Characterisation of main classes of fuzzy relations using fuzzy modal operators

Fuzzy Sets and Systems 152 (2005) 223 – 247 www.elsevier.com/locate/fss

Characterisation of main classes of fuzzy relations using fuzzy modal operators Anna Maria Radzikowskaa,∗ , Etienne E. Kerreb a Faculty of Mathematics and Information Science, Warsaw University of Technology, Plac Politechniki 1,

00-661 Warsaw, Poland b Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), B-9000 Gent, Belgium

Received 4 February 2002; received in revised form 23 July 2004; accepted 10 September 2004 Available online 2 October 2004

Abstract Fuzzy modal operators express interactions between binary fuzzy relations and fuzzy sets. Each of these operators is determined by a binary fuzzy relation (on a given universe) and transforms one fuzzy set (in this universe) to another one. In this paper, we define several fuzzy modal operators and provide characterisations of main classes of binary fuzzy relations by means of these operators. Interpretation of these characterisations is presented in the context of fuzzy modal logics. We show that these characterisations constitute the basis for determining characteristic axioms of particular classes of fuzzy modal logics. © 2004 Elsevier B.V. All rights reserved. Keywords: Fuzzy relations; Modal operators; Fuzzy logical operators; Modal operators; Modal logics; Fuzzy rough sets; Fuzzy mathematical morphology

1. Introduction and motivation In many research domains we deal with a set of objects related to each other in some ways. These relationships reflect in themselves some information and, on the other hand, are tools for deriving new information. Interactions between binary relations (on a universe X of discourse) and subsets of X may be expressed by means of modal operators. Each of these operators, based on a binary relation on X, transforms a subset of X into other subset of X, i.e. this is a mapping
: R(X) × 2X → 2X , where ∗ Corresponding author. Tel.: +48 22 660 7307; fax: +48 22 625 7460.

E-mail addresses: [email protected] (A.M. Radzikowska), [email protected] (E.E. Kerre). 0165-0114/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2004.09.005

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R(X) stands for the family of all binary relations on X. A general theory of classical abstract operators of this form was developed in [6,10–12,31,32] in the context of a broadly understood theory of information. These operators, however, are widely used in many other domains related to theoretical as well as practical aspects of computer science. Recall two very popular modal operators: for R ⊆ W × W and for any A⊆W

[R]A = {x ∈ W : [x]R ⊆ A},

(1)

R A = {x ∈ W : [x]R ∩ A = ∅},

(2)

where [x]R = {x  ∈ W : (x, x  ) ∈ R}. Note that (1) and (2) straightforwardly correspond to box and diamond operators of modal logics [6,26], dilation and erosion operations well-known in mathematical morphology [41], lower and upper rough approximation operations used in the theory of rough sets [34,35], images (of subsets of X) under relation on X studied in relational calculus, as well as interior and closure topological operators. Depending on particular application domain, modal operators are considered with respect to specific classes of relations. In particular, in the theory of rough sets we usually deal with operators (1)–(2) based on equivalence (or tolerance) relations. Similarly, in mathematical morphology some specific properties of structuring elements B (underlying erosion and dilation operations) are often assumed, which in turn determine properties of the relations derived from B. As regards modal logics, however, different classes of these systems are determined by the specific classes of their accessibility relations. It is worthwhile then to consider characterisations of various classes of relations in order to obtain characteristic axioms of the respective logics. When we deal with real-life problems, fuzzy structures often provide much more adequate models of information than classical structures. If the available data are fuzzy then relationships among objects (in the domain of discourse) should be naturally modelled by binary fuzzy relations. This, in turn, leads to fuzzy modal operators. Some pioneering work on similar structures were due to Bandler and Kohout [1,2] and Zadeh [46]. However, particular classes of modal operators were investigated in the literature in different contexts (e.g. [3,4,21]). In particular, they are applied in many-valued modal logics [15,16], fuzzy modal logics [13,17,19], fuzzy mathematical morphology [7,28,29], fuzzy rough sets [9,14,27,38,40,42], fuzzy topology [22,24]. In particular domains these operators are considered in different underlying algebraic structures. To be precise, Fitting discussed many-valued logics on the basis of finite Heyting algebras [15,16]. Therefore, the box operator is defined there with respect to the residuum of the lattice meet operation. On the other hand, Thiele [42,43] considered box and diamond operators taking specific classes of fuzzy relations, namely symmetric, reflexive and fuzzy transitive relations. Similar approaches can be found in fuzzy mathematical morphology or the theory of fuzzy rough sets. Finally, some authors considered these operators taking specific logical connectives. For example, Godo and Rodriguez [17] discussed fuzzy modal logics taking the Łukasiewicz connectives for interpretation of conjunction and implication. It is worth emphasizing that, depending on the domain of application, specific properties of fuzzy modal operators are investigated. In consequence, although fuzzy generalisations of these operators are very popular in the literature, they are discussed in different algebraic frameworks and from different points of view. It is well-known in the literature that modal operators may be used to characterise properties of binary relations [8,20,31,32]. In particular, for every binary relation R on W and for every A ⊆ W , we have:

