined over signature ΣBA,with Boolean algebras homomorphisms as their morphisms.

Proposition 4.2.4. Let {Ai}i∈I be a family of Boolean algebras in BAlg such that, forevery i ∈ I, Ai = �Ai, ∧i, ∨i, →i, 0i, 1i�. Let A = �

i∈I Ai be the standard construction ofthe cartesian product of the family of sets {Ai}i∈I with canonical projections πi : A → Ai

for every i ∈ I.3 Let A = �A, ∧, ∨, →, 0, 1� be an algebra such that its operations are givenby:

(i) (a#b)(i) = a(i)#ib(i), for every a, b ∈ A and # ∈ {∧, ∨, →};

(ii) 0A(i) = 0i;

(iii) 1A(i) = 1i.

Then:

(a) A = �A, ∧, ∨, →, 0, 1� is a Boolean algebra;

(b) the canonical projections πi : A → Ai are homomorphisms of Boolean algebras;

(c) �A, {πi}i∈I� is the product of the family {Ai}i∈I in BAlg.

Proof. It is a well-known result that the family of Boolean algebras {Ai}i∈I has productin BAlg and that �A, {πi}i∈I� as described above is its product.

Notation 4.2.5. The Boolean algebra A will be denoted by �i∈I Ai.

Consider again a family F = {Ai}i∈I of Boolean algebras such that I �= ∅,and let A = �

i∈I Ai be its product in BAlg as described above. We want to showthat the product B = �

i∈I BmbCAi

in MAlg(Σ) (recall Proposition 2.5.8) of the familyof multialgebras {BmbC

Ai}i∈I is isomorphic in MAlg(Σ) (recall Proposition 2.5.3) to the

multialgebra BmbCA .

To begin with, some notation is required: For i ∈ I and 1 ≤ j ≤ 3 let πij :

(Ai)3 → Ai be the canonical projections. Observe that, if a ∈ |B| = �i∈I BmbC

Aiand i ∈ I

then a(i) ∈ BmbCAi

⊆ (Ai)3. Thus, for every 1 ≤ j ≤ 3 let zj ∈ �i∈I Ai such that, for every

i ∈ I , zj(i) = πij(a(i)). Then z = (z1, z2, z3) belongs to |A|3. Moreover, it can be proven that

z belongs to BmbCA . Indeed, for every i ∈ I, z1(i) ∨i z2(i) = πi

1(a(i)) ∨i πi2(a(i)) = 1i since

a(i) ∈ BmbCAi

. From this, z1 ∨ z2 = 1A. Analogously it can be proven that z1 ∧ z2 ∧ z3 = 0A.

This allows to define a mapping fF : �i∈I BmbC

Ai→ BmbC�

i∈IAi

such that, for everya ∈ �

i∈I BmbCAi

, fF(a) = z where z = (z1, z2, z3) is defined as above.3 That is, A =

�a ∈

� �i∈I Ai

�I : a(i) ∈ Ai for every i ∈ I�

, if I �= ∅; And A is a singleton otherwise.

Proposition 4.2.6. Let F = {Ai}i∈I be a family of Boolean algebras such that I �= ∅.Then, the mapping fF : �

i∈I BmbCAi

→ BmbC�i∈I

Aiis an isomorphism in MAlg(Σ).

Proof. Clearly fF is a bijective mapping such that its inverse mapping is given by f−1F :

BmbC�i∈I

Ai→ �

i∈I BmbCAi

where f−1F (z1, z2, z3) = a, with a(i) = (z1(i), z2(i), z3(i)) for every

i ∈ I. It is also clear that, for every a, b ∈ �i∈I BmbC

Aiand # ∈ {∧, ∨, →}:

(i) fF [a#b] = fF(a)#fF(b);

(ii) fF [¬a] = ¬fF(a); and

(iii) fF [◦a] = ◦fF(a)

Thus, the result follows from Proposition 2.5.3.

Proposition 4.2.7. The category SWmbC has arbitrary products.

Proof. Let F = {Bi}i∈I be a family of swap structures for mbC, and assume that I �= ∅ (thecase I = ∅ is trivial). By definition of KmbC, for each i ∈ I there is a Boolean algebra Ai suchthat Bi ⊆ BmbC

Ai. Since SWmbC is a subcategory of MAlg(Σ) (where Σ is the signature of

mbC), and the latter has arbitrary products (cf. Proposition 2.5.8), there exists the product�B, {πi}i∈I� of F in MAlg(Σ). By the proof of Proposition 2.5.8, it is possible to define B insuch a way that B ⊆ �

i∈I BmbCAi

, where the multialgebra �i∈I BmbC

Aiis also constructed as

in the proof of Proposition 2.5.8. Let h : B → �i∈I BmbC

Aibe the inclusion homomorphism.

