on the characterizations of fuzzy implications satisfying

Information Sciences 177 (2007) 2954–2970

www.elsevier.com/locate/ins

On the characterizations of fuzzy implicationssatisfying Iðx; yÞ ¼ Iðx; Iðx; yÞÞ

Y. Shi a,*, D. Ruan a,b, E.E. Kerre a

a Fuzziness and Uncertainty Modelling Research Unit, Department of Applied Mathematics and Computer Science, Ghent University,

Krijgslaan 281 (S9), 9000 Gent, Belgiumb Belgium Nuclear Research Centre (SCK•CEN), 2400 Mol, Belgium

Received 27 October 2005; received in revised form 20 December 2006; accepted 26 January 2007

Abstract

Iterative boolean-like laws in fuzzy logic have been studied by Alsina and Trillas [C. Alsina, E. Trillas, On iterativeboolean-like laws of fuzzy sets, in: Proc. 4th Conf. Fuzzy Logic and Technology, Barcelona, Spain, 2005, pp. 389–394]for functional equations with boolean background in which only fuzzy conjunctions, fuzzy disjunctions and fuzzy nega-tions are contained. In this paper we study an iterative boolean-like law with fuzzy implications, more precisely we derivecharacterizations of some classes of fuzzy implications satisfying Iðx; yÞ ¼ Iðx; Iðx; yÞÞ, for all ðx; yÞ 2 ½0; 1�2. Our discussionmainly focuses on the three important classes of implications: S-implications, R-implications and QL-implications. Weprove the sufficient and necessary conditions for an S-implication generated by any t-conorm and any fuzzy negation,an R-implication generated by a left-continuous t-norm, a QL-implication generated by a continuous t-conorm, a contin-uous t-norm and a strong fuzzy negation to satisfy Iðx; yÞ ¼ Iðx; Iðx; yÞÞ, for all ðx; yÞ 2 ½0; 1�2.� 2007 Elsevier Inc. All rights reserved.

Keywords: Iterative boolean-like law; Fuzzy implications; S-implications; R-implications; QL-implications

1. Introduction

In [14, Section 4], the authors analyzed some non-standard aspects in the constructions of fuzzy set theoriesand dealt with the derived boolean properties of fuzzy operations, among which the iterative boolean-like laws

[1] were considered as derived boolean laws not valid in any standard fuzzy set theories. In [1], the authorsstudied a class of functional equations [1, Definition 4.1] with the boolean background [1, Definition 4.2], namedas iterative boolean-like laws which are formulated in fuzzy logic where some variables appear several timesbecause they come from boolean identities where no simplifications such as the application of idempotency ordistributivity, absorption, etc. have been made. In [1, Section 5], the authors analyzed some standard iterativeboolean-like laws in fuzzy logic such as ðA [ AÞ \ ðA \ AÞco ¼ ;, A [ B ¼ ðA \ BÞ [ ½ðA [ BÞ \ ðA \ BÞco�,

0020-0255/$ - see front matter � 2007 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2007.01.026

* Corresponding author. Tel.: +32 9 264 47 72; fax: +32 9 264 49 95.E-mail address: [email protected] (Y. Shi).

mailto:[email protected]

Y. Shi et al. / Information Sciences 177 (2007) 2954–2970 2955

ðA [ B [ BÞ [ ðA \ B \ BÞ ¼ A [ B, and solve the functional equations derived from them. But only laws withfuzzy conjunctions, fuzzy disjunctions and fuzzy negations were considered. Since fuzzy implications are alsofuzzy operations that play an important role in fuzzy logic, we will analyze some iterative boolean-like lawswith fuzzy implications.

In this paper we will consider the derived boolean law p ! ðp! qÞ ¼ p! q. In classical logic, the impli-cation operator ‘!’ used to gain the value of the proposition ‘if p, then q’ (p ! q) is defined through ‘:p _ q’(in which �denotes the negation and _denotes the disjunction) uniquely and the proposition ‘p ! ðp ! qÞ’ isequivalent to ‘p ! q’. However, this equivalence does not hold for every fuzzy implication derived from clas-sical logic, which is generated by t-norms, t-conorms and fuzzy negations. Our main concern of this paper is tofind out the conditions under which the following iterative boolean-like law in fuzzy logic holds:

Iðx; yÞ ¼ Iðx; Iðx; yÞÞ; 8x; y 2 ½0; 1�2; ð1Þ

where I denotes a fuzzy implication from the classes of S-, R- or QL-implications.In Section 2 we give some basic notations and definitions of fuzzy conjunctions, fuzzy disjunctions and

fuzzy negations. In Section 3 we focus on some general properties of fuzzy implications and list a table of somewidely used implications from the three classes to see the properties including (1) they satisfy. In Section 4 wework out the sufficient and necessary conditions for the three classes of fuzzy implications to satisfy (1).Finally we conclude our main results in Section 5.

2. Preliminaries

2.1. Fuzzy negations

In fuzzy logic, a mapping n: ½0; 1� ! ½0; 1� is a fuzzy negation if it satisfies:

n1. boundary conditions: nð0Þ ¼ 1 and nð1Þ ¼ 0,n2. monotonicity: x 6 y ) nðxÞP nðyÞ.

For any continuous fuzzy negation n, there exists a unique equilibrium point e 2�0; 1½ such that nðeÞ ¼ e andfor all x < e, nðxÞ > e > x, for all x > e, nðxÞ < e < x [4, Section 1.1]. A fuzzy negation is said to be strict if n isa continuous and strictly decreasing mapping. A fuzzy negation is said to be involutive if nðnðxÞÞ ¼ x, for allx 2 ½0; 1�. Fuzzy negations that are involutive are called strong fuzzy negations. A strong fuzzy negation isstrict and it is a continuous mapping.

Next we introduce concepts of automorphism and conjugate, which will be useful in this paper.An automorphism of the interval ½a; b� � R is a continuous, strictly increasing mapping u from ½a; b� to

½a; b� with boundary conditions uðaÞ ¼ a, uðbÞ ¼ b [4, Definition 0].Two mappings F ;G : ½0; 1�2 ! ½0; 1� are conjugate, if there exists an automorphism u of the unit interval

such that G ¼ F u, where F uðx; yÞ ¼ u�1ðF ðuðxÞ;uðyÞÞÞ, x; y 2 ½0; 1� [2, Definition 2].

Proposition 1 [18]. A fuzzy negation n is strong iff there exists an automorphism u of the unit interval such that

nðxÞ ¼ u�1ð1� uðxÞÞ 8x 2 ½0; 1�:

2.2. Triangular norms

The conjunction in fuzzy logic is often represented by a triangular norm (t-norm for short). A mappingT : ½0; 1�2 ! ½0; 1� is a t-norm if for all x; y; z 2 ½0; 1� it satisfies:

T1. boundary condition: T ðx; 1Þ ¼ x,T2. monotonicity: y 6 z implies T ðx; yÞ 6 T ðx; zÞ,T3. commutativity: T ðx; yÞ ¼ T ðy; xÞ,T4. associativity: T ðx; T ðy; zÞÞ ¼ T ðT ðx; yÞ; zÞ.

2956 Y. Shi et al. / Information Sciences 177 (2007) 2954–2970

There are also some additional requirements of t-norms. Here we mention seven of them:

T5. continuity: T is continuous,T6. left-continuity: the partial mappings of T are left-continuous,T7. idempotency: T ðx; xÞ ¼ x, for all x 2 ½0; 1�,T8. subidempotency: T ðx; xÞ < x, for all x 2�0; 1½ ,

T9. Archimedean property: for all ðx; yÞ 2 �0; 1½2, there exists n 2 N such that T ðx; x; . . . ; x|fflfflfflfflfflffl{zfflfflfflfflfflffl}n times

Þ < y [10, Defini-tion 2.9 (iv)],

T10. strictness: T is continuous and 0 < y < z < 1) T ðx; yÞ < T ðx; zÞ, for all x; y; z 2�0; 1�,T11. nilpotency: T is continuous and for all x 2�0; 1½ , there exists y 2�0; 1½ such that T ðx; yÞ ¼ 0.

Remark 1. According to [10, Definition 2.1 (i)], an element x 2 ½0; 1� such that T ðx; xÞ ¼ x is called anidempotent element of T. The numbers 0 and 1, which are idempotent elements for each t-norm T, are calledtrivial idempotent elements of T. Each idempotent element in �0; 1½ will be called a non-trivial idempotentelement of T. Thus T satisfying subidempotency is equivalent to T having only trivial idempotent elements.Hence according to [10, Fig. 2.2], the subidempotency is equivalent to the Archimedean property only forcontinuous t-norms.

