Completions of Basic Algebras

Majid Alizadeh�

School of Mathematics, Statistics and Computer Science, College of Science,University of Tehran, P. O. Box 14155-6455, Tehran, Iran

Research center for Integrated Science, Japan Advanced Institute of Science andTechnology, Asahidai, Nomi, Ishikawa, 923-1292, Japan

Abstract. We discuss completions of basic algebras. We prove that theideal completion of a basic algebra is also a basic algebra. It will beshown that basic algebras are not closed under MacNeille completions.By adding the join-infinite distributive law to basic algebras, we willshow that these kind of basic algebras are closed under the closed idealcompletion and moreover any other regular completions of these alge-bras are isomorphic to the closed ideal completion. As an applicationwe establish an algebraic completeness theorem for a logic weaker thanVisser’s basic predicate logic, BQL, a proper subsystem of intuitionisticpredicate logic, IQL.

Keywords: Heyting algebra, Basic algebra, Completion, Visser’s basiclogic, Intuitionistic logic.

1 Introduction

In the present paper, we will discuss completions of algebras for Visser’s basiclogic and a completeness theorem for it which is obtained using completions.Completions of basic algebras and some of its subvarieties are discussed in thesecond and third sections. In the second section, we will first give a brief overviewof the canonical extensions of basic algebras and then we will introduce the idealcompletion of a basic algebra. We will see that the ideal completion is alsoa Heyting algebra and consequently we can embed every basic algebra into acomplete Heyting algebra as a lattice. None of these completions of basic algebrasare regular, i.e., the embedding of a basic algebra into these two kinds of completealgebras does not preserve existing infinite joins and meets in general. In thethird section, we first show that the variety of basic algebras, a subvariety of thisvariety called Lob algebras and all the subvarieties of the latter are not closedunder MacNeille completions. Then, we introduce the closed ideal completionof a basic algebra satisfying the join-infinite distributive law and show that the� I would like to thank Prof. Mohammad Ardeshir for his valuable comments. This

paper was completed during my stay at Japan Advanced Institute of Science andTechnology, JAIST, as a visiting researcher. I am grateful to Prof. Hiroakira Ono forhis invitation to my attendance at JAIST also for reading early draft of the paperand giving helpful comments and suggestions.

H. Ono, M. Kanazawa, and R. de Queiroz (Eds.): WoLLIC 2009, LNAI 5514, pp. 72–83, 2009.c© Springer-Verlag Berlin Heidelberg 2009

Completions of Basic Algebras 73

class of basic algebras which satisfies the join-infinite distributive law is closedunder this completion and moreover any other regular completions of these kindsof algebras are isomorphic to the closed ideal completion. In the last section,we apply the regular completion of basic algebras that satisfy the join-infinitedistributive law to prove an algebraic completeness theorem for a logic weakerthan Visser’s basic predicate logic.

2 Canonical Completion and Ideal Completion

In this section first we give a brief overview of canonical extension of basicalgebras and then we will introduce the ideal completion of these algebras.

Definition 1. A basic algebra B = 〈B,∧,∨,→, 0, 1〉 is a structure with con-stants 0 and 1, and binary functions ∧, ∨, and →, such that

1. with respect to 0, 1, ∧, and ∨ we have a bounded distributive lattice, and2. for → we have the additional identities and quasi-identities

a→ b ∧ c = (a→ b) ∧ (a→ c);

b ∨ c→ a = (b→ a) ∧ (c→ a);

a→ a = 1;

a ≤ 1 → a; and

(a→ b) ∧ (b→ c) ≤ a→ c.

The relation ≤ is expressible in term of equations with ∧ or ∨ in the standardway, i.e.,

a ≤ b iff a ∧ b = a iff a ∨ b = b.

So this class of algebras forms a variety. A basic algebra B is called a Lob algebraiff for all x ∈ B, (1 → x) → x = 1 → x.

Lemma 1. [1]. Let B be a basic algebra. Then for a, b,∈ B,

1. if a ≤ b, then a ∧ (b→ c) = a ∧ (1 → c),2. if a ≤ b, then b→ c ≤ a→ c, and c→ a ≤ c→ b,3. if a ∧ b ≤ c, then a ≤ b→ c.