Now, let G = {Ai}i∈I and let fG : �i∈I BmbC

Ai→ BmbC�

i∈IAi

be the isomorphism in MAlg(Σ)of Proposition 4.2.6. Then, the homomorphism fG ◦h : B → BmbC�

i∈IAi

is an injective function

B � � h ����

fG◦h ��

�i∈I BmbC

Ai

fG��

BmbC�i∈I

Ai

and so it induces an isomorphism fG ◦ h in MAlg(Σ) between B and the submultialgebraB� = (fG ◦h)(B) of BmbC�

i∈IAi

, by Proposition 2.5.9. This means that �B�, {πi ◦ (fG ◦ h)−1}i∈I�is another realization of the product of F in MAlg(Σ).

Bπi

��

� � fG◦h �� BmbC�i∈I

Ai

Bi B�(fG◦h)−1

��

��

��

πi◦(fG◦h)−1��

Given that SWmbC is a full subcategory of MAlg(Σ) and by observing that B� is anobject of SWmbC, it follows that �B�, {πi ◦ (fG ◦ h)−1}i∈I� is a construction for the product

The assignment A ∈ BAlg �→ BmbCA ∈ SWmbC is functorial, as it will be

stated in Corollary 4.2.9 below.

Proposition 4.2.8. Let f : A → A� be a homomorphism between Boolean algebras. Thenit induces a homomorphism f∗ : BmbC

A → BmbCA� of multialgebras given by f∗(z1, z2, z3) =

(f(z1), f(z2), f(z3)). Moreover, (f ◦ g)∗ = f∗ ◦ g∗ and (idA)∗ = idBmbCA

, where idA : A → Aand idBmbC

A: BmbC

A → BmbCA are the corresponding identity homomorphisms.

Proof. Given a homomorphism f : A → A� between Boolean algebras, let f∗ : BmbCA →

BmbCA� be the mapping such that f∗(z) = (f(z1), f(z2), f(z3)) for every z = (z1, z2, z3) ∈

BmbCA . If z = (z1, z2, z3), w = (w1, w2, w3) ∈ BmbC

A and # ∈ {∧, ∨, →} then, for everyu = (u1, u2, u3) ∈ (z#w), u1 = z1#w1 and so f(u1) = f(z1)#f(w1). That is, (f∗(u))1 =(f∗(z))1#(f∗(w))1. This means that f∗[z#w] = {f∗(u) : u ∈ (z#w)} ⊆ {u� ∈ BmbC

A� :u�

1 = (f∗(z))1#(f∗(w))1} = f∗(z)#f∗(w).

On the other hand, if z = (z1, z2, z3) ∈ BmbCA and u = (u1, u2, u3) ∈ ¬z then

u1 = z2 whence (f∗(u))1 = f(u1) = f(z2) = (f∗(z))2. This means that f∗(u) ∈ {u� ∈BmbC

A� : u�1 = (f∗(z))2} = ¬f∗(z) and so f∗[¬z] ⊆ ¬f∗(z). Analogously it can be proven

that f∗[◦z] ⊆ ◦f∗(z). This shows that f∗ is indeed a homomorphism f∗ : BmbCA → BmbC

A�

in SWmbC. The rest of the proof is immediate, by the very definition of f∗.

Corollary 4.2.9. There is a functor F : BAlg → SWmbC given by F (A) = BmbCA for

every Bolean algebra A, and F (f) = f∗ for every homomorphism f : A → A� in BAlg.

Proposition 4.2.10. The functor F : BAlg → SWmbC preserves arbitrary products.

Proof. It is an immediate consequence of Proposition 4.2.6 and the fact that SWmbC is afull subcategory of MAlg(Σ).

Proposition 4.2.11. The functor F : BAlg → SWmbC preserves subalgebras in thefollowing sense: if A is a subalgebra of A� in BAlg then BmbC

A ⊆ BmbCA� according to

Definition 2.2.1.

Proof. Let A and A� be two Boolean algebras in BAlg such that A is subalgebra of A� andlet f : A → A� be the inclusion morphism in BAlg given by f(a) = a for every a ∈ |A|.By Proposition 4.2.8, f induces a homomorphism f∗ : BmbC

A → BmbCA� of multialgebras

given by f∗(z1, z2, z3) = (f(z1), f(z2), f(z3)) = (z1, z2, z3). By definition of |BmbCA | and

|BmbCA� | and since |A| ⊆ |A�| we have |BmbC

A | ⊆ |BmbCA� |. Moreover, BmbC

A ⊆ BmbCA� .

Proposition 4.2.12. The functor F : BAlg → SWmbC preserves monomorphisms.