There are four very important t-norms [10, Example 1.2]:

(1) T Mðx; yÞ ¼ minðx; yÞ, (minimum)(2) T P ðx; yÞ ¼ xy, (product)(3) T –Lðx; yÞ ¼ maxðxþ y � 1; 0Þ, (Łukasiewicz t-norm)

(4) T Dðx; yÞ ¼minðx; yÞ x ¼ 1 or y ¼ 10 otherwise:

�(drastic product)

The standard fuzzy conjunction TM is the only idempotent t-norm [11, Theorem 3.9].

Theorem 1 [11, Theorem 3.11]. T is a continuous Archimedean t-norm iff there exists a decreasing generator f

such that

T ðx; yÞ ¼ f ð�1Þðf ðxÞ þ f ðyÞÞ; 8x; y 2 ½0; 1�: ð2Þ
In this theorem, a decreasing generator f is defined as a continuous and strictly decreasing mapping from
½0; 1� to ½0;1� such that f ð1Þ ¼ 0. The pseudo-inverse of f, f ð�1Þ: ½0;1� ! ½0; 1� is defined as:

f ð�1ÞðxÞ ¼ f �1ðxÞ x 2 ½0; f ð0Þ�;0 x 2�f ð0Þ;1�;

�ð3Þ

where f�1 denotes the ordinary inverse of f.It has been proved that a continuous Archimedean t-norm is either strict or nilpotent [10, p. 33].

Theorem 2 [10, Corollary 5.7]

(i) A t-norm T is strict iff it is conjugate with the product TP.

(ii) A t-norm T is nilpotent iff it is conjugate with the Łukasiewicz t-norm T –L.

Theorem 3 [4, Theorem 2]. A continuous t-norm T is such that T ðx; nðxÞÞ ¼ 0 holds for all x 2 ½0; 1� with a strict

fuzzy negation n iff T is conjugate with the Łukasiewicz t-norm T Ł, i.e., there exists an automorphism / of the unit

interval such that

T ðx; yÞ ¼ /�1ðmaxð/ðxÞ þ /ðyÞ � 1; 0ÞÞ
and
nðxÞ 6 /�1ð1� /ðxÞÞ:


Continuous t-norms have been well studied. We have the next definition and theorem. Suppose thatf½am; bm�g is a countable family of non-overlapping, closed, proper subintervals of ½0; 1�, denoted by I andwith each ½am; bm� 2 I associate a continuous Archimedean t-norm Tm. Let T be a mapping defined on½0; 1�2 via

T ðx; yÞ ¼ am þ ðbm � amÞT mx�am

bm�am; y�am

bm�am

� �ðx; yÞ 2 ½am; bm�2;

minðx; yÞ otherwise:

(ð4Þ

Then T is called the ordinal sum of f½am; bm�; T mg and each Tm is called a summand.

Theorem 4 [8, Section 1.3.4], [10, Theorem 5.11]. T is a continuous t-norm iff

(i) T ¼ T M or

(ii) T is continuous Archimedean or

(iii) there exists a family f½am; bm�; T mg where f½am; bm�g is a countable family of non-overlapping, closed, proper

subintervals of ½0; 1� with each Tm being a continuous Archimedean t-norm such that T is the ordinal sum of

this family.

2.3. Triangular conorms

The disjunction in fuzzy logic is often represented by a triangular conorm (t-conorm for short). A mappingS : ½0; 1�2 ! ½0; 1� is a t-conorm if for all x; y; z 2 ½0; 1� it satisfies:

S1. boundary condition: Sðx; 0Þ ¼ x,S2. monotonicity: y 6 z implies Sðx; yÞ 6 Sðx; zÞ,S3. commutativity: Sðx; yÞ ¼ Sðy; xÞ,S4. associativity: Sðx; Sðy; zÞÞ ¼ SðSðx; yÞ; zÞ. Similar to t-norms, we mention here six other properties of t-

conorms:S5. continuity: S is continuous,S6. idempotency: Sðx; xÞ ¼ x, for all x 2 ½0; 1�,S7. subidempotency: Sðx; xÞ < x, for all x 2�0; 1½ ,S8. Archimedean property: for all ðx; yÞ 2 �0; 1½ 2, there exists n 2 N such that Sðx; x; . . . ; x|fflfflfflfflfflffl{zfflfflfflfflfflffl}

n times

Þ > y [10, Remark2.20 (AP*)],

S9. strictness: S is continuous and 0 < y < z < 1) Sðx; yÞ < Sðx; zÞ, for all x; y; z 2 ½0; 1½ ,S10. nilpotency: S is continuous and for all x 2�0; 1½ , there exists y 2�0; 1½ such that Sðx; yÞ ¼ 1.

Remark 2. Similar to that for t-norms, the subidempotency is equivalent to the Archimedean property onlyfor continuous t-conorms. [10, Remark 2.20, Fig. 2.2].

There are four very important t-conorms [10, Example 1.14]:

(1) SMðx; yÞ ¼ maxðx; yÞ, (maximum)(2) SP ðx; yÞ ¼ xþ y � xy, (probabilistic sum)(3) S–Lðx; yÞ ¼ minðxþ y; 1Þ, (Łukasiewicz t-conorm, bounded sum)

(4) SDðx; yÞmaxðx; yÞ x ¼ 0 or y ¼ 01 otherwise:

�(drastic sum)

The standard fuzzy disjunction SM is the only idempotent t-conorm [11, Theorem 3.14].

Theorem 5 [11, Theorem 3.16]. S is a continuous Archimedean t-conorm iff there exists an increasing generator

g such that

Sðx; yÞ ¼ gð�1ÞðgðxÞ þ gðyÞÞ; 8x; y 2 ½0; 1�: ð5Þ


In this theorem, an increasing generator g is defined as a continuous and strictly increasing mapping from½0; 1� to ½0;1� such that gð0Þ ¼ 0. The pseudo-inverse of g, gð�1Þ : ½0;1� ! ½0; 1� is defined as:

gð�1ÞðxÞ ¼ g�1ðxÞ x 2 ½0; gð1Þ�;1 x 2�gð1Þ;1�;

�ð6Þ

where g�1 denotes the ordinary inverse of g.Similar to t-norms, it has been concluded that a continuous Archimedean t-conorm is either strict or nil-

potent [10, Remark 2.20].

Theorem 6 [8, Theorem 1.8, Theorem 1.9]

(i) A t-conorm S is strict iff it is conjugate with the probabilistic sum SP.(ii) A t-conorm S is nilpotent iff it is conjugate with the Łukasiewicz t-conorm S–L.

Theorem 7 [4, Theorem 1]. A continuous t-conorm S is such that Sðx; nðxÞÞ ¼ 1 holds for all x 2 ½0; 1� with a

strict fuzzy negation n iff S is conjugate with the Łukasiewicz t-conorm S–L, i.e., there exists an automorphism

/ of the unit interval such that

Sðx; yÞ ¼ /�1ðminð/ðxÞ þ /ðyÞ; 1ÞÞ
and
nðxÞP /�1ð1� /ðxÞÞ:
And similar to continuous t-norms, we have the next definition and theorem for continuous t-conorms.
Suppose that f½am; bm�g is a countable family of non-overlapping, closed, proper subintervals of ½0; 1�, denotedby I. With each ½am; bm� 2 I associated a continuous Archimedean t-conorm Sm. Let S be a mapping definedon ½0; 1�2 via

Sðx; yÞ ¼ am þ ðbm � amÞSmx�am

bm�am; y�am

bm�am

� �ðx; yÞ 2 ½am; bm�2;

maxðx; yÞ otherwise:

(ð7Þ

S is called the ordinal sum of f½am; bm�; Smg and each Sm is called a summand.

Theorem 8 [8, Section 1.4.4], [10, Corollary 3.58]. S is a continuous t-norm iff

(i) S ¼ SM or

(ii) S is continuous Archimedean or(iii) there exists a family f½am; bm�; Smg where f½am; bm�g is a countable family of non-overlapping, closed, proper

subintervals of ½0; 1� with each Sm being a continuous Archimedean t-conorm such that S is the ordinal sum

of this family.

If for all x; y; z 2 ½0; 1�, Sðx; T ðy; zÞÞ ¼ T ðSðx; yÞ; Sðx; zÞÞ, we say the t-conorm S is distributive over the t-normT [10, Proposition 2.22]. Similarly, if for all x; y; z 2 ½0; 1�, T ðx; Sðy; zÞÞ ¼ SðT ðx; yÞ; T ðx; zÞÞ, we say the t-norm T

is distributive over the t-conorm S. The only distributive pair is TM and SM [10, Proposition 2.22].