A basic algebra B is called complete if it is complete as a lattice. A variety ofbasic algebras is closed under a completion method if the completion of everyalgebra in the variety according to the method belongs to the variety too.

Canonical extensions of basic algebras, introduced by Ardeshir [4], are brieflydescribed as follows. For a basic algebra B, let W = WB be the set of all primefilters of B. Define a binary relation ≺ on W as follows: F ≺ F

′iff b ∈ F

′

whenever a → b ∈ F and a ∈ F′. A subset X of W is called an upset of W if

and only if for each F,G ∈ W such that F ∈ X and F ⊆ G, we have G ∈ X .Then the structure 〈UP (W ),∩,∪,→, ∅,W 〉 of upsets, UP (W ), of W is a basicalgebra, where

74 M. Alizadeh

X → Y = {F ∈ W : ∀G � F (if G ∈ X then G ∈ Y )}.One can embed the original algebra B in its canonical extension by a mapping

f defined by

f(a) = {P | P is a prime filter and a ∈ P}, for each a ∈ B.

It is well-known that the above canonical embedding works well for distribu-tive lattices but it does not preserve existing infinite joins and meets in general.In the rest of this section we introduce the ideal completion of basic algebras.First, let us recall some known definitions.

A subset I ⊂ B is an ideal on B if 0 ∈ I; if a, b ∈ I, then a∨b ∈ I and if a ≤ band b ∈ I, then a ∈ I. An ideal I on B is called prime, if a ∧ b ∈ I implies thata ∈ I or b ∈ I. As usual we use the symbol (a] for the ideal generated by theelement a of the basic algebra B, i.e., (a] = {x ∈ B : x ≤ a}. For S ⊆ B, putI(S) = (S] = {x ∈ B : x ≤ (a1 ∨ ... ∨ an) for some elements a1, ..., an in S}.It is easy to show that (S] is the minimal ideal containing S. For ideals I and Jin B we define I ∧ J = I ∩ J . For a set Γ of ideals of B, ∨Γ is defined by (∪Γ ],in particular I ∨ J = (I ∪ J ].

For a basic algebra B the set I(B) of all ideals of B forms a complete dis-tributive lattice with respect to ∧ and ∨. We call it the ideal completion of B.

Theorem 1. Let B be a basic algebra. Then I(B) = 〈I(B),∧,∨,→∗, (0], (1]〉where, for I, J ∈ I(B),

I →∗ J = {x ∈ B : for every i in I there is a j in J such that x ≤ i→ j}is a basic algebra.

Proof. First, we show that I →∗ J is an ideal, for every ideal I and J . Easily onecan see that 0 ∈ I →∗ J also if x ≤ y and y ∈ I →∗ J , then x ∈ I →∗ J . Now letx, y ∈ I →∗ J , we show that x∨ y ∈ I →∗ J . Let i ∈ I, then there are j1, j2 ∈ Jsuch that x ≤ i→ j1 and y ≤ i→ j2. So, x∨y ≤ (i→ j1)∨(i → j2) ≤ i→ j1∨j2.Thus x ∨ y ∈ I →∗ J .

It is clear that for every ideal I and J , J ⊆ I →∗ J . For ideals I, J and K weonly show that, I ∨ J →∗ K = (I →∗ K) ∧ (J →∗ K).

Let x ∈ (I →∗ K)∧(J →∗ K) and u ∈ I∨J , then there exist a1, ..., at ∈ I and

b1, ..., bs ∈ J such that, u≤ (t∨

i=1

ai) ∨ (s∨

i=1

bi). Let a =t∨

i=1

ai and b =s∨

i=1

bi, then

a ∈ I and b ∈ J , so there exist v1, v2 ∈ K such that, x ≤ a → v1 ≤ a → v1 ∨ v2and x ≤ b→ v2 ≤ b → v1 ∨ v2. Then x ≤ a ∨ b → v1 ∨ v2 ≤ u→ v1 ∨ v2. Hencex ∈ I ∨ J →∗ K. The converse is trivial and by semilar arguments we can provethe others.