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225

• R is reflexive iff [R]A ⊆ A iff A ⊆ R A, • R is symmetric iff R [R]A ⊆ A iff A ⊆ [R] R A, • R is transitive iff [R]A ⊆ [R][R]A iff R R A ⊆ R A. In this paper, we essentially address the following problem: Can fuzzy modal operators be used to characterise properties of (some) fuzzy relations? To what extent classical characterisations can be generalised to the fuzzy case? Extending preliminary results presented in [36,39], we define six fuzzy modal operators and provide characterisations of main classes of binary fuzzy relations by means of these operators. These operators are defined using an arbitrary left-continuous t-norm T , its residuum I and a negator N(x) = I (x, 0). As an example of applications of these characterisations, we present their interpretations in fuzzy modal logics. We formulate characteristic axioms for particular classes of fuzzy modal logics determined by specific classes of accessibility relations. The paper is structured as follows. In Section 2 we recall basic definitions and properties of fuzzy logical operators and binary fuzzy relations. In Section 3, we define fuzzy modal operators, investigate their basic properties and show their interpretations in some research areas: fuzzy rough sets, fuzzy morphology, fuzzy topology and fuzzy modal logics. The main results of this paper are presented in Section 4, where characterisations of main classes of binary fuzzy relations are provided. The characteristic axioms of the respective classes of fuzzy modal logics are given. Some concluding remarks complete the paper. To increase the readability of the paper, proofs of propositions from Section 4 were relegated to Appendix.

2. Preliminaries In this section, we recall definitions of basic fuzzy logical operators and binary fuzzy relations (see, e.g. [23,25]). 2.1. Fuzzy logical operators A triangular norm (t-norm, for short) is an increasing, associative, and commutative mapping T : [0, 1]2 → [0, 1] that satisfies the boundary condition T (x, 1) = x for every x ∈ [0, 1]. The most popular continuous t-norms are: • the standard min operator TZ (x, y) = min(x, y) (the largest t-norm), • the algebraic product TP (x, y) = x · y, • the Łukasiewicz t-norm TL (x, y) = max(0, x + y − 1). A t-norm T is called left-continuous iff it has left-continuous partial mappings. The class of all leftcontinuous t-norms will be denoted by Tlc . An implicator is a mapping I : [0, 1]2 → [0, 1] with decreasing first and increasing second partial mappings, satisfying I (0, 0) = I (0, 1) = I (1, 1) = 1 and I (1, 0) = 0. The most popular implicators are: • • • •

the Łukasiewicz implicator IL (x, y) = min(1, 1 − x + y), the Kleene–Dienes implicator IKD (x, y) = max(1 − x, y), the Reichenbach implicator IR (x, y) = 1 − x + x · y, the Gödel implicator IG (x, y) = 1 for x
y and IG (x, y) = y elsewhere,

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• the Gaines implicator Ig (x, y) = 1 for x
y and Ig (x, y) =

y x

elsewhere.

For T ∈ Tlc , the residual implicator based on T (the residuum of T) is defined as IT (x, y) = sup{ ∈ [0, 1] : T (x, )
y}

for all x, y ∈ [0, 1].

For example, IL , IG and Ig are residual implicators based on TL , TZ and TP , respectively. A negator is a decreasing mapping N : [0, 1] → [0, 1] that satisfies N(1) = 0 and N(0) = 1. For an implicator I , the mapping given by: NI (x) = I (x, 0), x ∈ [0, 1], is a negator, called a negator induced by I. If I is the residuum of a left-continuous t-norm T then the negator induced by I will be denoted by NT . Let us recall several basic properties of residual implicators (see, for example [18,44]). Lemma 1. Let T ∈ Tlc , I be its residuum and let N = NT . Then for all x, y, z ∈ [0, 1] and for any families (xi )i∈I , (yi )i∈I ⊆ [0, 1], we have (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) (xiii) (xiv) (xv)

I (1, x) = x, I (x, y) = 1 iff x
y, T (x, I (x, y))
y, I (y, inf i∈I xi ) = inf i∈I I (y, xi ), I (supi∈I xi , y) = inf i∈I I (xi , y), I (inf i∈I xi , y)  supi∈I I (xi , y), I (inf i∈I xi , supi∈I yi )  supi∈I I (xi , yi ), T (supi∈I xi , supi∈I xi ) = supi∈I T (xi , xi ), I (x, I (y, z)) = I (T (x, y), z), I (I (x, y), y)  x, I (x, T (x, y))  y, I (x, N(y)) = N(T (x, y)), I (x, y)
I (N(y), N(x)), N(N(x))  x, N(supi∈I xi ) = inf i∈I N(xi ).