3. Fuzzy implications

3.1. Definitions and properties

Fuzzy implications have been widely studied in the fuzzy literature and they play important roles in differ-ent domains. In Zadeh’s composition rule of fuzzy inference (fuzzy modus ponens, fuzzy modus tollens, etc.),a fuzzy implication represents the fuzzy relation between two variables [9, Chapter 5,12,13,16,20]. In imageprocessing, a fuzzy implication is used to define the subsethood measure between two fuzzy sets [5, Theorem1]. And in data mining, a fuzzy implication is used to express the relationship between two items in an asso-ciation rule [22, Eq. (2)]. Also the set of axioms of fuzzy implications has been well studied [2,17].


Originally, a fuzzy implication is a fuzzy operator which is used to represent IF–THEN rules in fuzzy logic.The truth value of the proposition ‘if p, then q’ could be obtained through IðvðpÞ; vðqÞÞ, in which I refers to afuzzy implication and vðpÞ and vðqÞ refer to the truth value of p and q, respectively.

In contrast to t-norms and t-conorms, there exist several definitions of fuzzy implications [4, Section 2], [8,Definition 1.15]. We define a fuzzy implication as a ½0; 1�2 ! ½0; 1� mapping at least satisfying the boundaryconditions:

I1. Ið0; 0Þ ¼ Ið0; 1Þ ¼ Ið1; 1Þ ¼ 1, Ið1; 0Þ ¼ 0.

Hence a fuzzy implication extends the classical implication.Many other potential properties could be found in [2,4,7,9,19,21]. The most important properties of fuzzy

implications are [4,8,15,19]:

I2. first place antitonicity: x 6 y ) Iðx; zÞP Iðy; zÞ, for all x; y; z 2 ½0; 1�,I3. second place isotonicity: y 6 z) Iðx; yÞ 6 Iðx; zÞ, for all x; y; z 2 ½0; 1�,I4. left neutrality of truth: Ið1; xÞ ¼ x, for all x 2 ½0; 1�,I5. exchange principle: Iðx; Iðy; zÞÞ ¼ Iðy; Iðx; zÞÞ, for all x; y; z 2 ½0; 1�,I6. ordering property: Iðx; yÞ ¼ 1 iff x 6 y, for all x; y 2 ½0; 1�,I7. contrapositive law: Iðx; yÞ ¼ IðnðyÞ; nðxÞÞ, for all x; y 2 ½0; 1�, where n denotes a strong fuzzy negation,I8. continuity.

These properties are not independent from each other, it was proved in [2, Section 4] that a fuzzy implica-tion that satisfies I5, I6 and I8 also satisfies the other five properties I1, I2, I3, I4 and I7.

Some implications do not satisfy the property I8, but they are right continuous mappings in the second argu-

ment [2, Section 3].

3.2. Implications generated by t-norms, t-conorms and fuzzy negations

It was mentioned in [4, Section 2] that all fuzzy implications are obtained by generalizing the implications ofclassical logic. There are three important ways to generate fuzzy implications through classical logic: S-impli-

cations, R-implications and QL-implications.Let S be a t-conorm and n be a fuzzy negation. An S-implication is defined by:

Iðx; yÞ ¼ SðnðxÞ; yÞ; 8x; y 2 ½0; 1�: ð8Þ
It should be mentioned here that in some references, n is assumed to be a strong fuzzy negation [4, Section 4],[8, Definition 1.16]. But in other references, n is not necessary supposed to be strong, even not necessary to becontinuous [10, Definition 11.5], [3] (where the authors call the S-implications generated by a fuzzy negation N
and a t-conorm S (S,N)-implications). In this paper, we assume n to be any fuzzy negation, so as [10, Defini-tion 11.5].

Let T be a t-norm. An R-implication is defined by:

Iðx; yÞ ¼ supft 2 ½0; 1�jT ðx; tÞ 6 yg; 8x; y 2 ½0; 1�: ð9Þ
The definition of R-implication comes from a residuation concept in intuitionistic logic. Actually, this defini-tion is only reasonable for left-continuous t-norms [2, Section 3]. And when T is a left-continuous t-norm, (9)is equivalent to [2, Section 3]:
Iðx; yÞ ¼ maxft 2 ½0; 1�jT ðx; tÞ 6 yg; 8x; y 2 ½0; 1�: ð10Þ

Let I be an R-implication generated by a left-continuous t-norm via (9) (or (10)), then a t-norm T can be gen-erated by I through:

T ðx; yÞ ¼ infft 2 ½0; 1�jIðx; tÞP yg; 8x; y 2 ½0; 1�: ð11Þ
The following important theorem and its corollary will be useful in our further discussion.


Theorem 9 [8, Theorem 1.14]. A mapping I : ½0; 1�2 ! ½0; 1� is an R-implication based on some left-continuous t-

norm iff I satisfies I3, I5, I6 and it is right continuous in the second argument.

If we denote the R-implication generated by a fixed left-continuous t-norm T through (9) (or (10)) as IT andthe t-norm generated by an R-implication I, which satisfies I3, I5, I6 and right-continuity in the second argu-ment through (11) as TI, then we will find in the proof of Theorem 1.14 in [8] that IT I ¼ I and T IT ¼ T . Thefollowing corollary is based on these facts.

Corollary 1 [2, Corollary 10]. A mapping T : ½0; 1�2 ! ½0; 1� is a left-continuous t-norm iff T can be represented

by:

TableFuzzy

Name

Kleene

Reiche

Most S

Least

Łukas

R0

Godel

Gogue

Early

Klir an

Klir anYua

T ðx; yÞ ¼ minft 2 ½0; 1�jIðx; tÞP yg ð12Þ
for some mapping I : ½0; 1�2 ! ½0; 1� which satisfies I3, I5, I6 and right-continuity in the second argument.
Let S be a t-conorm, n be a strong fuzzy negation and T be a t-norm. A QL-implication is defined by:

Iðx; yÞ ¼ SðnðxÞ; T ðx; yÞÞ; 8x; y 2 ½0; 1�: ð13Þ

Some authors assume n to be any fuzzy negation, not necessary strong, e.g., [6, Eq. (1.1)]. But in this paper weassume n to be strong.

3.3. An overview of some fuzzy implications and their properties

In this paper, we will discuss fuzzy implications of the three important classes that satisfy Eq. (1). In Table 1we list several widely used fuzzy implications from the three classes and in the last column we indicate theirproperties. Actually an S-implication satisfies I1, I2, I3, I4 and I5 [3, Proposition 1]. An R-implication basedon a left-continuous t-norm satisfies I1, I2, I3, I4 I5 and I6 [8, Lemma 1.3, Theorem 1.14]. A QL-implicationsatisfy I1, I3 and I4 [4, Proposition 2(i)].

1implications and their properties

Symbol Iðx; yÞ Class Properties

-Dienes Ib maxð1� x; yÞ S, QL I1, I2, I3I4, I5, I7, I8, (1)

nbach Ir 1� xþ xy S, QL I1, I2, I3I4, I5, I7, I8

trict IM1 x ¼ 0y otherwise

�S I1, I2, I3

I4, I5, (1)

ILS

y x ¼ 11� x y ¼ 01 otherwise

8<: S I1, I2, I3

I4, I5, I7

Strict ILRy x ¼ 11 otherwise

�S, R, QL I1, I2, I3

I4, I5, I6, (1)iewicz Ia minð1� xþ y; 1Þ S, R, QL I1, I2, I3

I4, I5, I6, I7, I8Imin0

1 x 6 ymaxð1� x; yÞ otherwise

�S, R I1, I2, I3

I4, I5, I6, I7Ig

1 x 6 yy otherwise

�R I1, I2, I3

I4, I5, I6, (1)n ID

1 x 6 yy=x otherwise

�R I1, I2, I3

I4, I5, I6Zadeh Im maxð1� x; minðx; yÞÞ QL I1, I3, I4

I8, (1)d Yuan 1 Ip 1� xþ x2y QL I1, I3, I4, I8

d Iq

y x ¼ 11� x x 6¼ 1; y 6¼ 11 x 6¼ 1; y ¼ 1

8<: QL I1, I3, I4, (1)

n 2


4. Solutions of (1) in S-, R- and QL-implications

Now we analyze under which conditions the three classes of fuzzy implications satisfy (1).

4.1. S-implications

Theorem 10. An S-implication I generated by a t-conorm S and a fuzzy negation n satisfies (1) iff the range of n is

a subset of the idempotent elements of S.

Proof. (: According to (8), Iðx; yÞ ¼ SðnðxÞ; yÞ and by S4,

Iðx; Iðx; yÞÞ ¼ SðnðxÞ; SðnðxÞ; yÞÞ ¼ SðSðnðxÞ; nðxÞÞ; yÞ:
If the range of n is a subset of the idempotent elements of S, then for all x 2 ½0; 1�, SðnðxÞ; nðxÞÞ ¼ nðxÞ andhence I satisfies (1).): By S1, for all x 2 ½0; 1�, Iðx; 0Þ ¼ SðnðxÞ; 0Þ ¼ nðxÞ and
Iðx; Iðx; 0ÞÞ ¼ SðnðxÞ; SðnðxÞ; 0ÞÞ ¼ SðnðxÞ; nðxÞÞ:
If I satisfies (1), then Iðx; 0Þ ¼ Iðx; Iðx; 0ÞÞ, and hence nðxÞ ¼ SðnðxÞ; nðxÞÞ, for all x 2 ½0; 1�. So for ally 2 rngðnÞ, Sðy; yÞ ¼ y. h
The next corollary is a strong result for the condition that n refers to a continuous fuzzy negation in theabove theorem.