Proof. Let f : A → A� be a monomorphism between Boolean algebras, and let f∗ :BmbC

A → BmbCA� be the induced homomorphism of multialgebras given by f∗(z1, z2, z3) =

(f(z1), f(z2), f(z3)). It is well-known that every monomorphism in BAlg is an injectivefunction, and then f is injective. From this it is immediate to see that f∗ is also an injectivefunction. As a consequence of Proposition 2.5.4, f∗ is a monomorphism in the categoryMAlg(Σ). Given that SWmbC is a full subcategory of MAlg(Σ), it follows that f∗ is amonomorphism in the category SWmbC.

As it was done in Definition 4.1.5, each B ∈ KmbC induces naturally a non-deterministic matrix M(B) = (B, DB). Moreover, in (CARNIELLI; CONIGLIO, 2016) itwas proven that the class Mat(KmbC) = {M(B) : B ∈ KmbC} semantically characterizesmbC:

Theorem 4.2.13. (CARNIELLI; CONIGLIO, 2016, Theorem 6.4.8) Let Γ∪{α} ⊆ For(Σ)be a set of formulas of mbC. Then: Γ �mbC α iff Γ |=Mat(KmbC) α.

The non-deterministic matrix MmbC5 induced by the swap structure BmbC

A2

defined over the two-element Boolean algebra A2 was originally introduced by A. Avronin (AVRON, 2005)4 in order to semantically characterize the logic mbC . The domain of themultialgebra BmbC

A2 is the set BmbCA2 =

�t, I, tI , f, fI

�such that t = (1, 0, 1), I = (1, 1, 0),

tI = (1, 0, 0), f = (0, 1, 1), and fI = (0, 1, 0). Let D be the set of designated elements of thenon-deterministic matrix MmbC

5 = M�BmbC

A2

�. Then, D = {t, I, tI}. Let ND =

�f, fI

�

be the set of non-designated truth-values. The multioperations proposed by Avron over theset BmbC

A2 (see the Example 2.1.6) corresponds exactly with that for BmbCA2 described after

Proposition 4.2.3. It was proved in (AVRON, 2005) that mbC is adequate for MmbC5 :

Theorem 4.2.14. (AVRON, 2005, Theorem 3.6) For every set of formulas Γ ∪ {α} ⊆For(Σ): Γ �mbC α iff Γ |=MmbC

5α.

As observed in (CARNIELLI; CONIGLIO, 2016, Chapter 6), Avron’s resultmeans that the non-deterministic matrix induced by the swap structure BmbC

A2 definedover the two-element Boolean algebra A2 is sufficient for characterizing the logic mbC,and so it represents, in a certain way, the whole class KmbC of swap structures for mbC.One interesting question is to prove that the 5-element multialgebra BmbC

A2 generates (insome sense to be determined) the class KmbC, in analogy to the fact that the 2-elementBoolean algebra A2 generates the class of Boolean algebras.4 In (AVRON, 2005) Avron denoted the system mbC by B, so the non-deterministic matrix for mbC is

Recall that the power set ℘(I) of a given set I is a Boolean algebra, wherethe operations are the usual set-theoretic ones. A field of sets is any subalgebra of apower set Boolean algebra ℘(I). Birkhoff proves in 1935 (see (BIRKHOFF, 1935)) thatevery Boolean algebra A is isomorphic to a field of sets. Taking into account that ℘(I) isisomorphic, as a Boolean algebra, to the product �

i∈I A2 of Boolean algebras, Birkhoff’sresult is equivalent to the following: for every Boolean algebra A, there exists a set I anda monomorphism of Boolean algebras h : A → �

i∈I A2.

From Birkhoff’s representation theorem for Boolean algebras, and taking intoaccount the properties of the functor F : BAlg → SWmbC, a representation theorem forthe class KmbC of swap structures for mbC can be obtained:

Theorem 4.2.15 (Representation Theorem for KmbC). Let B be a swap structure formbC. Then, there exists a set I and a monomorphism of multialgebras h : B → �

i∈I BmbCA2 .

Proof. Let B be a swap structure for mbC. Then, there is a Boolean algebra A such thatB ⊆ BmbC

A . Let g : B → BmbCA be the inclusion monomorphism in SWmbC. Using Birkhoff’s

representation theorem for Boolean algebras5, there exists a set I and a monomorphismh : A → �

i∈I A�i of Boolean algebras, where A�

i = A2, for every i ∈ I . By Proposition 4.2.12,there is a monomorphism h∗ : BmbC

A → BmbC�i∈I

A�i. Let fG : �

i∈I BmbCA�

i→ BmbC�

i∈IA�

ibe the

isomorphism in MAlg(Σ) of Proposition 4.2.6, where G = {A�i}i∈I . By definition of A�

i itfollows that BmbC

A�i

= BmbCA2 , for every i ∈ I . Then h : B → �

i∈I BmbCA2 is a monomorphism

in MAlg(Σ), where h = f−1G ◦ h∗ ◦ g.