2.2. Fuzzy sets and fuzzy relations 2.2.1. Fuzzy sets Let X be a nonempty universe. By a fuzzy set in X we mean any mapping F : X → [0, 1]. The family of all fuzzy sets in X will be denoted by F (X). The operations of classical intersection, union, and complementation are generalized in the following way. Let T and N be a t-norm and a negator, respectively. By a T-intersection of A, B ∈ F (X) we mean the fuzzy set A ∩T B in X defined by (A ∩T B)(x) = T (A(x), B(x)) for every x ∈ X. The Zadeh union of A, B ∈ F (X) is the fuzzy set A ∪ B given by: (A ∪ B)(x) = max(A(x), B(x)) for every x ∈ X.

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For an arbitrary indexed family (Ai )i∈I of fuzzy sets in X, we will write denote the fuzzy sets  
Ai (x) = sup Ai (x) i∈I i∈I  for every x ∈ X.  Ai (x) = inf Ai (x) i∈I




i∈I

227

Ai and



i∈I

Ai to

i∈I

An N-complementation of A ∈ F (X) is the fuzzy set coN A ∈ F (X) defined by (coN A)(x) = N(A(x))

for every x ∈ X.

For an implicator I and two fuzzy sets A, B ∈ F (X), we will write A ⇒I B to denote the following fuzzy set in X: (A ⇒I B)(x) = I (A(x), B(x)) for every x ∈ X. Given two fuzzy sets A and B in X, we will write A ⊆ B iff for every x ∈ X, A(x)
B(x) (⊆ is also called Zadeh’s inclusion). 2.2.2. Fuzzy relations A fuzzy relation on X is any mapping R : Xn → [0, 1], n  2. If n = 2 then R is called a binary fuzzy relation. The family of all binary fuzzy relations on X will be denoted by R(X). Definition 1. Let R be a binary fuzzy relation on X. We say that R is • serial iff supy∈X R(x, y) = 1 for every x ∈ X, • reflexive iff R(x, x) = 1 for every x ∈ X, • weakly reflexive iff supy∈X R(x, y) = R(x, x) for every x ∈ X, • irreflexive iff R(x, x) = 0 for every x ∈ X, • weakly irreflexive iff inf y∈X R(x, y) = R(x, x) for every x ∈ X, • symmetric iff R(x, y) = R(y, x) for all x, y ∈ X. Let T and N be a t-norm and a negator, respectively. R is called • right N-asymmetric iff R(x, y)
N(R(y, x)) for all x, y ∈ X, • left N-asymmetric iff N(R(x, y))
R(y, x) for all x, y ∈ X, • T -Euclidean iff T (R(z, x), R(z, y))
R(x, y) for all x, y, z ∈ X, • T -transitive iff T (R(x, z), R(z, y))
R(x, y) for all x, y, z ∈ X, • T -cotransitive iff coN R is T -transitive, where T ∈ Tlc and N = NT . If R is reflexive, symmetric, and T -transitive then it is called a T-equivalence relation. It is worth noting that every reflexive relation is serial, but not conversely. Also, every symmetric and T -Euclidean relation is T -transitive. Given a binary fuzzy relation R on X and x ∈ X, we will write xR to denote the R-afterset of x [23], that is the fuzzy set (xR)(y) = R(x, y), y ∈ X.

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Let I and T be an implicator and a t-norm, respectively. For all A, B ∈ F (X), let us define the following two binary fuzzy relations on F (X): I nclI (A, B) = inf I (A(x), B(x)), x∈X

Comp T (A, B) = sup T (A(x), B(x)). x∈X

I nclI (A, B) reflects the degree to which A is included in B, whereas Comp T (A, B) represents the degree to which A and B overlap. The binary fuzzy relations I nclI and CompT are called I-inclusion and Tcompatibility, respectively. If I is the residuum of a left-continuous t-norm T then I -inclusion will be referred to as T -inclusion and denoted by I nclT .

3. Fuzzy modal operators Definition 2. A fuzzy modal operator is a mapping
: R(X) × F (X) → F (X). In the following we will focus on a subclass of fuzzy modal operators determined by left-continuous t-norms. These operators will be referred to as T-modal operators. Definition 3. Let T ∈ Tlc , I be its residuum and N = NT . Moreover, let R ∈ R(X). Define the following T-modal operators: for every A ∈ F (X) and for every x ∈ X, (O.1) (O.2) (O.3) (O.4) (O.5) (O.6)

([R]T A)(x) = inf y∈X I (R(x, y), A(y)), ( R T A)(x) = supy∈X T (R(x, y), A(y)), ('R (T A)(x) = inf y∈X I (A(y), R(x, y)), ( R

T A)(x) = supy∈X T (N(R(x, y)), N(A(y))), ('R]T A)(x) = inf y∈X I (N(R(x, y)), A(y)), R T A)(x) = supy∈X T (N(R(x, y)), A(y)).

It is easily noted that the first two operators are fuzzy counterparts of the necessity and possibility operators well-known in modal logics [6]. Also, these two operators correspond to I -lower and T -upper fuzzy rough approximators of A in the space (X, R) [38]. Furthermore, T -modal operators (O.3) and (O.4) are fuzzy generalisations of the sufficiency and impossibility operators introduced in [20]. Note also that for every R ∈ R(X), for every A ∈ F (X), and for every x ∈ X, ([R]T A)(x) = I nclT (xR, A) ('R (T A)(x) = I nclT (A, xR) ('R]T A)(x) = I nclT (xR N , A)

( R T A)(x) = CompT (xR, A), R

T A(x) = CompT (coN A, xR N ), ( R T A)(x) = CompT (xR N , A),

where R N = coN R. The following lemma provides basic properties of T -modal operators defined above.