Corollary 2. An S-implication I generated by a t-conorm S and a continuous fuzzy negation n satisfies (1) iff

S ¼ SM .

Proof. (: Straightforward.): If n is continuous, then the range of n is ½0; 1�. According to Theorem 10, I satisfies (1) iff the subset of

the idempotent elements of S is ½0; 1� i.e., S is an idempotent t-conorm. Since max is the only idempotent t-conorm, S ¼ SM . h

Example 1. Consider the Godel fuzzy negation NGðxÞ ¼1 x ¼ 00 otherwise

�.

Because the range of n is f0; 1g and for every t-conorm, 0 and 1 are idempotent elements, when the S-implication I is generated by the Godel fuzzy negation and any t-conorm, I ¼ IM and (1) holds.

Example 2. The dual t-conorm of the well-known left-continuous t-norm nilpotent minimum [7, Section 3] with

the strong fuzzy negation nðxÞ ¼ 1� x is Sðx; yÞ ¼ 1 xþ y P 1maxðx; yÞ otherwise

�.

The set of its idempotent elements is ½0; 0:5½[f1g. Consider the fuzzy negation8
nðxÞ ¼
1 x ¼ 0;

f ðxÞ x 2�0; 0:5�;0 x 2�0:5; 1�;

><>:

where f denotes a strictly decreasing mapping satisfying f ð0Þ ¼ 0:5, f ð0:5Þ ¼ 0, for all x 2 ½0; 0:5�. The range ofn is equal to the set of the idempotent elements of S. So the S-implication generated by S and n satisfies (1).

4.2. R-implications

Theorem 11. A mapping I : ½0; 1�2 ! ½0; 1� satisfies I3, I5, I6, right-continuity in the second argument and (1) iff Iis the Godel implication i.e.,

Iðx; yÞ ¼ Igðx; yÞ ¼1 x 6 y;

y otherwise:

�


Proof. (: It is easy to see that the Godel implication Ig satisfies I3, I5, I6, right-continuity in the second argu-ment and (1).

): Since I satisfies I3, I5, I6 and right-continuity in the second argument, according to Corollary 1, a left-continuous t-norm T can be generated through:

T ðx; yÞ ¼ minft 2 ½0; 1�jIðx; tÞP yg; 8ðx; yÞ 2 ½0; 1�2:

And if we denote IT ðx; yÞ ¼ supft 2 ½0; 1�jT ðx; tÞ 6 yg by (9), then according to the proof of Theorem 1.14 in[8], IT ¼ I .

We will first show that T ðx; xÞ ¼ x, for all x 2 ½0; 1�.Indeed, from the formula (12), we have for all x 2 ½0; 1�, T ðx; xÞ ¼ minft 2 ½0; 1�jIðx; tÞP xg. Obviously,

when t 2 ½x; 1�, Iðx; tÞ ¼ 1 P x holds by I6. Assume that there exists t0 2 ½0; x½ such that Iðx; t0ÞP x. Then byI6, Iðx; Iðx; t0ÞÞ ¼ 1. Since we have for all ðx; yÞ 2 ½0; 1�2, Iðx; yÞ ¼ Iðx; Iðx; yÞÞ, Iðx; t0Þ ¼ Iðx; Iðx; t0ÞÞ ¼ 1. Andby I6, x 6 t0. This is a contradiction with our assumption that t0 < x. So if t0 2 ft 2 ½0; 1�jIðx; tÞP xg, thent0 P x i.e.,

T ðx; xÞ ¼ minft 2 ½0; 1�jIðx; tÞP xg ¼ x; 8x 2 ½0; 1�:

That is to say, T ¼ T M . Hence,

Iðx; yÞ ¼ supft 2 ½0; 1�jminðx; tÞ 6 yg ¼1 x 6 y

y otherwise

�¼ Igðx; yÞ: �

Corollary 3. An R-implication I generated by a left-continuous t-norm T satisfies (1) iff T ¼ T M .

Proof. (: The R-implication generated by T ¼ T M through (9) is the Godel implication Ig and hence it sat-isfies (1).): According to Theorem 9, an R-implication I satisfying I3, I5, I6 and right-continuity in the second

argument is generated by a left-continuous t-norm T through (9). From Theorem 11, if I satisfies (1)additionally, it is the Godel implication Ig. Denote T Ig as the t-norm generated by Ig via (11) and IT Ig

as the R-implication generated by T Ig via (9), then according to Theorem 9 and its corollary, Ig ¼ IT Ig

i.e., Ig isgenerated by T Ig through (9) where T Ig is the t-norm generated by Ig through (11), that is T Ig ¼ T M . h

Remark 3. A continuous t-norm is of course left-continuous. So Theorem 11 and its corollary are strongresults for R-implications generated by a left-continuous t-norm. They are also proper for those R-implica-tions generated by a continuous t-norm.

Remark 4. In [2], Theorem 1, which was first proposed by Smets and Magrez [17], shows that a mappingI : ½0; 1�2 ! ½0; 1� is continuous and satisfies I3, I5 and I6 iff I is conjugate with Ia, which means the fuzzy impli-cations being conjugate with Ia are the ones and only the ones to be continuous and to satisfy I3, I5 and I6.Since TM is conjugate only with itself, by Proposition 12 in [2], we know that Ig is also conjugate with itself.Thus Theorem 11 shows that the fuzzy implications being conjugate with Ig are the ones and only the ones tobe right-continuous in the second argument and to satisfy I3, I5 and I6. Thus Theorem 11 is an analogousresult to Theorem 1 in [2].

4.3. QL-implications

According to (13), a QL-implication I is generated by a t-conorm S, a t-norm T and a strong fuzzy negationn. In this section, let n denote any strong fuzzy negation. We suppose both S and T are continuous and intendto investigate all possible combinations of S and T and find the sufficient and necessary conditions for I tosatisfy (1). In Table 2 we list all possible combinations of S and T.

Table 2Nine possible combinations of S and T to generate a QL-implications

S ¼ SM S is continuous Archimedean S is ordinal sum

T ¼ T M 1 4 7T is continuous 2 5 8ArchimedeanT is ordinal sum 3 6 9


Theorem 12. A QL-implication I generated by the t-conorm SM, a continuous t-norm T and a strong fuzzy

negation n satisfies (1) iff

(i) T ¼ T M or

(ii) T is the ordinal sum of a family f½am; bm�; T mg where f½am; bm�g is a countable family of non-overlapping,

closed, proper subintervals of ½0; 1� with each Tm being a continuous Archimedean t-norm, and for all

½am; bm�, bm 6 e, where e denotes the equilibrium point of n.

Proof. Here we have for all ðx; yÞ 2 ½0; 1�2, Iðx; yÞ ¼ maxðnðxÞ; T ðx; yÞÞ and

Iðx; Iðx; yÞÞ ¼ maxðnðxÞ; T ðx;maxðnðxÞ; T ðx; yÞÞÞÞ:
(:
(i) Because SM is distributive over TM, we obtain Iðx; Iðx; yÞÞ ¼ maxðnðxÞ;minðx;maxðnðxÞ;minðx; yÞÞÞÞ ¼minðmaxðnðxÞ; xÞ;maxðnðxÞ;maxðnðxÞ;minðx; yÞÞÞÞ ¼ minðmaxðnðxÞ; xÞ;maxðnðxÞ;minðx; yÞÞÞ by S4 andthe idempotency of max, ¼ minðmaxðnðxÞ; xÞÞ;minðmaxðnðxÞ; xÞ;maxðnðxÞ; yÞÞ ¼ minðmaxðnðxÞ; xÞ;maxðnðxÞ; yÞÞ by T4 and the idempotency of min ¼ maxðnðxÞ;minðx; yÞÞ ¼ Iðx; yÞ.