From the previous result, it is natural to ask about the possibility of theclass KmbC being a variety of multialgebras, that is, a class closed under products,submultialgebras and homomorphic images. We known that KmbC is closed under products(by Proposition 4.2.7) and submultialgebras (by the very definitions). Unfortunately, theclass is not closed under homomorphic images:

Proposition 4.2.16. The class KmbC of multialgebras is closed under submultialgebrasand (direct) products, but it is not closed under homomorphic images.

Proof. Recall the notions of multicongruence (Definition 2.4.1), quotient multialgebra(Definition 2.4.5) and the canonical map p : A → A/Θ for every multicongruence Θ(Proposition 2.4.8). Now, let D = {z1, z2, z3} and ND =

�z4, z5

�be an enumeration of

the elements of the domain BmbCA2 = D ∪ ND of the multialgebra BmbC

A2 . Let Θ be theequivalence relation asociated to the partition {a, b} of BmbC

A2 such that a = {z1, z4} andb = {z2, z3, z5}. The relation Θ has the following property: for every z ∈ D there exists somew ∈ ND such that (z, w) ∈ Θ, and vice versa. From this, and by observing the definition of5 In (DUNN; HARDEGREE 2001 Theorem 8.11.7, p. 307).

the multioperations in the multialgebra BmbCA2 , it follows that Θ is a multicongruence over

BmbCA2 . It is easy to prove that the multioperations in the quotient multialgebra BmbC

A2 /Θ

are trivial, that is: for every x, y ∈ {a, b} and # ∈ {∧, ∨, →}, (x#y) = ¬x = ◦x = {a, b}.Clearly BmbC

A2 /Θ is not a swap structure for mbC: otherwise, it would generate a trivialnon-deterministic matrix where the set of designated values is the whole domain. Thiswould contradict (CARNIELLI; CONIGLIO, 2016, Proposition 6.4.5(ii)), where it wasproven that no non-deterministic matrix in the class Mat(KmbC) is trivial. This showsthat BmbC

A2 /Θ, the homomorphic image of the canonical map p : BmbCA2 → BmbC

A2 /Θ, doesnot belong to the class KmbC, despite its domain BmbC

A2 is in KmbC.

From this last result, a question that arises is: Why don’t we change/adapt thehomomorphic images or the congruence definitions in order to show that KmbC is a varietyof multialgebras? The problem is that, if we do this, we will lose some important resultssuch as the method of completeness that we apply in the main theorems of this Thesis.

4.3 Swap structures for some extensions of mbCIn (CARNIELLI; CONIGLIO, 2016, Chapter 6) the concept of swap structure

for mbC was generalized to some axiomatic extensions of mbC. As it was done in theSection 4.2, these structures will be reintroduced here in a slightly modified form, moresuitable for an algebraic study of them.

Definition 4.3.1. (CARNIELLI; CONIGLIO, 2016, Definition 3.1.1) The logic mbCciwis obtained from mbC by adding the axiom schema

◦α ∨ (α ∧ ¬α) (ciw)

Definition 4.3.2. Let A be a Boolean algebra. The universe of swap structures formbCciw over A is the set Bciw

A = {(c1, c2, c3) ∈ BmbCA : c3 ∨ (c1 ∧ c2) = 1}.

Clearly, BciwA = {(c1, c2, c3) ∈ A3 : c1 ∨ c2 = 1 and c3 = ∼(c1 ∧ c2)}.6

Definition 4.3.3. Let A be a Boolean algebra. A swap structure for mbC over A is saidto be a swap structure for mbCciw over A if its domain is included in Bciw

A .

Let KmbCciw = {B ∈ KmbC : B is a swap structure for mbCciw} be theclass of swap structures for mbCciw. So:

Proposition 4.3.4. The following holds:

KmbCciw = {B ∈ KmbC : |=M(B) (ciw)}

= {B ∈ KCPL+e

: |=M(B) (Ax10) ∧ (bc1) ∧ (ciw)}.

Proof. The proof is very similar to that of Proposition 4.2.3.

For every Boolean algebra A there is a unique swap structure BmbCciwA for

mbCciw with domain BciwA such that, for every a = (a1, a2, a3) and b = (b1, b2, b3) in Bciw

A :

(i) (a1, a2, a3)#(b1, b2, b3) = {(c1, c2, c3) ∈ BciwA : c1 = a1#b1}, for # ∈ {∧, ∨, →};

(ii) ¬(a1, a2, a3) = {(c1, c2, c3) ∈ BciwA : c1 = a2};

(iii) ◦(a1, a2, a3) = {(c1, c2, c3) ∈ BciwA : c1 = a3}.

The full subcategory in SWCPL+e

of swap structures for mbCciw will bedenoted by SWmbCciw. By the very definitions, SWmbCciw is a full subcategory inSWmbC, and a full subcategory in MAlg(Σ). Hence, the class of objects of SWmbCciw

is KmbCciw, and the morphisms between two given swap structures for mbCciw are justthe homomorphisms between them as multialgebras over Σ.