Theorem 2. Let B be a basic algebra. Then the map x �−→ (x] embeds B intothe complete basic algebra I(B).

Proof. Clearly x �−→ (x] is one to one and preserves ∧,∨, 0 and 1. We showthat it also preserves ” → ”. Let a ∈ (x] →∗ (y], then for element u ≤ x,there is an element v ≤ y such that a ≤ u → v. Suppose that u = x, then

Completions of Basic Algebras 75

a ≤ x → v ≤ x → y, therefore a ∈ (x → y] and so (x] →∗ (y] ⊆ (x → y]. Theconverse is obvious.

In the following theorem we show the relation between the ideal completion andHeyting algebras

Proposition 1. The basic algebra I(B) is a Heyting algebra if and only if B isa Heyting algebra.

Proof. It is known that a basic algebra B is a Heyting algebra if and only if forany element x we have 1 → x = x, see [1]. On the other hand, by Theorem 1,for any element x in basic algebra B we have (1 → x] = (1] →∗ (x]. Now we canget the desired result.

In the following theorem we give some basic properties of the ideal completionof basic algebras.

Theorem 3. LetB be a basic algebra. Then I(B) satisfies the following properties:

1. I ∧ ∨t∈T Jt =

∨t∈T (I ∧ Jt),

2. I ∨ ∧t∈T Jt ≤

∧t∈T (I ∨ Jt),

3.∨t∈T (It →∗ J) ≤ ∧

t∈T It →∗ J ,4.

∨t∈T (I →∗ Jt) ≤ I →∗ ∨

t∈T Jt,5.

∧t∈T (It →∗ J) =

∨t∈T It →∗ J ,

6.∧t∈T (It →∗ Jt) ≤

∨t∈T It →∗ ∨

t∈T Jt.

Proof. We only prove clauses 1 and 5. Similar arguments work for the others.1. Let x ∈ I ∧ ∨

t∈T Jt, then there exist elements a1, . . . , an ∈ ⋃t∈T Jt such

that x ≤ ∨ni=1 ai. Without loss of generality we can assume that ai ∈ Ji, so x ∈∨n

i=1 Ji. By distributivity of I(B), we have x ∈ ∨ni=1(I∧Ji), so x ∈ ∨

t∈T (I∧Jt).The converse is trivial.

5. Let x ∈ ∧t∈T (It →∗ J), then for every t ∈ T , x ∈ It →∗ J . Let u ∈∨

t∈T It, then there exist elements a1, ..., an, such that u ≤ ∨ni=1 ai. Without

loss of generality we can assume that ai ∈ Ii. For each ai there is bi in J suchthat x ≤ ai → bi ≤ ai → ∨n

i=1 bi, then x ≤ (a1 → ∨ni=1 bi) ∧ ... ∧ (an →∨n

i=1 bi) ≤ u → ∨ni=1 bi. So x ∈ ∨

t∈T It →∗ J , since∨ni=1 bi ∈ J . On the

other hand, we have It ≤ ∨t∈T It, so

∨t∈T It →∗ J ≤ It →∗ J and hence∨

t∈T It →∗ J ≤ ∧t∈T (It →∗ J).

Definition 2. A lattice L is called join-infinite distributive, if the equality a ∧(∨t bt) =

∨t(a ∧ bt) holds in the case that

∨t bt exists.

It is well-known that a complete bounded distributive lattice is a Heyting algebraif and only if it is join-infinite distributive. From Theorem 2 and 3 it follows thatthe ideal completion of every basic algebra A has (implicity) a structure of aHeyting algebra. In this algebra the Heyting implication is the operation definedfor every a, b by:

a ↪→ b =∨

{c : a ∧ c ≤ b}This implication may be different from the basic algebra implication defined inTheorem 1, as the following example shows.

76 M. Alizadeh

Example 1. Let B be a bounded distributive lattice with universeB = {0, a, b, 1}such that a ∧ b = 0, a ∨ b = 1. We define an implication on B as follows:1 → a = b → a = a, 1 → b = a → b = 1, x → 0 = a, for x > 0 and x → y = 1,for x ≤ y. It is easy to see that 〈B,→〉 is a basic algebra.