(ii) Suppose T is the ordinal sum of the corresponding family f½am; bm�; T mg with each bm 6 e. For an arbi-trary pair ðx0; y0Þ 2 ½0; 1�

2, if for all ½am; bm�, ðx0; y0Þ 62 ½am; bm�2, then according to (4), T ðx0; y0Þ ¼minðx0; y0Þ, from the proof above we can see that Iðx0; y0Þ ¼ Iðx0; Iðx0; y0ÞÞ. If on the contrary there exists½am; bm� such that ðx0; y0Þ 2 ½am; bm�2, then since for x0 6 bm 6 e, nðx0ÞP x0 holds, we havenðx0ÞP x0 P T ðx0; y0Þ and nðx0ÞP x0 P T ðx0;maxðnðx0Þ; T ðx0; y0ÞÞÞ. Thus Iðx0; Iðx0; y0ÞÞ ¼ nðx0Þ ¼Iðx0; y0Þ. Hence Iðx; Iðx; yÞÞ ¼ Iðx; yÞ, for all ðx; yÞ 2 ½0; 1�2.

): According to Theorem 4, if T is a continuous t-norm, then T is either TM, or continuous Archimedean,or the ordinal sum of a family f½am; bm�; T mg. Thus we prove this part through proving that if T is a continuousArchimedean t-norm or the ordinal sum of a family f½am; bm�; T mg and there exists ½am; bm� such that bm > e,then (1) does not hold.

(i) Let T be continuous Archimedean. Since n is continuous, there always exists an x0 2 �0; 1½ such thatnðx0Þ < x0. For y = 1, we have Iðx0; 1Þ ¼ maxðnðx0Þ; x0Þ ¼ x0 and Iðx0; Iðx0; 1ÞÞ ¼ maxðnðx0Þ; T ðx0; x0ÞÞ.Since T is continuous Archimedean, T ðx0; x0Þ < x0. Thus (1) does not hold.

(ii) Let T be the ordinal sum of a family f½am; bm�; T mg and suppose there exists ½am; bm� such that bm > e.Take y ¼ bm and x0 2 �maxðam; eÞ; bm½, then nðx0Þ < e < x0. Thus according to (4), Iðx0; bmÞ ¼maxðnðx0Þ; T ðx0; bmÞÞ ¼ max nðx0Þ; am þ ðbm � amÞT m

x0�ambm�am

; bm�ambm�am

� �� ¼ maxðnðx0Þ; x0Þ ¼ x0.

Since Tm is continuous Archimedean and according to (4), T ðx0; x0Þ ¼ am þ ðbm � amÞT mx0�ambm�am

; x0�ambm�am

� �< am þ ðbm � amÞ x0�am

bm�am¼ x0. Thus Iðx0; Iðx0; bmÞÞ ¼ maxðnðx0Þ; T ðx0; x0ÞÞ < x0. Hence (1) does not

hold. h

Remark 5. Theorem 12 treats the 3 possible combinations in the first column of Table 2:

CASE 1: a QL-implication generated by SM and TM satisfies (1);CASE 2: a QL-implication generated by SM and a continuous Archimedean T does not satisfy (1);


CASE 3: a QL-implication generated by SM and an ordinal sum T satisfies (1) iff T satisfies some extra con-dition as mentioned in Theorem 12.

Theorem 13. A QL-implication I generated by a continuous t-conorm S, the t-norm TM and a strong fuzzy nega-

tion n satisfies (1) iff

(i) S ¼ SM or

(ii) S is the ordinal sum of a family f½am; bm�; Smg where f½am; bm�g is a countable family of non-overlapping,

closed, proper subintervals of ½0; 1� with each Sm being a continuous Archimedean t-conorm, and for all

½am; bm�, am P e, where e denotes the equilibrium point of n.

Proof. Here we have for all ðx; yÞ 2 ½0; 1�2, Iðx; yÞ ¼ SðnðxÞ;minðx; yÞÞ and

Iðx; Iðx; yÞÞ ¼ SðnðxÞ;minðx; SðnðxÞ;minðx; yÞÞÞÞ

(:

(i) Straightforward from the proof of Theorem 12.(ii) Suppose S is the ordinal sum of the corresponding family f½am; bm�; Smg with each am P e. For an arbi-

trary nðx0Þ 2 ½0; 1�, if for all ½am; bm�, nðx0Þ 62 ½am; bm�, then according to (7), for all y 2 ½0; 1�, we haveIðx0; yÞ ¼ maxðnðx0Þ;minðx0; yÞÞ and

Iðx0; Iðx0; yÞÞ ¼ maxðnðx0Þ;minðx0;maxðnðx0Þ;minðx0; yÞÞÞÞ:
Thus according to the proof of Theorem 12, Iðx0; yÞ ¼ Iðx0; Iðx0; yÞÞ. If on the contrary there exists ½am; bm�
such that nðx0Þ 2 ½am; bm�, then we have minðx0; yÞ 6 x0 6 e 6 am 6 nðx0Þ, for all y 2 ½0; 1�.

(a) If minðx0; yÞ ¼ e, then x0 ¼ e ¼ nðx0Þ ¼ am. Thus according to (7), Iðx0; yÞ ¼ Sðe; eÞ ¼ e andIðx0; Iðx0; yÞÞ ¼ Sðe;minðx0; eÞÞ ¼ Sðe; eÞ ¼ e ¼ Iðx0; yÞ.

(b) If minðx0; yÞ < e, then minðx0; yÞ < am. According to (7), Iðx0; yÞ ¼ maxðnðx0Þ;minðx0; yÞÞ ¼ nðx0Þ andIðx0; Iðx0; yÞÞ ¼ Sðnðx0Þ;minðx0; nðx0ÞÞÞ ¼ Sðnðx0Þ; x0Þ. If x0 ¼ e, then am ¼ e ¼ nðx0Þ. Thus according to(7) Iðx0; Iðx0; yÞÞ ¼ Sðe; eÞ ¼ e ¼ nðx0Þ ¼ Iðx0; yÞ. If x0 < e, then x0 < am. Thus according to (7)Iðx0; Iðx0; yÞÞ ¼ maxðnðx0Þ; x0Þ ¼ nðx0Þ ¼ Iðx0; yÞ.

Hence Iðx; yÞ ¼ Iðx; Iðx; yÞÞ, for all ðx; yÞ 2 ½0; 1�2.): According to Theorem 8, if S is a continuous t-conorm, then S is either SM, or continuous

Archimedean, or the ordinal sum of a family f½am; bm�; Smg. Thus we prove this part through proving that if Sis continuous Archimedean or the ordinal sum of a family f½am; bm�; Smg and there exists ½am; bm� such thatam < e, then (1) does not hold.

(i) Let S be continuous Archimedean. Since n is continuous, there always exists nðx0Þ 2 �0; 1½ such thatnðx0Þ < x0. For y ¼ 0, we have Iðx0; 0Þ ¼ nðx0Þ and Iðx0; Iðx0; 0ÞÞ ¼ Sðnðx0Þ;minðx0; nðx0ÞÞÞ ¼Sðnðx0Þ; nðx0ÞÞ. Since S is continuous Archimedean, Sðnðx0Þ; nðx0ÞÞ > nðx0Þ. Thus (1) does not hold.

(ii) Let S be the ordinal sum of a family f½am; bm�; Smg and suppose there exists ½am; bm� such that am < e.Since n is continuous, there always exist nðx0Þ 2 �am;minðbm; eÞ½ and x0 > e > nðx0Þ. Take y ¼ 0, sinceSm is continuous Archimedean and according to (7),

Iðx0; Iðx0; 0ÞÞ ¼ Sðnðx0Þ;minðx0; nðx0ÞÞÞ ¼ Sðnðx0Þ; nðx0ÞÞ ¼ am þ ðbm � amÞSmnðx0Þ � am

bm � am;nðx0Þ � am

bm � am

� �

> am þ ðbm � amÞnðx0Þ � am

bm � am¼ nðx0Þ:

Since Iðx0; 0Þ ¼ nðx0Þ, (1) does not hold. h


Remark 6. Theorem 13 treats the three possible choices in the first row of Table 2: CASE 1 (so as mentioned inRemark 5); CASE 4: a QL-implication generated by a continuous Archimedean S and TM does not satisfy (1);CASE 7: a QL-implication generated by an ordinal sum S and TM satisfies (1) iff S satisfies some extra con-dition as mentioned in Theorem 13.

Lemma 1. A necessary condition for a QL-implication I generated by a continuous Archimedean t-conorm S, a

continuous t-norm T and a strong fuzzy negation n to satisfy (1) is T ðx; nðxÞÞ ¼ 0, for all x 2 ½0; 1�.