The class Mat(KmbCciw) of non-deterministic matrices associated to swapstructures for mbCciw is defined analogously to the class Mat(KCPL+

e) introduced in

Definition 4.1.5.

Theorem 4.3.5. (CARNIELLI; CONIGLIO, 2016, Theorem 6.5.4) Let Γ ∪ {α} ⊆ For(Σ)be a set of formulas. Then: Γ �mbCciw α iff Γ |=Mat(KmbCciw) α.

Now, stronger extensions of mbC will be analized:

Definition 4.3.6. Consider the following extensions of mbC:

(1) The logic mbCci (CARNIELLI; CONIGLIO, 2016, Definition 3.1.7) is obtained frommbC by adding the axiom schema:

¬◦α → (α ∧ ¬α) (ci)

(2) The logic CPLe is obtained from mbC by adding the axiom schema:

◦α (cons)

Proposition 4.3.7. The following holds:(1) The logic mbCci properly extends mbCciw.(2) The logic CPLe is an expansion of CPL by a connective ◦ such that ◦α is a validschema. Thus, it properly extends mbCci, and it is semantically characterized by the usual2-valued truth-tables for CPL plus the operator ◦(x) = 1 for every x ∈ {0, 1}.

Proof. (1) See (CARNIELLI; CONIGLIO, 2016, Proposition 3.1.10).(2) Observe that, by (cons), (bc1) and MP, the negation ¬ is explosive in CPLe and so itcoincides with the classical negation. Since CPL+ is included in CPLe then, by axiom(Ax10), this logic is nothing more than an expansion of CPL by adding as theorems allthe formulas of the form ◦α.

Definition 4.3.8. A swap structure for mbCci is any B ∈ KmbCciw such that:

◦(a1, a2, a3) def= {(∼(a1 ∧ a2), a1 ∧ a2, 1)}.

The class of swap structures for mbCci will be denoted by KmbCci.

The class Mat(KmbCci) of non-deterministic matrices is defined analogously tothe class Mat(KCPL+

e) introduced in Definition 4.1.5.

Theorem 4.3.9. (CARNIELLI; CONIGLIO, 2016, Theorem 6.5.11) Let Γ∪{α} ⊆ For(Σ)be a set of formulas. Then: Γ �mbCci α iff Γ |=Mat(KmbCci) α.

Proposition 4.3.10. The following holds:

KmbCci = {B ∈ KmbCciw : |=M(B) (ci)}

= {B ∈ KmbC : |=M(B) (ci)}

= {B ∈ KCPL+e

: |=M(B) (Ax10) ∧ (bc1) ∧ (ci)}.

Proof. The proof is very similar to that of Proposition 4.2.3.

Definition 4.3.11. Let A be a Boolean algebra with domain A. The universe of swapstructures for CPLe over A is the set:

BCPLeA = {(c1, c2, c3) ∈ Bciw

A : c2 = ∼c1} = {(a, ∼a, 1) : a ∈ A} � A.

Definition 4.3.12. A swap structure for CPLe is any B ∈ KmbCci such that |B| ⊆ BCPLeA .

The class of swap structures for CPLe will be denoted by KCPLe .

The class Mat(KCPLe) of non-deterministic matrices is defined analogously tothe class Mat(KCPL+

e) introduced in Definition 4.1.5.

Proposition 4.3.13. The following holds:

KCPLe = {B ∈ KmbCci : |=M(B) (cons)}

= {B ∈ KmbC : |=M(B) (cons)}

= {B ∈ KCPL+e

: |=M(B) (Ax10) ∧ (bc1) ∧ (cons)}.

For every Boolean algebra A the swap structure BmbCciA for mbCci and BCPLe

Afor CPLe are defined as expected. The full subcategory in SWCPL+

eof swap structures for

mbCci and for CPLe will be denoted by SWmbCci and SWCPLe , respectively. By thevery definitions, they are full subcategories in SWmbC, and full subcategories in MAlg(Σ).

Remark 4.3.14.(1) If B ∈ KCPLe then B can be seen as a Boolean algebra isomorphic to the Booleanalgebra π1[|B|].(2) Observe that

KCPLe ⊂ KmbCci ⊂ KmbCciw ⊂ KmbC ⊂ KCPL+e

whileCPLe ⊃ mbCci ⊃ mbCciw ⊃ mbC ⊃ CPL+

e .