Now the ideal completion of B is a finite distributive lattice, so it is a Heytingalgebra too. In this case we haveB ↪→∗ (b] = (b]. But by basic algebra implicationdefined in Theorem 1 we have B →∗ (b] = B.

Theorem 2 implies only the following:

Theorem 4. Every basic algebra can be embedded in a complete Heyting algebraas a lattice.

Similar to canonical extensions of basic algebras, also the ideal completion ofthese algebras does not preserve the existing infinite joins and infinite meet ingeneral. In fact, it does not preserve infinite joins. We end this section by atheorem which says that the variety of Lob algebras is not closed under the idealcompletion. First we note the following

Lemma 2. [2]. Let B be a Lob algebra. Then for any element x ∈ B, 1 → x = xif and only if x = 1.

Theorem 5. The variety of Lob algebras is not closed under the ideal comple-tion.

Proof. Take the linear order structure 〈N + N∗, <〉, where N is the set of allnatural numbers with the strict order, followed by a copy of N with reverseorder, i.e., {0 < 1 < 2 < · · · < 2∗ < 1∗ < 0∗}. Let B be the algebra withuniverse N+N∗ and with greatest and least elements 0∗ and 0, respectively. Foreach element x ∈ N − {0}, the successor s(x∗) of x∗ is defined to be (x − 1)∗.For every element x, y in B, x→ y is equal to s(y), if y < x and 0∗ otherwise.

It can easily be seen that this structure is a Lob algebra. Now for ideal N =(N ] in the ideal completion of B we have B →∗ N = N . So by pervious Lemmathe ideal completion of B is not a Lob algebra.

3 MacNeille Completions and Closed Ideal Completions

In the previous section we saw that the canonical extension and the ideal com-pletion of a basic algebra do not necessarily preserve existing infinite joins andmeets in the original algebra. In this section we will discuss two other completionsof basic algebras, the MacNeille completion and the closed ideal completion. Itis well-known that the MacNeille completion of a distributive lattice preservesexisting infinite joins and meets but does not necessarily preserve distributivity.But MacNeille completions of Heyting algebras are still Heyting algebras.TheMacNeille completion of a distributive lattice is briefly described as follows. Fora distributive lattice A and B ⊆ A, let L(B) be the collection of all lower boundsof B, U(B) be the collection of all upper bounds of B, and call B a normal ideal

Completions of Basic Algebras 77

of A if B = LU(B). Then the collection of all normal ideals of A is the MacNeillecompletion of A. Unlike the variety of Heyting algebras, in the following theo-rem we will see that the variety of basic algebras is not closed under MacNeillecompletions.

Theorem 6. The variety of basic algebras is not closed under MacNeille com-pletions.

Proof. It is well-known [6] that the variety of bounded distributive lattices is notclosed under MacNeille completions. Now suppose that L is a bounded latticewhich is distributive and → is defined as the function constantly equal to 1. Then〈L,→〉 is a basic algebra, indeed a Lob algebra. Now if L is a distributive latticewhose MacNeille completion is not distributive, then independently of how oneextends implication, it is clear that the result can not be a basic algebra.

A basic algebra B is called an L1-algebra if it satisfies 1 → 0 = 1. Note that inevery basic algebra B, 1 → 0 = 1 if and only if for any element a and b in B wehave a→ b = 1. It is easy to see that every L1-algebra is a Lob algebra. In [3] itwas shown that the minimal subvarieties of the variety of basic algebras are onlythe variety of Boolean algebras and the variety of L1-algebras, and consequentlythe latter is the single minimal variety of the variety of Lob algebras. Two famousvarieties of basic algebras are the variety of Heyting algebras and the variety ofLob algebras and all of their subvarieties. It was shown [7] that the only varietiesof Heyting algebras which are closed under MacNeille completions are the trivialvariety, the variety of Boolean algebras, and the variety of Heyting algebras. Wehave the following theorem for the variety of Lob algebras.

Theorem 7. 1. If a subvariety of basic algebras contains the two elements L1-algebra, then it is not closed under MacNeille completions.