Proof. Suppose there exists x0 2 ½0; 1� such that T ðx0; nðx0ÞÞ > 0, then x0 > 0 and nðx0Þ > 0. Since n is strong,x0 > 0 implies nðx0Þ < 1. According to (5) and (6), there exists a strictly increasing generator g and its pseudo-inverse gð�1Þ such that

Sðnðx0Þ; T ðx0; nðx0ÞÞÞ ¼ gð�1Þðgðnðx0ÞÞ þ gðT ðx0; nðx0ÞÞÞÞ ¼g�1ðgðnðx0ÞÞ þ gðT ðx0; nðx0ÞÞÞÞ gðnðx0ÞÞ þ gðT ðx0; nðx0ÞÞÞ 2 ½0; gð1Þ�1 otherwise:

�

If Sðnðx0Þ; T ðx0; nðx0ÞÞÞ ¼ 1, then Sðnðx0Þ; T ðx0; nðx0ÞÞÞ > nðx0Þ.If Sðnðx0Þ; T ðx0; nðx0ÞÞÞ < 1, then

Sðnðx0Þ; T ðx0; nðx0ÞÞÞ ¼ g�1ðgðnðx0ÞÞ þ gðT ðx0; nðx0ÞÞÞÞ:
Since g and g�1 are strictly increasing, T ðx0; nðx0ÞÞ > 0 implies
g�1ðgðnðx0ÞÞ þ gðT ðx0; nðx0ÞÞÞÞ > g�1ðgðnðx0ÞÞÞ ¼ nðx0Þ;

namely, Sðnðx0Þ; T ðx0; nðx0ÞÞÞ > nðx0Þ. According to (13), Iðx0; 0Þ ¼ nðx0Þ and Iðx0; Iðx0; 0ÞÞ ¼ Sðnðx0Þ;T ðx0; nðx0ÞÞÞ. That is to say, Iðx0; 0Þ < Iðx0; Iðx0; 0ÞÞ, and hence, I does not satisfy (1). h

Remark 7. According to Theorem 3, a continuous t-norm T satisfying T ðx; nðxÞÞ ¼ 0, for all x 2 ½0; 1� iff T isconjugate with the Łukasiewicz t-norm T –L. Thus Lemma 1 treats the case 6 of Table 2: a QL-implication gen-erated by a continuous Archimedean S and an ordinal sum T does not satisfy (1).

Lemma 2. A necessary condition for a QL-implication I generated by a continuous Archimedean t-conorm S, a

continuous Archimedean t-norm T and a strong fuzzy negation n to satisfy (1) is Sðx; nðxÞÞ ¼ 1, for all x 2 ½0; 1�.

Proof. Suppose there exists x0 2 ½0; 1� such that Sðx0; nðx0ÞÞ < 1, then x0 2 �0; 1½. According to (2) and (3),

T ðx0; Sðnðx0Þ; x0ÞÞ ¼ f ð�1Þðf ðx0Þ þ f ðSðnðx0Þ; x0ÞÞÞ ¼f �1ðf ðx0Þ þ f ðSðnðx0Þ; x0ÞÞÞ f ðx0Þ þ f ðSðnðx0Þ; x0ÞÞ 2 ½0; f ð0Þ�;0 otherwise:

�

Since f is strictly decreasing and f ð1Þ ¼ 0, Sðnðx0Þ; x0Þ < 1 implies f ðSðnðx0Þ; x0ÞÞ > 0. If f ðx0Þþf ðSðnðx0Þ; x0ÞÞ 2 ½0; f ð0Þ�, then

T ðx0; Sðnðx0Þ; x0ÞÞ ¼ f �1ðf ðx0Þ þ f ðSðnðx0Þ; x0ÞÞÞ < f �1ðf ðx0ÞÞ ¼ x0:

If f ðx0Þ þ f ðSðnðx0Þ; x0ÞÞ 2 �f ð0Þ;1�, then T ðx0; Sðnðx0Þ; x0ÞÞ ¼ 0 < x0. Thus T ðx0; Sðnðx0Þ; x0ÞÞ < x0 alwaysholds.

According to (5) and (6), since Sðnðx0Þ; x0Þ < 1, it can be expressed by g�1ðgðnðx0ÞÞ þ gðx0ÞÞ and we havegðnðx0ÞÞ þ gðx0Þ 2 ½0; gð1Þ½.

Because T ðx0; Sðnðx0Þ; x0ÞÞ < x0 and g is strictly increasing, gðT ðx0; Sðnðx0Þ; x0ÞÞÞ < gðx0Þ. Thus

gðnðx0ÞÞ þ gðT ðx0; Sðnðx0Þ; x0ÞÞÞ < gðnðx0ÞÞ þ gðx0Þ;

which implies gðnðx0ÞÞ þ gðT ðx0; Sðnðx0Þ; x0ÞÞÞ 2 ½0; gð1Þ½ . Thus we obtain

Sðnðx0Þ; T ðx0; Sðnðx0Þ; x0ÞÞÞ ¼ g�1ðgðnðx0ÞÞ þ gðT ðx0; Sðnðx0Þ; x0ÞÞÞÞ:


Since g�1 is also strictly increasing and gðnðx0ÞÞ þ gðx0Þ 2 ½0; gð1Þ½,

g�1ðgðnðx0ÞÞ þ gðx0ÞÞ > g�1ðgðnðx0ÞÞ þ gðT ðx0; Sðnðx0Þ; x0ÞÞÞÞ;

namely, Sðnðx0Þ; x0Þ > Sðnðx0Þ; T ðx0; Sðnðx0Þ; x0ÞÞÞ. According to (13), Iðx0; 1Þ ¼ Sðnðx0Þ; x0Þ and Iðx0; Iðx0; 1ÞÞ ¼Sðnðx0Þ; T ðx0; Sðnðx0Þ; x0ÞÞÞ. That is to say, Iðx0; 1Þ > Iðx0; Iðx0; 1ÞÞ, and hence, I does not satisfy (1). h

Corollary 4. If a QL-implication I generated by a continuous Archimedean t-conorm S, a continuous Archime-dean t-norm T and a strong fuzzy negation n satisfies (1), then neither S nor T can be strict.

Proof. Straightforward from Lemmas 1, 2, Theorems 3 and 7. h

Lemma 3. If a QL-implication I generated by a continuous t-conorm S, a continuous Archimedean t-norm T and a

strong fuzzy negation n satisfies (1), then there does not exist a countable family f½am; bm�g of non-overlapping,

closed, proper subintervals of ½0; 1� such that S can be the ordinal sum of the corresponding family

f½am; bm�; Smg with each Sm being a continuous Archimedean t-norm.

Proof. According to (13), take y ¼ 1, then Iðx; 1Þ ¼ SðnðxÞ; xÞ and

Iðx; Iðx; 1ÞÞ ¼ SðnðxÞ; T ðx; SðnðxÞ; xÞÞÞ:
We will prove this lemma by contraposition.Let e denote the equilibrium point of n and suppose there exists a countable family f½am; bm�g such that S
can be the ordinal sum of the corresponding family f½am; bm�; Smg with each Sm being a continuousArchimedean t-norm.

If for all ½am; bm�, ½0; e� 6� ½am; bm�, then since n is continuous, there always exists nðx0Þ 2 �0; e½ such thatnðx0Þ 62 ½am; bm� and x0 > e > nðx0Þ. Thus Iðx0; 1Þ ¼ Sðnðx0Þ; x0Þ ¼ maxðnðx0Þ; x0Þ ¼ x0. And since T iscontinuous Archimedean and x0 2 �e; 1½, T ðx0; x0Þ < x0. Thus by (13) and (7), Iðx0; Iðx0; 1ÞÞ ¼Sðnðx0Þ; T ðx0; x0ÞÞ ¼ maxðnðx0Þ; T ðx0; x0ÞÞ < x0.

Hence (1) does not hold.If on the contrary there exists ½am; bm� such that ½0; e� � ½am; bm�, then we have am ¼ 0 and bm 2 ½e; 1½ . Take

x1 2 �bm; 1½ , then nðx1Þ < nðbmÞ 6 bm < x1.And since T is continuous Archimedean, T ðx1; x1Þ < x1.Thus by (13) and (7), we have Iðx1; 1Þ ¼ Sðnðx1Þ; x1Þ ¼ maxðnðx1Þ; x1Þ ¼ x1, and

Iðx1; Iðx1; 1ÞÞ ¼ Iðx1; x1Þ ¼ Sðnðx1Þ;T ðx1; x1ÞÞ ¼am þ ðbm � amÞSmðnðx1Þ�am

bm�am; T ðx1;x1Þ�am

bm�amÞ T ðx1; x1Þ 2 ½am; bm�;

maxðnðx1Þ;T ðx1; x1ÞÞ otherwise:

(

If T ðx1; x1Þ 2 ½am; bm�, then Iðx1; Iðx1; 1ÞÞ 2 ½am; bm�, i.e., Iðx1; Iðx1; 1ÞÞ < x1.If T ðx1; x1Þ 62 ½am; bm�, then Iðx1; Iðx1; 1ÞÞ ¼ maxðnðx1Þ; T ðx1; x1ÞÞ < x1. Therefore we always have

Iðx1; Iðx1; 1ÞÞ < Iðx1; 1Þ.Hence (1) does not hold. h

Remark 8. Lemma 3 treats the case 8 of Table 2: a QL-implication generated by an ordinal sum S and a con-tinuous Archimedean T does not satisfy (1).