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Lemma 2. For every T ∈ Tlc , (i) Specific cases: [R]T X = 'R]T X = X, R T ∅ = R T ∅ = ∅, 'R (T ∅ = X, R

T X = ∅. (ii) Monotonicity: For every A, B ∈ F (X), if A ⊆ B then [R]T A ⊆ [R]T B

R T A ⊆ R T B, R

T A ⊇ R

T B, R T A ⊆ R T B.

'R (T A ⊇ 'R (T B 'R]T A ⊆ 'R]T B

(iii) Interaction with union and intersection For every family (Ai )i∈I of fuzzy sets in X, [R]T R T 'R (T




i∈I




i∈I




R

T

i∈I




Ai ⊇ Ai = Ai =

 i∈I 

[R]T Ai

[R]T

R T Ai

R T

' R ( T Ai

'R (T

i∈I 

i∈I 

Ai ⊆ R

T Ai i∈I 
'R]T 'R]T Ai i∈I Ai ⊇ i∈I 
R T i∈I Ai = R T Ai i∈I

i∈I



i∈I



i∈I



R

T

i∈I



Ai = Ai ⊆ Ai ⊇

 i∈I 

[R]T Ai , R T Ai ,

i∈I 

'R (T Ai ,

i∈I 

Ai ⊇ R

T Ai , i∈I   'R]T 'R]T Ai , i∈I Ai = i∈I   R T i∈I Ai ⊆ R T Ai . i∈I

i∈I

Proof. (i) For every x ∈ X, ([R]T ∅)(x) = inf y∈X T (R(x, y), 0) = 0. The remaining equalities can be shown in the analogous way. (ii) Follows directly from monotonicity of T , its residuum I , and NT . (iii) By way of example we show the conditions for the fuzzy sufficiency operator. For every x ∈ X,    Ai )(x) = inf I sup Ai (y), R(x, y) ('R (T i∈I y∈X i∈I    = inf inf I (Ai (y), R(x, y)) = 'R (T Ai (x) i∈I y∈X

i∈I

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by Lemma 1(v). Moreover,    ('R (T Ai )(x) = inf I inf Ai (y), R(x, y) i∈I

y∈X

i∈I

 inf sup I (Ai (y), R(x, y)) y∈X i∈I

 sup inf I (Ai (y), R(x, y)) = i∈I y∈X







'R (T Ai

(x)

i∈I

by Lemma 1(vi).  Let
T be any T -modal operator introduced in Definition 3. Note first that for every binary crisp relation R on X, for every crisp subset A ⊆ X and for any T1 ∈ Tlc and T2 ∈ Tlc it holds
T1 (R)(A) =
T2 (R)(A). In other words,
T (R) gives the same crisp subset regardless of the choice of the underlying t-norm. In such a case we actually deal with (crisp) modal operators. Furthermore, observe that for every binary crisp relation R, for every crisp subset A ⊆ X and for every T ∈ Tlc , R T and [R]T , as well as 'R (T and R

T are dual in the sense R T A = coN ([R]T coN A), 'R (T A = coN ( R

T coN A), where N = NT . In the fuzzy case duality does not hold in general. We have, however, the following weaker property. First, let us introduce the following notion. Definition 4. Let T ∈ Tlc and let N = NT . T -modal operators
1 and
2 are called T -semidual iff for any A ∈ F (X),
i (A) ⊆ coN
j (coN A) for i = j and i, j ∈ {1, 2}. Proposition 1. The following are pairs of T-semidual modal operators: [ ]T and T , ' (T , and

T ,

' ]T , and T .

Proof. By way of example we show that ' ]T and T are T -semidual. For every A ∈ F (X) and for every x ∈ X, we have ('R]T A)(x) = inf I (N(R(x, y)), A(y)) y∈X


inf I (N(R(x, y)), N(N(A(y)))) y∈X

Lemma 1(xiv)

= inf N(T (N(R(x, y)), N(A(y)))) Lemma 1(xii) y∈X   = N sup T (N(R(x, y)), N(A(y))) y∈X

Lemma 1(xv)

= (coN R T coN A)(x). Whence 'R]T A ⊆ coN R T coN A. In the similar way one can show that R T A ⊆ coN 'R]T coN A. 

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Remark 1. In general, the following condition does not hold: [R]T (A ⇒I B) ⊆ [R]T A ⇒I [R]T B. For example, let T  0.4 R= 0 0

= TP and I = Ig . Next, let X = {a, b, c}, R ∈ R(X) and A, B ∈ F (X) be as below.      0.4 0.1 a b c a b c  0.5 0.3 , A = , B= . 0.7 0.8 0.5 0 0.6 0.1 0.5 0.4

By simple calculations we obtain   a b c [R]T (A ⇒I B) = , 0 23 21

 [R]T A ⇒I [R]T B =

a 0

b 1 3

c 1 4

 .