2. The variety of Lob algebras and all of its subvarieties are not closed underMacNeille completions.

Proof. It was shown [3] that the two elements L1-algebra generates the varietyof all L1-algebras. Now by using the same argument of Theorem 6 one can getthe desired result. 2 is a corollary of 1.

Let us recall some definitions to introduce the closed ideal completion for basicalgebras. An ideal I of a basic algebra B is closed if for every subset S in I we have∨S ∈ I, whenever

∨S exists. Suppose that Ic(B) is the set of closed ideals of B.

For Γ ⊆ Ic(B) we define∨c

Γ = Ic(⋃Γ ), i.e., the minimal closed ideal including⋃

Γ . Note that for every subset X of a basic algebra, there exists a uniqueminimal closed ideal Ic(X) including X . In fact Ic(X) is the intersection of allclosed ideals including X . For I, J ∈ Ic(B) we define I →c J = Ic(I →∗ J). Notethat in any Heyting algebra, according to the presence of the law of residuation,i.e.,

a ∧ b ≤ c if and only if a ≤ b→ c

78 M. Alizadeh

we can show that an ideal is normal if and only if it is closed under existingjoins. So in the variety of Heyting algebras, MacNeille completions and closedideal completions coincide. We will show that the analogous result is true forsome kind of basic algebras. For convenience we include a proof of the followingknown result.

Proposition 2. Let B be a join-infinite distributive basic algebra and X ⊆ B.Then every element x of Ic(X) is characterized as follows:

x =∨{y | y ≤ x and y ≤ z for some z ∈ X}.

Proof. Let J be the set of all x’s in the proposition. First we show that it isan ideal. Suppose that x ∈ J and a ≤ x, then by the following sequences ofidentities we can get a ∈ J .a = a ∧ x = a ∧ ∨{y | y ≤ x and y ≤ z for some z ∈ X}

=∨{y ∧ a | y ≤ x and y ≤ z for some z ∈ X}

=∨{y | y ≤ a and y ≤ z for some z ∈ X}.

In the third identity we used the join-infinite distributive law. It is fairly easyto show that if x and y are in J , then x ∨ y is also in J . Now assume that S isa subset of J and a =

∨{s | s ≤ a and s ∈ S}. For s ∈ S put As = {x | x ≤s and x ≤ y for some y ∈ X}. Then s =

∨As for s in S. So

a =∨{∨As | s ≤ a and s ∈ S} =

∨{x | x ≤ a and x ≤ y for some y ∈ X}.Therefore a is in J . J also includes X and the minimality of J is clear.

Let us recall that an embedding of an algebra B to a complete algebra B∗ iscalled regular embedding if it preserves the existing infinite joins and meets inB. In this case the complete algebra B∗ is called a regular completion of B.

Theorem 8. Let B be a join-infinite distributive basic algebra. Then

Ic(B) = 〈Ic(B),∧,∨c,→c, (0], (1]〉is a join-infinite and complete basic algebra and the embedding f : B −→ Ic(B);f(x) = (x] is a regular embedding.

Proof. Clearly Ic(B) is a bounded lattice. we show that it is also join-infinitedistributive. Obviously

∨cJ∈Γ I∧J ≤ I∧∨c

Γ , for every closed ideal I and everysubset Γ of Ic(B) with

∨cΓ = Ic(∪Γ ). Let x be an element of I ∧ ∨c

Γ , thenx ∈ I and x =

∨{y|y ≤ x and y ∈ J for some J ∈ Γ}, by previous Proposition.Therefore x ∈ Ic(∪J∈Γ I ∧ J) =

∨cJ∈Γ I ∧ J .

It is clear that for ideals I and J , J ⊆ I →c J . For ideals I, J and K we showthat I ∨c J →c K = (I →c K) ∧ (J →c K).

Let x ∈ (I →c K) ∧ (J →c K). By the previous proposition we have

x =∨{y|∃z ∈ I →∗ K with y ≤ z}

=∨{y′|∃z′ ∈ J →∗ K with y′ ≤ z′}

=∨{t|∃u ∈ I ∨ J →∗ K with t ≤ u}.