As analyzed in (CARNIELLI; CONIGLIO, 2016, Chapter 6), the logic mbCciwcan be characterized by a single 3-valued non-deterministic matrix, by considering the two-element Boolean algebra A2. Indeed the non-deterministic matrix MmbCciw

3 induced by theswap structure BmbCciw

A2 , that is MmbCciw3 = M

�BmbCciw

A2

�, was originally considered by

A. Avron in (AVRON, 2005) obtaining so a semantical characterization of mbCciw. Thedomain of the multialgebra BmbCciw

A2 is the set BmbCciwA2 =

�t, I, f

�such that t = (1, 0, 1),

I = (1, 1, 0) and f = (0, 1, 1), where D3 = {t, I} is the set of designated elements of thenon-deterministic matrix MmbCciw

3 . The multioperations are defined as follows:

∧ t I f

t {t, I} {t, I} {F}I {t, I} {t, I} {F}f {F} {F} {F}

∨ t I f

t {t, I} {t, I} {t, I}I {t, I} {t, I} {t, I}f {t, I} {t, I} {F}

→ t I f

t {t, I} {t, I} {f}I {t, I} {t, I} {f}f {t, I} {t, I} {t, I}

¬t {f}I {t, I}f {t, I}

◦t {t, I}I {f}f {t, I}

It is clear that BmbCciwA2 is a submultialgebra of BmbC

A2 . Moreover, by an analysissimilar to the one presented above, it is possible to prove representation theorem forKmbCciw analogous to that for KmbC (recall Theorem 4.2.15). In order to do this, considerthe following lemma:

Lemma 4.3.15. Let an assignment FmbCciw : BAlg → SWmbCciw, with FmbCciw(A) =BmbCciw

A , and FmbCciw(f) = f∗ for every morphism f : A → A� in BAlg, where f∗ :BmbCciw

A → BmbCciwA� and f∗(z) = (f(z1), f(z2), f(z3)), for every z ∈ BmbCciw

A . ThenFmbCciw is a functor which preserves monomorphisms and arbitrary products.

Proof. From BmbCciwA ⊆ BmbC

A (as submultialgebra) and by proof of the Proposition 4.2.8(in the case of F : BAlg → SWmbC), it is easy to see that f∗ is a morphism in SWmbCciw.So, FmbCciw is a functor.

By Proposition 4.2.6 and the fact that SWmbCciw is a full subcategory ofSWmbC that is a full subcategory of MAlg(Σ), then the functor FmbCciw : BAlg →SWmbCciw preserves arbitrary products.

Let f : A → A� be a monomorphism in BAlg. It is well-known that everymonomorphism in BAlg is an injective function, and then f is injective. From this itis immediate to see that f∗ is also an injective function. As a consequence of Proposi-tion 2.5.4, f∗ is a monomorphism in the category MAlg(Σ). Given that SWmbCciw is afull subcategory of SWmbC that is a full subcategory of MAlg(Σ), it follows that f∗ is amonomorphism in SWmbCciw. So, the functor FmbCciw : BAlg → SWmbCciw preservesmonomorphisms.

Theorem 4.3.16 (Representation Theorem for KmbCciw). Let B be a swap structurefor mbCciw. Then, there exists a set I and a monomorphism of multialgebras h : B →�

i∈I BmbCciwA2 .

Proof. Let B be a swap structure for mbCciw. Then, there is a Boolean algebra A suchthat B ⊆ BmbCciw

A . Let g : B → BmbCciwA be the inclusion monomorphism in SWmbCciw.

Using Birkhoff’s representation theorem for Boolean algebras7, there exists a set I and amonomorphism h : A → �

i∈I A�i of Boolean algebras, where A�

i = A2, for every i ∈ I.

Consider the functor FmbCciw : BAlg → SWmbCciw such that FmbCciw(h) =h∗ which preserves monomorphisms (by Lemma 4.3.15). So, there is a monomorphismh∗ : BmbCciw

A → BmbCciw�i∈I

A�i.

Recall that SWmbCciw is a full subcategory in SWmbC and SWmbC is afull subcategory in MAlg(Σ), and we have that fG : �

i∈I BmbCciwA�

i→ BmbCciw�

i∈IA�

iis an

isomorphism in MAlg(Σ), where G = {A�i}i∈I (by Proposition 4.2.6). By definition of

A�i it follows that BmbCciw

A�i

= BmbCciwA2 , for every i ∈ I. Then h : B → �

i∈I BmbCciwA2 is a

monomorphism in MAlg(Σ), where h = f−1G ◦ h∗ ◦ g.

7 In (DUNN; HARDEGREE, 2001, Theorem 8.11.7, p. 307).

Concerning mbCci and CPLe, similar results can be obtained. Indeed, A.Avron has proven in (AVRON, 2005) that mbCci can be characterized by a single 3-valuednon-deterministic matrix MmbCci

3 .