3.1. Fuzzy rough sets Let W be a non-empty universe and let R ∈ R(W ). A pair FAS = (W, R) is called a fuzzy approximation T space. Next, let T and I be a t-norm and an implicator, respectively. We define two mappings FASI , FAS : F (W ) → F (W ) given by: for any A ∈ F (W ) and for any x ∈ W FASI (A)(x) = inf I (R(x, y), A(y)), y∈W

T

FAS (A)(x) = sup T (R(x, y), A(y)). y∈W

T

For any A ∈ F (W ), FASI (A) (resp. FAS (A)) is called an I-lower (resp. T-upper) fuzzy rough approximation of A in FAS. T It is easily noted that FASI (A) = [R]T A and FAS (A) = R T A. 3.2. Fuzzy morphological and fuzzy topological operators 3.2.1. Fuzzy morphological operators Fuzzy mathematical morphology is an extension of binary morphology to gray-scale morphology using techniques from fuzzy set theory. The basic tools of mathematical morphology are the morphological operations. A fuzzy morphological operation P transforms an image A ∈ F (Rn ) by means of a structuring element B ∈ F (Rn ) into a new image P (A, B) ∈ F (Rn ) using a translation Ty (B) of B by y ∈ Rn . The basic fuzzy morphological operations are fuzzy erosion and fuzzy dilation defined as follows. Let T and I be a t-norm and an implicator, respectively, and let A, B ∈ F (Rn ). Then for every x ∈ Rn , EI (A, B)(x) = infn I (B(y − x), A(y)), y∈R

DT (A, B)(x) = sup T (B(y − x), A(y)). y∈Rn

EI (A, B) and DT (A, B) are called an I -erosion and T -dilation of A wrt B.

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Given a structuring element B ∈ F (Rn ), let us define a binary fuzzy relation on Rn as follows: RB (x, y) = B(y − x) for all x, y ∈ Rn . Then one easily derives that EI (A, B) = [RB ]T A, DT (A, B) = RB T A. In other words, using the relation RB , I -erosion and T -dilation of A can be defined by means of fuzzy modal operators (O.1) and (O.2), respectively. 3.2.2. Fuzzy topological operations In classical settings it is well-known that box and diamond operators coincide with topological interior and closure operations. The analogous property holds in the fuzzy case. First, let us recall the definitions of fuzzy interior and fuzzy closure operators. A mapping I : F (X) → F (X) is called a fuzzy interior operator iff it satisfies the following properties: (I1) (I2) (I3) (I4)

I(X) = X, I(A) ⊆ A for every A ∈ F (X), I(I(A)) = I(A) for every A ∈ F (X), I(A ∩ B) = I(A) ∩ I(B) for all A, B ∈ F (X).

A mapping C : F (X) → F (X) is called a fuzzy closure operator iff it satisfies the following properties: (C1) (C2) (C3) (C4)

C(∅) = ∅, A ⊆ C(A) for every A ∈ F (X), C(C(A)) = C(A) for every A ∈ F (X), C(A ∪ B) = C(A) ∪ C(B) for all A, B ∈ F (X).

Now we have the following. Proposition 2 (Boixader et al. [5]). Let T be a left-continuous t-norm, I be its residuum and let R be a T-equivalence relation. Then [R]T is a fuzzy interior operator and R T is a fuzzy closure operator. 3.3. Fuzzy modal operators In modal logics modal operators are traditionally used for interpretation of modal expressions. Clearly, this is also the case in fuzzy modal logics. Below we show how modalities are interpreted in fuzzy modal logics. We take into account fuzzy modal logic based on left-continuous t-norms, where conjunction, implication and negation are interpreted by a left-continuous t-norm T , its residuum I and NT , respectively. To begin with, let us briefly recall several basic notions and terminology. Consider a propositional fuzzy modal logic, which language is determined by a denumerable set Var of propositional variables, the truth constant ⊥ and basic logical connectives ∧ (min-conjunction), & (strong conjunction), → (implication) and six modal operators , ♦, , ,  and . Other non-modal connectives def def def are defined as: ¬ =  →⊥, ∨ = (( → ) → )∧(( → ) → ),  ≡  = ( → )&( → ). The set L of all formulae (language) is inductively defined by: (i) V ar ⊂ L, (ii) ⊥∈ L, and (iii) if ,  ∈ L then  ∧ , &,  → , , ♦, , , ,  ∈ L. The operators  and ♦ are the standard modal operators of necessity and possibility, respectively, while  and  are the sufficiency and the impossibility operators, which are fuzzy counterparts of the respective operators introduced in [20].