Note that in the third equality we used both join-infinite distributivity andthe fact that z ∧ z′ ∈ (I →∗ K) ∧ (J →∗ K) = I ∨ J →∗ K, since z ∈ I →∗ K

Completions of Basic Algebras 79

and z′ ∈ J →∗ K. The converse inclusion is trivial. We can prove the otherproperties of ” → ” by similar arguments.

By Theorem 2, f is an embedding. Now we show that f preserves all infinitejoins. Suppose that

∨S exists for S ⊆ B. Let U = {f(s) | s ∈ S}, then

∨S ∈

Ic(⋃U). Therefor f(

∨S) = Ic(

⋃U) =

∨cU =

∨cs∈S f(s).

Similar to the ideal completion of basic algebras we have

Proposition 3. The basic algebra Ic(B) is a Heyting algebra if and only if Bis a Heyting algebra.

Proof. Note that for every closed ideal I, (1] →c I = I if and only if (1] →∗ I = I.Now apply Proposition 1.

Proposition 4. Every join-infinite distributive basic algebra can be regularlyembedded in a complete Heyting algebra as a lattice.

As we mentioned before, in the case of Heyting algebras the ideal completion andMacNeille completions coincide. Here we will generalize this fact for join-infinitedistributive basic algebras. Let us recall the following definition

Definition 3. Let B and B∗ be join-infinite distributive basic algebras which B∗

is also a regular completion of B with related embedding f . B∗ is called a joindense completion of B if for any element a in B∗ we have a =

∨{f(x)| f(x) ≤a and x ∈ B}.

Theorem 9. Let B be a join-infinite distributive basic algebra. Then every joindense completion of B is lattice isomorphic to Ic(B).

Proof. Let f be the related embedding of B into Ic(B) and B∗ be a join densecompletion of B with related embedding g. For ever closed ideal I we defineh(I) =

∨x∈I g(x). We show that h is the required isomorphism. Let I and J be

two closed ideals. Since B∗ is join-infinite distributive we have h(I) ∧ h(J) =∨x∈I g(x)∧

∨y∈J g(y) =

∨x∈I

∨y∈J g(x∧y) ≤

∨z∈I∧J g(z) = h(I∧J). The other

direction is trivial. Now we show that h preserves infinite joins. Let Γ be a setof closed ideal, it is sufficient to show that h(

∨cΓ ) ≤ ∨

I∈Γ h(I). For x ∈ ∨cΓ ,

x =∨{y|y ≤ x and y ∈ ⋃

Γ}. Then g(x) =∨{g(y)|y ≤ x and y ∈ ⋃

Γ}, sinceg is regular. Now for every y ∈ ⋃

Γ there exists an ideal I in Γ such that y ∈ I,so g(x) ≤ ∨

z∈I g(z) = h(I). Now one can get the required inequality.For injectivity suppose that h(I) = h(J). Then

∨x∈I g(x) =

∨y∈J g(y). So for

every x ∈ I, g(x) ≤ ∨y∈J g(y) which implies that g(x) =

∨y∈J g(x∧y). We show

that x =∨y∈J x∧y and so x ∈ J . Clearly x is a upper bound. Suppose that z is

an element such that for any y ∈ J , x∧y ≤ z. Then g(x) =∨y∈J g(x∧y) ≤ g(z),

it implies that x ≤ z, since g is injective. Hence, I = J .For surjectivity of h suppose that x ∈ B∗, then x =

∨{g(y)| g(y) ≤ x and y ∈B}. It is easy to see that h(

∨c{f(y)| g(y) ≤ x and y ∈ B}) = x, since h and gpreserve infinite joins.

80 M. Alizadeh

In fact the above proof shows that h o f = g. It means that our isomorphism isunique, since suppose that h′ is another isomorphism with the same properties.Then for any closed ideal I we have h(I) = h(

∨c{f(a)| a ∈ I}) =∨{h(f(a))| a ∈

I} =∨{h′(f(a))| a ∈ I} = h′(I). As a corollary we have

Corollary 1. Let B be a join-infinite distributive basic algebra and I be an idealin B. I is a closed ideal if and only if it is a normal ideal.