Hence besides the theorems and lemmas above, there are only two combinations of S and T to be investi-gated: the case 5 and the case 9. And from Corollary 4, we know that for the case 5, it is necessary that both S

and T are nilpotent.

Theorem 14. Let / denote an automorphism of the unit interval. A QL-implication I generated by the nilpotent

t-conorm which is expressed by Sðx; yÞ ¼ /�1ðminð/ðxÞ þ /ðyÞ; 1ÞÞ, the nilpotent t-norm which is expressed by

T ðx; yÞ ¼ /�1ðmaxð/ðxÞ þ /ðyÞ � 1; 0ÞÞ and a strong fuzzy negation n satisfies (1) iff nðxÞ ¼ /�1ð1� /ðxÞÞ, for

all x 2 ½0; 1�.


Proof. According to Lemmas 1 and 2, I satisfies (1) iff for all x 2 ½0; 1�, T ðx; nðxÞÞ ¼ 0 and SðnðxÞ; xÞ ¼ 1. Andaccording to Theorems 3 and 7, T ðx; nðxÞÞ ¼ 0 and SðnðxÞ; xÞ ¼ 1 iff nðxÞ ¼ /�1ð1� /ðxÞÞ. h

Remark 9. Actually, if there exists an automorphism / of the unit interval such that the QL-implication I isgenerated by the t-conorm Sðx; yÞ ¼ /�1ðminð/ðxÞ þ /ðyÞ; 1ÞÞ, the t-norm T ðx; yÞ ¼ /�1ðmaxð/ðxÞþ/ðyÞ � 1; 0ÞÞ and the strong fuzzy negation nðxÞ ¼ /�1ð1� /ðxÞÞ, then Iðx; yÞ ¼ SðnðxÞ; T ðx; yÞÞ ¼maxð/�1ð1� /ðxÞÞ; yÞ ¼ maxðnðxÞ; yÞ, i.e., I is an S-implication generated by SM and the strong fuzzy nega-tion n, and hence according to Corollary 2, it satisfies (1).

Theorem 15. Let u and / denote two different automorphisms of the unit interval. The QL-implication I gener-

ated by the nilpotent t-conorm which is expressed by Sðx; yÞ ¼ u�1ðminðuðxÞ þ uðyÞ; 1ÞÞ, the nilpotent t-norm

which is expressed by T ðx; yÞ ¼ /�1ðmaxð/ðxÞ þ /ðyÞ � 1; 0ÞÞ and a strong fuzzy negation n satisfies (1) iff n

satisfies

u�1ð1� uðxÞÞ 6 nðxÞ 6 /�1ð1� /ðxÞÞ; 8x 2 ½0; 1� ð14Þ
and I is expressed by
Iðx; yÞ ¼nðxÞ T ðx; yÞ ¼ 0;

y 0 < T ðx; yÞ < u�1ð1� uðnðxÞÞÞ;1 T ðx; yÞP u�1ð1� uðnðxÞÞÞ:

8><>: ð15Þ

Proof. Here we have for all ðx; yÞ 2 ½0; 1�2, Iðx; yÞ ¼ SðnðxÞ; T ðx; yÞÞ and

Iðx; Iðx; yÞÞ ¼ SðnðxÞ; T ðx; SðnðxÞ; T ðx; yÞÞÞÞ: ð16Þ
): Suppose that I satisfies (1), then according to Lemmas 1, 2, Theorems 3 and 7, we have
u�1ð1� uðxÞÞ 6 nðxÞ 6 /�1ð1� /ðxÞÞ:
Hence (14) holds. Next we will prove that I satisfy (15).
According to (13), T ðx; yÞ ¼ 0 implies Iðx; yÞ ¼ nðxÞ. And T ðx; yÞP u�1ð1� uðnðxÞÞÞ implies Iðx; yÞ ¼ 1.Thus in order to prove that I satisfies (15), we only need to prove:

0 < T ðx; yÞ < u�1ð1� uðnðxÞÞÞ ) Iðx; yÞ ¼ y:

Indeed, if T ðx; yÞ < u�1ð1� uðnðxÞÞÞ, then

Iðx; yÞ ¼ u�1ðuðnðxÞÞ þ uðT ðx; yÞÞÞ < 1:

If I satisfies (1), then Iðx; Iðx; yÞÞ < 1. Thus according to (16), we have

Iðx; Iðx; yÞÞ ¼ u�1ðuðnðxÞÞ þ uðT ðx;u�1ðuðnðxÞÞ þ uðT ðx; yÞÞÞÞÞÞ:
Since Iðx; yÞ ¼ Iðx; Iðx; yÞÞ and both u and u�1 are strictly increasing mappings, we have
T ðx; yÞ ¼ T ðx;u�1ðuðnðxÞÞ þ uðT ðx; yÞÞÞÞ:
Because T ðx; yÞ > 0, T ðx; yÞ ¼ /�1ð/ðxÞ þ /ðyÞ � 1Þ. And
T ðx;u�1ðuðnðxÞÞ þ uðT ðx; yÞÞÞÞ ¼ /�1ð/ðxÞ þ /ðu�1ðuðnðxÞÞ þ uðT ðx; yÞÞÞÞ � 1Þ:
Since both / and /�1 are strictly increasing mappings, we have y ¼ u�1ðuðnðxÞÞ þ uðT ðx; yÞÞÞ, namelyIðx; yÞ ¼ y. Hence (15) holds.(: Suppose n and I satisfy (14) and (15), then we will prove that I satisfies (1). According to (15), if
T ðx; yÞ ¼ 0, then Iðx; yÞ ¼ nðxÞ. And

Iðx; Iðx; yÞÞ ¼ Iðx; nðxÞÞ ¼ SðnðxÞ; T ðx; nðxÞÞÞ:
Since n satisfies (14), T ðx; nðxÞÞ ¼ 0, for all x 2 ½0; 1�. Thus
SðnðxÞ; T ðx; nðxÞÞÞ ¼ SðnðxÞ; 0Þ ¼ nðxÞ ¼ Iðx; yÞ:


If T ðx; yÞP u�1ð1� uðnðxÞÞÞ, then Iðx; yÞ ¼ 1. Thus Iðx; Iðx; yÞÞ ¼ Iðx; 1Þ ¼ SðnðxÞ; xÞ. Since n satisfies (14),SðnðxÞ; xÞ ¼ 1 for all x 2 ½0; 1�. Thus Iðx; Iðx; yÞÞ ¼ 1 ¼ Iðx; yÞ.

If 0 < T ðx; yÞ < u�1ð1� uðnðxÞÞÞ, then Iðx; yÞ ¼ y. Thus Iðx; Iðx; yÞÞ ¼ Iðx; yÞ. Hence I satisfies (1). h

Remark 10. Corollary 4, Theorems 14 and 15 treat the case 5 of Table 2: a QL-implication generated by acontinuous Archimedean S and a continuous Archimedean T satisfies (1) providing both are nilpotent andsatisfy some extra conditions as stated.

Theorem 16. Let f½am; bm�g and f½cm; dm�g denote two countable families of non-overlapping, closed, proper sub-

intervals of ½0; 1�. And let T be a t-norm which is the ordinal sum of the corresponding family f½am; bm�; T mg with

each Tm being a continuous Archimedean t-norm and S be a t-conorm which is the ordinal sum of the correspond-

ing family f½cm; dm�; Smg with each Sm being a continuous Archimedean t-conorm. Then the QL-implication I gen-

erated by S, T and a strong fuzzy negation n satisfies (1) iff for all ½am; bm�, bm 6 e and for all ½cm; dm�, cm P e,where e denotes the equilibrium point of n.

Proof. We will first derive some special instances of Iðx; Iðx; yÞÞ.According to (13) and (16), for y ¼ 1, we obtain Iðx; 1Þ ¼ SðnðxÞ; xÞ and Iðx; Iðx; 1ÞÞ ¼ SðnðxÞ;

T ðx; SðnðxÞ; xÞÞÞ. And for y ¼ 0, we obtain Iðx; 0Þ ¼ nðxÞ and Iðx; Iðx; 0ÞÞ ¼ SðnðxÞ; T ðx; nðxÞÞÞ.(: In order to prove the ‘(’ part, we will consider three cases according to the position of x w.r.t. e:

(i) For all x < e, we have nðxÞ > e, x < nðxÞ, and x < cm, for all cm. Since for all y 2 ½0; 1�, T ðx; yÞ 6 x < cm

and T ðx; SðnðxÞ; T ðx; yÞÞÞ 6 x < cm, according to (7), we have:

Iðx; yÞ ¼ SðnðxÞ; T ðx; yÞÞ ¼ maxðnðxÞ; T ðx; yÞÞ ¼ nðxÞand

Iðx; Iðx; yÞÞ ¼ SðnðxÞ; T ðx; SðnðxÞ; T ðx; yÞÞÞÞ ¼ maxðnðxÞ; T ðx; SðnðxÞ; T ðx; yÞÞÞÞ ¼ nðxÞ:Thus Iðx; yÞ ¼ Iðx; Iðx; yÞÞ.