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233

An interpretation of L is a fuzzy Kripke structure  = (W, R, m, T ), where (i) W is a non-empty set of possible worlds (ii) R : W × W → [0, 1] is a fuzzy accessibility relation (iii) m : V ar → F (W ) is a meaning function, and (iv) T is an arbitrary left-continuous t-norm. Intuitively, for every proposition p ∈ V ar and for every world w ∈ W , m(p)(w) is the degree to which p is true in w. For a left-continuous t-norm T , we will write KT to denote the class of all fuzzy Kripke structures with the t-norm T . A valuation in  is an extension of m to the set L of all formulae. Namely, for any  ∈ L we write val () to denote the truth value of  in  inductively defined by • • • • • • • • • •

val (p) = m(p) for every p ∈ V ar val ( ∧ ) = val () ∩ val () val (&) = val () ∩T val () val ( → ) = val () ⇒I val () val () = [R]T val () val (♦) = R T val () val () = 'R (T val () val () = R

T val () val () = 'R]T val () val () = R T val ().

A formula  ∈ L is true in , written   , iff val () = W . For a class K of fuzzy Kripke structures,  is called a K-tautology, in symbols K , iff    for every  ∈ K;  is called a tautology iff K  for every class K of fuzzy Kripke structures. For a class K of fuzzy Kripke structures, by a logic L(K) we mean the set of all K-tautologies. In view of Lemma 1(ii) we obtain:

Proposition 3. Let K be a class of fuzzy Kripke structures for the language L and let ,  ∈ L. Then K  →  iff for every  ∈ K, val () ⊆ val (). Note also that in view of Remark 1, in general case we have: / ( → ) → ( → )  which means that axiom K, well-known in traditional modal logics, is not in general a tautology in fuzzy modal logics. 4. Characterisations of binary fuzzy relations In this section, we show how fuzzy modal operators can be used to characterise properties of particular classes of fuzzy binary relations. We also show how these characterisations are interpreted in terms of characteristic axioms of particular classes of fuzzy modal logics. Proofs of all propositions were relegated to Appendix. 4.1. Serial fuzzy relations The following proposition provides characterisation of fuzzy serial relations.

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Proposition 4. For every R ∈ R(X) the following conditions are equivalent: (i) R is serial (ii) [R]T A ⊆ R T A for every A ∈ F (X) and for some T ∈ Tlc . Remark 2. Since seriality does not depend on any t-norm, it directly follows from the above proposition that (ii) holds for every T ∈ Tlc in case of serial fuzzy relations. Let us consider a class KD of fuzzy Kripke structures which accessibility relations are fuzzy serial relations. By Propositions 3 and 4 we obtain: KD  → ♦,

that is, for every formula ,  → ♦ is a KD -tautology. Observe that this is the fuzzy analogon of the basic modal postulate in two-valued deontic logic, well-known as D-schema. Also, this is the tautology for any fuzzy Kripke structure , regardless of the form of the respective left-continuous t-norm T . 4.2. Reflexive and irreflexive fuzzy relations In this section, we will give characterisations of fuzzy (ir)reflexive and weakly (ir)reflexive relations. Proposition 5. For every R ∈ R(X) the following conditions are equivalent: (i) R is reflexive, (ii) [R]T A ⊆ A for every A ∈ F (X) and for some T ∈ Tlc , (iii) A ⊆ R T A for every A ∈ F (X) and for some T ∈ Tlc . Proposition 6. For every R ∈ R(X) the following conditions are equivalent: (i) R is weakly reflexive, (ii) R T X ∩T A ⊆ R T A for every A ∈ F (X) and for some T ∈ Tlc , (iii) R T X ∩T [R]T A ⊆ A for every A ∈ F (X) and for some T ∈ Tlc . Let KT (resp. KwT ) stand for the class of fuzzy Kripke structures with fuzzy reflexive (resp. weakly reflexive) relations. Then, by Propositions 3, 5 and 6 we obtain: KT  →  KwT ♦! ∧  → ♦

 K T  → ♦ KwT ♦! ∧  → .

Note that the schema  →  corresponds to the well-known T-schema in KT-systems of modal logic, its dual  → ♦ is sometimes called T ♦-schema (see [6]). Examples of irreflexive relations can be found in the rough-set style data analysis in information systems. In particular, diversity relations, reflecting differences among objects in information systems, are irreflexive. Fuzzy generalisations of these relations were considered, e.g. in [36,37,39], and the respective fuzzy information logics were discussed.

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The following two propositions provide characterisations of fuzzy (weakly) irreflexive relations. Proposition 7. For every R ∈ R(X) the following conditions are equivalent: (i) R is irreflexive, (ii) 'R (T A ⊆ coN A for every A ∈ F (X), for some T ∈ Tlc , and for N = NT , (iii) coN A ⊆ R

T A for every A ∈ F (X), for some T ∈ Tlc , and for N = NT . Proposition 8. For every R ∈ R(X), (i) R is weakly irreflexive iff A ∩T 'R (T A ⊆ 'R (T X for every A ∈ F (X), for some T ∈ Tlc , and for N = NT , (ii) if R is weakly irreflexive then A ∩T R

T ∅ ⊆ R

T coN A for every A ∈ F (X), for some T ∈ Tlc , and for N = NT . Again, since (weak) (ir)reflexivity does not depend on t-norms, Propositions 5–8 imply that the respective characterisations hold for all left-continuous t-norms T . Let KI and KwI denote classes of fuzzy Kripke structures with irreflexive and weakly irreflexive fuzzy accessibility relations, respectively. Propositions 3, 7 and 8 imply the following characteristic axioms of logics L(KI ) and L(KwI ): KI  → ¬ KwI  ∧  → !.