Proof. First note that our related regular embedding in the MacNeille comple-tions is g(x) = (x]. We show that the above unique isomorphism h is the identityfunction, i.e., for any closed ideal I, h(I) = I.h(I) = h(

∨c{f(x)| x ∈ I}) =∨c{h(f(x))| x ∈ I} =

∨c{∨y≤x g(y)| x ∈ I} =∨c{(x]| x ∈ I} = I. So every closed ideal is a normal ideal.For the converse, suppose I is a normal ideal, S ⊆ I and

∨S = a. If x ∈ U(I),

then a ≤ x, and hence a ∈ LU(I) = I.

Corollary 2. The class of all join-infinite distributive basic algebras is closedunder MacNeille completions.

4 Algebraic Completeness of Visser’s Predicate Logic

Visser’s basic predicate logic is a predicate logic with intuitionistic languagewhich is interpreted in Kripke models with transitive accessibility relation. Thepropositional part of the logic was first introduced by Visser [9], called basicpropositional logic, BPL, and developed by Ardeshir and Ruitenburg [5]. Basicpredicate logic, BQL, as a predicate extension of BPL was first introduced byRuitenburg in [8].

The language of BQL, L, contains a denumerable set of predicate symbols ofeach finite arity, a denumerable set V of variable symbols, parentheses, logicalconstants � and ⊥, the logical connectives ∧, ∨, → and quantifiers ∃ and ∀. Ourlanguage is freed of function symbols. We usually include the binary predicate =for equality. Formulas are defined as usual, except for the universally quantifiedformulas. A universally quantified formula is of the form ∀x(φ(x) → ψ(x)), inwhich x = (x1, x2, · · · , xn), a finite sequence of distinct variables. A sequent isan expression of the form φ⇒ ψ, in which φ and ψ are formulas. We often writeφ for � ⇒ φ. A rule with a double horizontal line means a two direction rule.For more details see [8]. We give an axiomatization with the axioms and rules

1) φ⇒ φ, 2) ⊥ ⇒ φ, 3) φ⇒ �, 4) � ⇒ x = x,

5) φ ∧ (ψ ∨ η) ⇒ (φ ∧ ψ) ∨ (φ ∧ η),

6) ∃xφ ∧ ψ ⇒ ∃x(φ ∧ ψ), x is not free in ψ,

7) x = y ∧ φ⇒ φ[x/y], A is atomic, 8) ∀x(φ → ψ) ∧ ∀x(ψ → η) ⇒ ∀x(φ→ η),

Completions of Basic Algebras 81

9) ∀x(φ → ψ) ∧ ∀x(φ → η) ⇒ ∀x(φ → ψ ∧ η),

10) ∀x(φ→ η) ∧ ∀x(ψ → η) ⇒ ∀x(φ ∨ ψ → η),

11) ∀x(φ→ ψ) ⇒ ∀y(φ→ ψ), no variable in y is free on the left hand side,

12) ∀x(φ → ψ) ⇒ ∀x(φ[x/t] → ψ[x/t]), no variable in t is bounded in φor ψ,

13) φ⇒ψ ψ⇒ηφ⇒ψ , 14)

φ⇒ψ φ⇒η

φ⇒ψ∧η , 16)φ⇒η ψ⇒η

φ∨ψ⇒η ,

17) φ∧ψ⇒ηφ⇒∀x(ψ→η) , no variable in x is free in φ,

18) φ⇒ψ

∃xφ⇒ψ, x is not free in ψ,

19) φ⇒ψφ[x/t]⇒ψ[x/t] , no variable in t is bounded in the denominator,

20) ∀yx(φ → ψ) ⇒ ∀y(∃xφ → ψ), x is not free in ψ.

BQL without the last axiom above is shown by BQL−. In the following, weprove the algebraic completeness and soundness for BQL−.