(ii) For all x > e, we have nðxÞ < e, x > nðxÞ, x > bm, for all bm and nðxÞ < cm, for all cm. Then by (7) and (4),for all y 2 ½0; 1�, Iðx; yÞ ¼ maxðnðxÞ;minðx; yÞÞ and

Iðx; Iðx; yÞÞ ¼ maxðnðxÞ;minðx;maxðnðxÞ;minðx; yÞÞÞÞ:Thus according to the proof of Theorem 12, Iðx; yÞ ¼ Iðx; Iðx; yÞÞ.

(iii) For x ¼ e, we have nðxÞ ¼ e and Iðx; yÞ ¼ Iðe; yÞ ¼ Sðe; T ðe; yÞÞ.Two subcases will be considered depend-ing now on the position of the variable y:
(1) y 6 e.If for all ½am; bm�, ðe; yÞ 62 ½am; bm�2, then according to (4), T ðe; yÞ ¼ minðe; yÞ ¼ y.If on the contrary there exists ½am; bm� such that ðe; yÞ 2 ½am; bm�2, then bm ¼ e. According to (4),
T ðe; yÞ ¼ am þ ðe� amÞT mðe� am

e� am;y � am

e� amÞ ¼ y:

That is to say, T ðe; yÞ ¼ y; for all y 6 e. Thus Iðx; yÞ ¼ Sðe; yÞ:And if for all ½cm; dm�, ðe; yÞ 62 ½cm; dm�2, then according to (7), Sðe; yÞ ¼ maxðe; yÞ ¼ e.If on the contrary there exists ½cm; dm� such that ðe; yÞ 2 ½cm; dm�2, then cm ¼ y ¼ e. According to (7),

Sðe; yÞ ¼ eþ ðdm � eÞSme� e

dm � e;

y � edm � e

� �¼ y ¼ e:

That is to say, Iðx; yÞ ¼ Iðe; yÞ ¼ e, for all y 6 e.(2) y > e.In this case y > bm. Thus according to (4), T ðe; yÞ ¼ minðe; yÞ ¼ e and hence Iðx; yÞ ¼ Sðe; T ðe; yÞÞ ¼Sðe; eÞ.Now we look at the position of cm (P e) w.r.t. e.


If cm > e, then according to (7), Sðe; eÞ ¼ maxðe; eÞ ¼ e. If cm ¼ e, then according to (7),

Sðe; eÞ ¼ eþ ðdm � eÞSme� e

dm � e;

e� edm � e

� �¼ e:

That is to say, Iðe; yÞ ¼ e, for all y > e.Hence Iðe; yÞ ¼ e, for all y 2 ½0; 1�. Therefore we have

Iðx; Iðx; yÞÞ ¼ Iðe; Iðe; yÞÞ ¼ Sðe; T ðe; Iðe; yÞÞÞ ¼ Sðe; T ðe; eÞÞ:Finally we consider the position of the upper bound bm (6 e) w.r.t. e.If bm < e, then according to (4), T ðe; eÞ ¼ minðe; eÞ ¼ e. If bm ¼ e, then according to (4),

T ðe; eÞ ¼ am þ ðe� amÞT me� am

e� am;e� am

e� am

� �¼ e:

Thus Sðe; T ðe; eÞÞ ¼ Sðe; eÞ ¼ e, according to the proof above. Therefore Iðx; Iðx; yÞÞ ¼ e ¼ Iðe; yÞ ¼Iðx; yÞ.

Hence we can conclude that Iðx; Iðx; yÞÞ ¼ Iðx; yÞ, for all ðx; yÞ 2 ½0; 1�2, i.e., I satisfies (1).): The reverse implication (‘Rightarrow’) will be proved by contraposition.

(i) First assume there exists ½am; bm� such that bm > e.If for all ½cm; dm�, cm > nðbmÞ, then since n is continu-ous, there exists x0 such that nðx0Þ 2 �nðbmÞ;minðcm; eÞ½ and x0 2 �maxðam; eÞ; bm½ . Thus Iðx0; 1Þ ¼Sðnðx0Þ; x0Þ ¼ maxðnðx0Þ; x0Þ ¼ x0 and Iðx0; Iðx0; 1ÞÞ ¼ Sðnðx0Þ; T ðx0; Sðnðx0Þ; x0ÞÞÞ ¼ Sðnðx0Þ; T ðx0; x0ÞÞ ¼maxðnðx0Þ; T ðx0; x0ÞÞ:Since Tm is continuous Archimedean and according to (4),

T ðx0; x0Þ ¼ am þ ðbm � amÞT mx0 � am

bm � am;

x0 � am

bm � am

� �< x0:

Thus Iðx0; Iðx0; 1ÞÞ < x0. Hence (1) does not hold. If on the contrary there exists ½cm; dm� such thatcm 6 nðbmÞ, then cm < e. Since n is continuous, there exists x1 such that nðx1Þ 2 �cm;minðdm; eÞ½ andx1 2 �bm; 1½ . Thus Iðx1; Iðx1; 0ÞÞ ¼ Sðnðx1Þ; T ðx1; nðx1ÞÞÞ ¼ Sðnðx1Þ;minðx1; nðx1ÞÞÞ ¼ Sðnðx1Þ; nðx1ÞÞ. SinceSm is continuous Archimedean and according to (7),

Sðnðx1Þ; nðx1ÞÞ ¼ cm þ ðdm � cmÞSmnðx1Þ � cm

dm � cm;nðx1Þ � cm

dm � cm

� �> nðx1Þ:

And Iðx1; 0Þ ¼ nðx1Þ < Iðx1; Iðx1; 0ÞÞ. Thus (1) does not hold. So we have proved that if there exists½am; bm� such that bm > e, then (1) does not hold.

(ii) Second assume that there exists ½cm; dm� such that cm < e and for all ½am; bm�, bm 6 e.Since n is continuous, there exists x0 such that nðx0Þ 2 �cm;minðdm; eÞ½ and x0 > bm. Then Iðx0; 0Þ ¼ nðx0Þand according to (4), Iðx0; Iðx0; 0ÞÞ ¼ Sðnðx0Þ; T ðx0; nðx0ÞÞÞ ¼ Sðnðx0Þ;minðx0; nðx0ÞÞÞ ¼ Sðnðx0Þ; nðx0ÞÞ.Since Sm is continuous Archimedean and according to (7),

Sðnðx0Þ; nðx0ÞÞ ¼ cm þ ðdm � cmÞSmnðx0Þ � cm

dm � cm;nðx0Þ � cm

dm � cm

� �> nðx0Þ:

Thus (1) does not hold. h

Remark 11. Theorem 16 treats the case 9 of Table 2: a QL-implication generated by an ordinal sum S and anordinal sum T satisfies (1) providing both satisfy some extra conditions as stated.


5. Conclusions

Iterative boolean-like laws are proposed by the authors in [1]. In this paper, we studied an iterative boolean-like law in which fuzzy implications are concerned, namely whether Iðx; Iðx; yÞÞ ¼ Iðx; yÞ holds for allðx; yÞ 2 ½0; 1�2 or not where I denotes a fuzzy implication derived from classical logic, which is generated byt-norms, t-conorms and fuzzy negations.

In Section 4 we gave sufficient and necessary conditions for an S-implication generated by any t-conormand any fuzzy negation to satisfy (1) in Theorem 10 and its corollary, for an R-implication generated by a leftcontinuous t-norm in to satisfy (1) in Theorem 11 and its corollary and for a QL-implication generated by acontinuous t-conorm, a continuous t-norm and a strong fuzzy negation to satisfy (1) from Theorem 12–16.

The standard t-norm TM and the standard t-conorm SM play important roles in the results of the investi-gated Eq. (1). An S-implication generated by a t-conorm S and a continuous fuzzy negation n satisfies Eq. (1)iff S is SM. And an R-implication generated by a left-continuous t-norm T satisfies Eq. (1) iff T is TM. But for aQL-implication generated by a fuzzy negation n, a t-norm T and a t-conorm S to satisfy Eq. (1), it is sufficientbut not necessary for T to be TM and S to be SM. Instead, the equilibrium point of the strong fuzzy negationplay an important role in the solutions for QL-implications to satisfy (1).

Acknowledgement

The authors want to thank the referees for their helpful comments that substantially improved this paper.

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on the characterizations of fuzzy implications satisfying

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