KI ¬ → 

Weakly reflexive (resp. irreflexive) fuzzy relations are discussed when (fuzzy) information relations and (fuzzy) information logics are considered. In particular, orthogonality relations (measuring degrees to which two fuzzy sets are disjoint) are weakly irreflexive. The above characteristic axioms are then useful while fuzzy generalisations of information logics based on such relations are considered. 4.3. Symmetric and asymmetric fuzzy relations The following propositions give characterisations of fuzzy symmetric and left (resp. right) N-asymmetric relations. Proposition 9. For every R ∈ R(X) the following conditions are equivalent: (i) R is symmetric, (ii) R T [R]T A ⊆ A for every A ∈ F (X) and for some T ∈ Tlc , (iii) A ⊆ [R]T R T A for every A ∈ F (X) and for some T ∈ Tlc . Note that the characterisations (ii) and (iii) of the above proposition hold for every left-continuous t-norm due to the fact that symmetry does not dependent on any t-norm. Proposition 10. Let T ∈ Tlc and let N = NT . Then for every R ∈ R(X) the following conditions are equivalent:

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(i) R is right N-asymmetric, (ii) A ⊆ [R]T R

T coN A for every A ∈ F (X), (iii) R T 'R (T A ⊆ coN A for every A ∈ F (X). Proposition 11. Let T ∈ Tlc and let N = NT . Then for every R ∈ R(X) the following conditions are equivalent: (i) R is left N-asymmetric, (ii) A ⊆ 'R (T [R]T coN A for every A ∈ F (X). As before, Propositions 9–11 give characteristic axioms of the respective fuzzy modal logics. Namely, let KB be the class of all fuzzy Kripke structures with symmetric relations and for any T ∈ Tlc let T = {(W, R, m, T ) : R is left N -asymmetric}, KT = {(W, R, m, T ) : R is right N -asymmetric}. KlA T T rA We have the following: KB ♦ →  KT  → ¬ rA KT  → ¬.

KB  → ♦ KT ♦ → ¬ rA

lA

Note that the schemas  → ♦ and ♦ →  are fuzzy analogons of the famous Brouwer axioms. 4.4. Euclidean fuzzy relations Proposition 12. Let T ∈ Tlc . Then for every R ∈ R(X) the following conditions are equivalent: (i) R is T-Euclidean, (ii) R T [R]T A ⊆ [R]T A for every A ∈ F (X), (iii) R T A ⊆ [R]T R T A for every A ∈ F (X). Let KET be the class of fuzzy Kripke structures with a left-continuous t-norm T and T-Euclidean fuzzy accessibility relations. Then, by Propositions 3 and 12, we obtain: KT ♦ →  E

KT ♦ → ♦. E

Both schemas are fuzzy counterparts of well-known Euclidean axioms in two-valued modal logics. Euclidean relations are traditionally used in BDI systems (Belief-Desire-Intensions) to model agents’ beliefs [45]. The importance of the above characterisations are therefore important when fuzzy generalisations of these systems are to be discussed. 4.5. Transitive and cotransitive fuzzy relations Proposition 13. Let T ∈ Tlc . Then for every R ∈ R(X) the following conditions are equivalent: (i) R is T-transitive, (ii) [R]T A ⊆ [R]T [R]T A for every A ∈ F (X),

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(iii) R T R T A ⊆ R T A for every A ∈ F (X). Proposition 14. Let T ∈ Tlc and let N = NT . Then for every R ∈ R(X) the following conditions are equivalent: (i) R is T-cotransitive, (ii) 'R]T A ⊆ 'R]T 'R]T A for every A ∈ F (X), (iii) R T R T A ⊆ R T A for every A ∈ F (X). T ) the class of fuzzy Kripke structures with a left-continuous Finally, let us denote by K4T (resp. K∼4 t-norm T and T-transitive (resp. T-cotransitive) fuzzy relations. Proposition 3 together with Propositions 13 and 14 imply:

KT  →  4 KT  →  ∼4

KT ♦♦ → ♦ 4 KT  → . ∼4

L(K4T ),

i.e.  →  and ♦♦ → ♦ are fuzzy analogons of the respective axioms Two axioms of characteristic for the system S4 of Lewis, which are traditionally referred to as 4-schemas.