An algebraic model is a structure B = 〈B,D, I〉, where B is a join- infinitedistributive complete basic algebra and D is a non-empty set. Let D be the setof names of all elements in D. I is a map from atomic sentences of L(D), thelanguage L expanded with the constants in D, to B which satisfies the followingconditions: for any a and b in D, and for nay atomic formula P (x),I(a = a) = 1, and I(P (a)) ∧ I((a = b)) ≤ I(P (b)).The map I can be uniquely extended to the set of all sentences on L(D). For

quantified formulas we have:

1. I(∀x(φ → ψ)) =⋂a∈D I(φ[x/a] → ψ[x/a]), and

2. I(∃xφ) =⋃a∈D I(φ[x/a]).

For φ, ψ ∈ For(L), suppose that FV (φ) ∪ FV (ψ) = {x1, ..., xn}. A sequentφ⇒ ψ is satisfied in B, B |= φ⇒ ψ, if for any a1, ..., an ∈ D, I(φ[x1/a1, ..., xn/an]) ≤ I(ψ[x1/a1, ..., xn/an]). A sequent φ ⇒ ψ is valid, |= φ ⇒ ψ, if for everymodel B, B |= φ⇒ ψ.

Theorem 10. (Soundness and completeness) For any formulas φ and ψ,BQL− � φ⇒ ψ if and only if BQL− |= φ⇒ ψ.

Proof. The proof of the soundness part of the theorem proceeds by inductionon the height of the derivation. For the completeness part, we construct theLindenbaum algebra U of BQL. Consider the equivalence relation ∼ defined inthe set of all formulas by: φ ∼ ψ iff � φ ⇒ ψ and � ψ ⇒ φ. This relationis a congruence with respect to the operations associated with the connectives.Then U is the quotient algebra of the algebra of formulas given by the operations

82 M. Alizadeh

associated with the connectives, � and ⊥. One may easily check that the algebraU is a basic algebra whose lattice order is such that [φ] ≤ [ψ] iff � φ ⇒ ψ. Inaddition we have:

[∃xφ] =⋃

y∈V[φ[x/y]]; [∀x(φ→ ψ)] =

⋂

y∈V[φ[x/y] → ψ[x/y]].

We check the first equality. It is easy to show that � φ(x) ⇒ ∃xφ(x). So, �φ[x/y] ⇒ ∃xφ(x), for every variable y. Therefore [∃xφ] is an upper bound forthe set {[φ[x/y]]|y ∈ V }. To show that it is the least upper bound suppose thatψ is a formula, such that � φ[x/y] ⇒ ψ for all y ∈ V. If x �∈ FV (ψ), thentake x for y. So we have � φ(x) ⇒ ψ, then we can deduce � ∃xφ(x) ⇒ ψ,which implies [∃xφ] ≤ [ψ]. If x ∈ FV (ψ), take fresh variable z. Then, by ourassumption, we have � φ[x/z] ⇒ ψ. Therefore � ∃zφ[x/z] ⇒ ψ, which implies[∃zφ[x/z]] ≤ [ψ]. By axiom (6), the basic algebra U is countable join-infinitedistributive basic algebra but it does not have to be complete. By Theorem 8,U can be embedded into a complete basic algebra B, in which the infinite meetsand infinite joins of elements of U are preserved. Note that in this case thecomplete algebra B is also countable. Consider a model B = 〈B,V, I〉 where Vis the set of variables and I is the map from the atomic sentences of L(V) givenby I(φ(x1, . . . , xn)) = [φ(x1, . . . , xn)] for every atomic formula φ(x1, . . . , xn). Byinduction it follows that I(φ(x1, . . . , xn)) = [φ(x1, . . . , xn)] for every formulaφ(x1, . . . , xn). Now we are in a position to prove the completeness. Suppose|= φ ⇒ ψ. Then for any model C, C |= φ ⇒ ψ. Take B for C, then [φ] ≤ [ψ],which means � φ⇒ ψ.

By the same method as above we can prove one direction of the above theoremfor BQL.

Theorem 11. (Completeness) For any formulas φ and ψ, if BQL |= φ⇒ ψ,then BQL � φ⇒ ψ.

To prove the soundness for BQL we need to prove that the regular completionsof join-infinite basic algebras satisfy clause 5 of Theorem 3. At this moment,unfortunately, we don’t know if the closed ideal completion satisfies this propertyor not. It is an interesting problem to introduce a complete algebraic semanticsfor BQL.

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