In (CARNIELLI; CONIGLIO, 2016, Chapter 6) was proved that MmbCci3 is

the one obtained by the 3-valued swap structure BmbCciA2 , that is MmbCci

3 = M�BmbCci

A2

�,

with the same domain and multioperations than BmbCciwA2 , but now the multioperator ◦ is

single-valued, and it is defined as follows:

◦t {t}I {f}f {t}

Clearly, BmbCciA2 is a submultialgebra of BmbCciw

A2 and so of BmbCA2 . Moreover:

Theorem 4.3.17 (Representation Theorem for KmbCci). Let B be a swap structure formbCci. Then, there exists a set I and a monomorphism of multialgebras h : B →�

i∈I BmbCciA2 .

Proof. It is enough to observe that, let f : A → A� be a monomorphism in BAlgand let FmbCci : BAlg → SWmbCci such that FmbCci(f) = f∗ be a functor thatpreserves monomorphisms (by similar proof of the Lemma 4.3.15). If a ∈ BmbCci

A ,then ◦a

def= {(∼(a1 ∧ a2), a1 ∧ a2, 1)}. So, f∗[◦a] = {f∗(∼(a1 ∧ a2), a1 ∧ a2, 1)} ={(f(∼(a1 ∧ a2)), f(a1 ∧ a2), f(1))} = {(∼(f(a1) ∧ f(a2)), (f(a1) ∧ f(a2)), f(1))} = ◦f∗(a)and therefore f∗ is a monomorphism in SWmbCci. The remainder of this proof is similarto proof of Theorem 4.3.16.

Finally, the case of CPLe is quite simple. The swap structure BCPLeA2 has

domain {t, f} where t = (1, 0, 1) and f = (0, 1, 1). The multiperations are single-valued,producing a Boolean algebra isomorphic to A2.

Clearly BCPLeA2 ⊆ BmbCci

A2 ⊆ BmbCciwA2 ⊆ BmbC

A2 ⊆ BCPL+e

A2 . Additionally:

Theorem 4.3.18 (Representation Theorem for KCPLe). Let B be a swap structure forCPLe. Then, there exists a set I and a monomorphism of multialgebras h : B →�

i∈I BCPLeA2 .

Proof. It is similar to proof of Theorem 4.3.16.

The last theorem is just the original Birkhoff’s theorem for Boolean algebraspublished under differen

Final Considerations

This Thesis proposes a general study of non-deterministic matrices appliedto logic systems from the perspective of Universal algebra and category theory. Thenon-deterministic matrices differ from usual matrices (deterministic) by the use of whatis called today “multioperations”. That is, operations which assign, to some element ofthe domain, a non-empty subset of that domain. When this Thesis was conceived whilea research project, we aimed to develop a formal theory for non-deterministic matricessemantics in order to better understand the scope of the original proposal by Avron andhis collaborators. The main goal of the research was to propose an alternative way foralgebraization of logic systems, in order to deal with logics in which the usual algebraizationmethods cannot (or it is very hard to) be applied. Starting from this, the concept of“non-deterministic algebra” (or “ND-algebra”)8 arose naturally. However, soon we realizethat this notion was already proposed and intensively studied in the literature, underdifferent names: “hyperalgebras”, “multialgebras” and “non-deterministic algebras”, amongothers.

Since non-deterministic algebras constitute a generalization of standard algebras,it is natural that the generalization of the usual concepts for this framework is not unique.Being so, at least three different definitions of multialgebra were proposed, and thissituation is similar for more specific concepts, such as homomorphism, for which at leastfive distinct definitions are available. Thus, after adapting and organizing the diversenomenclature for these concepts, we present them in Chapter 2. Since there are manypossibilities for each concept, the choice of the “right” notion in each case was motivatedby purely pragmatical reasons: the notions better suited to our proposals (namely, itsapplication to the study of logic systems) were adopted.

It is important to note that most of the authors have studied and developedthe theory of hyperstructures independently, which could justify the different names forsimilar concepts. From this point of view, the historical analysis obtained in Chapter 1 wasextremely useful for us. In fact, this historical research leads us to the discovery of someinteresting facts: for instance, that Marty in 1935 (MARTY, 1935) already introduced adefinition of homomorphism between hyperstructures and so, probably he was the first topresent this concept. We also discovered that the Brazilian logician and mathematicianAntonio Antunes Mario Sette (from the Centre for Logic, Epistemology and the History ofScience –CLE at the University of Campinas – UNICAMP) already used in his Master’sthesis from 1971 the concept of hyperlattice under the name of “reticuloide”. We also found8 By analogy with the terms “non-deterministic matrix” (or ND-matrix) and “Nmatrix” coined by Avron

that the concept of non-deterministic matrices was applied by other authors much beforethe works of Avron and his colaborators (AVRON; LEV, 2001). This study provided severalpossibilities for several future developments in the study of multialgebras applied to Logic.For instance, the concepts presented in Chapter 2 enabled the development in Chapter 4of an incipient algebraic theory of swap structures (a particular class of multialgebrasintroduced by Carnielli and Coniglio in (CARNIELLI; CONIGLIO, 2016, Chapter 6)),by adapting concepts of universal algebra to multialgebras in a suitable way. Althoughwe found in the literature a large amount of theory and research about multialgebras, itsapplication to Logic and especially to the algebraization of logic systems was not enoughexplored, as far as we know.

In Chapter 3 we introduced an original class of swap structures as a suitablesemantics for a family of non-normal modal systems. This is the first example of swapstructures defined for logics outside the scope of Logics of Formal Inconsistency (LFIs),the class of paraconsistent logics for which swap structures were originally proposed. Basedon the ideas of the completeness proofs of LFIs with respect to Fidel structures semanticsfound in (CARNIELLI; CONIGLIO, 2016, Chapter 6), a quotient swap structure that wecalled of Lindenbaum-Tarski swap structure, as well as a canonical valuation over it, wereproposed.

After defining a suitable notion of swap structures, as well as the class ofnon-deterministic matrices naturally associated to them, it was possible to prove thesoundness theorems of the Hilbert calculi defining these non-normal modal systems withrespect to such Nmatrix semantics. In order to obtain the completeness theorem, themethod of Lindenbaum-Tarski swap structures mentioned above was employed.

The notion of Lindenbaum-Tarski swap structures proposed here generalizes ina quite natural way the classical Lindenbaum-Tarski method, allowing to deal with logicswhich are not algebraizable in the usual sense. We conjecture that this technique couldbe applied to a wide class of non-algebraizable logics (even by the general techniques ofBlok-Pigozzi), allowing an interesting and new paradigm for algebraizing logics by meansof multialgebras.

The Lindenbaum-Tarski swap structures method was also applied in Chapter 4in order to obtain completeness theorem for the system CPL+

e (see Theorems 4.1.8and 4.1.13). Its application to all the LFIs studied there should be immediate (observe thatthe adequacy of such LFIs with respect to swap structures semantics was already statedin (CARNIELLI; CONIGLIO, 2016, Chapter 6), by proving its equivalence with the Fidelstructures semantics). Besides this, in Chapter 4 other important results were obtained,concerning the algebraic and categorial properties of several categories of swap structures(seen as multialgebras), including representation theorems analogous to Birkhoff’s theoremfor

Finally, some suggestions for future work that arise naturally from what hasbeen developed in the present Thesis will be presented:

• In this Thesis, the method of Lindenbaum-Tarski swap structures was applied to twodistinct family of logic systems (namely, non-normal modal logics and LFIs). Theyhave in common the existence of at least one non-congruential operator, justifyingso this non-deterministic algebraic approach. We believe that the same methodcan be applied to other logical systems, with similar conditions (that is, where theapplication of the usual algebraic methods is hard or even impossible).

• The behavior of the logics presented in Chapter 3 and the definitions of swapstructures and of Lindenbaum-Tarski swap structures for them suggests that itis possible to obtain algebraic and categorial results, similar to those obtained inChapter 4 for the family of LFIs, to the family of non-normal modal systemspresented in Chapter 3 of this thesis. That is, a modular treatment of the algebraictheory of swap structures for these modal logics could be obtained in this way.

• In the paper (SCHWEIGERT, 1985) by Schweigert the proof of the Birkhoff’stheorem for multialgebras9 is clearly incomplete. Indeed, in that paper the authordoes not specify the basic definitions that are being adopted in order to obtain themain result. Given that, as discussed above (and in Chapter 2 of this Thesis), eachconcept from ordinary algebra can be generalized in several ways to the realm ofmultialgebras, this is not a minor issue. By its turn, Hansoul (HANSOUL, 1983)presented a detailed proof of a version of the Birkhoff’s theorem for multialgebras.However, the definitions adopted there are too restrictive for our purpose, namely,its application to Logic. For instance, the author assumes that any multioperationmust return a finite set of possible values for each argument. In view of this, it wouldbe interesting to investigate the validity of the Birkhoff’s theorem for multialgebrasusing the definitions presented in Chapter 2 of this thesis.

• Some results have been obtained for the category of multialgebras proposed inChapter 2. In Chapter 1 we presented some other general proposals and, in Chapter 4,we developed theoretic studies applied to some LFIs. We can still develop an algebraicand categorial study of swap structures (and other categories of multialgebrasassociated to logic systems) by changing the definition of homomorphism, similar towhat Nolan did in (NOLAN, 1979).

As mentioned above, it is important to observe that the theory of multialgebraswas not explored from the point of view of its logical application. In this sense, the present9 Every multialgebra can be represented as a sub-direct product of sub-directly irreducible multialgebras.

research could be considered as pioneering, and the results presented in chapters 3 and 4speak by themselves. Thus, this field has several open possibilities. This promises to be afruitful ground for new researches.

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