5. Conclusions In this paper, we have defined six fuzzy modal operators and provided their basic properties. Using these operators, characterisations of main classes of binary fuzzy relations have been given. It is interesting to note that these characterisations are fuzzy analogons of the respective characterisations for crisp relations. Based on these results we have shown characteristics axioms of several classes of fuzzy modal logics. The results presented here may be a starting point for further investigations of fuzzy information logics—fuzzy generalisations of information logics (see, e.g., [10,30,32,33]). These formal systems are in fact fuzzy modal logics with parameterised accessibility relations. These parameters correspond to properties of objects, parameterised accessibility relations reflect relationships among objects determined by these properties. Fuzzy information logics, therefore, are logical systems for representing and reasoning about both properties of objects and relationships among them in information systems. Some classes of these logics were proposed in [36,39] and will be further investigated.

Acknowledgements Anna M. Radzikowska was partially supported by the KBN Grant No. 8T11C01617. This work was carried out in the framework of COST Action 274 on Theory and Applications of Relational Structures as Knowledge Instruments.

Appendix. Proofs of propositions Proposition 4. For every R ∈ R(X) the following conditions are equivalent:

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(i) R is serial, (ii) [R]T A ⊆ R T A for every A ∈ F (X) and for some T ∈ Tlc . Proof. (i) ⇒ (ii) Let T be an arbitrary left-continuous t-norm and I be its residuum. Moreover, let A be a fuzzy set in X and x ∈ X. Then we have I (([R]T A)(x), ( R T A)(x))  =I



inf I (R(x, y), A(y)), sup T (R(x, y), A(y))

y∈X

y∈X

 sup I (I (R(x, y), A(y)), T (R(x, y), A(y))) y∈X

by Lemma 1(vii). Since from Lemma 1(iii), A(y)  T (R(x, y), I (R(x, y), A(y))), by monotonicity and associativity of T we obtain T (R(x, y), A(y))  T (R(x, y), T (R(x, y), I (R(x, y), A(y)))) = T (T (R(x, y), R(x, y)), I (R(x, y), A(y))). Therefore, sup I (I (R(x, y), A(y)), T (R(x, y), A(y))) y∈X

 sup I (I (R(x, y), A(y)), T (T (R(x, y), R(x, y)), I (R(x, y), A(y)))) y∈X

 sup T (R(x, y), R(x, y)) y∈X

by Lemma 1(xi). Finally, since R is serial, by Lemma 1(viii) we have  sup T (R(x, y), R(x, y)) = T y∈X



sup R(x, y), sup R(x, y) = T (1, 1) = 1. y∈X

y∈X

Then we obtain I (([R]T A)(x), ( R T A)(x)) = 1 for every x ∈ X, which by Lemma 1(ii) means that [R]T A(x)
R T A(x), x ∈ X. Hence [R]T A ⊆ R T A.

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(ii) ⇒ (i) Assume that R is not serial, i.e. supy∈X R(x0 , y) < 1 for some x0 ∈ X. Consider A = x0 R. Then we have I (([R]TA)(x0 ), ( R T A)(x0 )) =I



inf I (R(x0 , y), R(x0 , y)), sup T (R(x0 , y), R(x0 , y)) y∈X   y∈X

= I 1, sup T (R(x0 , y), R(x0 , y))

by Lemma 1(ii)

= sup T (R(x0 , y), R(x0 , y)) y∈ X

by Lemma 1(i)

y∈X

=T



sup R(x0 , y), sup R(x0 , y) y∈X


sup R(x0 , y).

y∈X

by Lemma 1(viii)

y∈X

Then we have: I (([R]T A)(x0 ), ( R T A)(x0 )) < 1. By Lemma 1(ii), this means that [R]T A(x0 ) > R T A(x0 ). Clearly, [R]T A
R T A.  Proposition 5. For every R ∈ R(X) the following conditions are equivalent: (i) R is reflexive, (ii) [R]T A ⊆ A for every A ∈ F (X) and for some T ∈ Tlc , (iii) A ⊆ R T A for every A ∈ F (X) and for some T ∈ Tlc . Proof. The proofs of implications (i) ⇒ (ii) and (i) ⇒ (iii) are given in [38]. Assume that R is not reflexive, i.e. R(x0 , x0 ) < 1 for some x0 ∈ X. (ii) ⇒ (i) For A = x0 R we have ([R]T A)(x0 ) = inf I (R(x0 , y), A(y)) y∈X

= inf I (R(x0 , y), R(x0 , y)) y∈X

=1 > R(x0 , x0 ) = A(x0 ), by Lemma 1(ii), whence [R]T A
A. (iii) ⇒ (i) For A = {x0 } we easily obtain ( R T A)(x0 ) = R(x0 , x0 ) < 1 = A(x0 ), so clearly A
R T A.  Proposition 6. For every R ∈ R(X) the following conditions are equivalent: (i) R is weakly reflexive, (ii) R T X ∩T A ⊆ R T A for every A ∈ F (X) and for some T ∈ Tlc , (iii) R T X ∩T [R]T A ⊆ A for every A ∈ F (X) and for some T ∈ Tlc . Proof. The proofs of (i) ⇒ (ii) and (i) ⇒ (iii) are straightforward.

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Assume that R is not weakly reflexive, i.e. R(x0 , x0 ) < supy∈X R(x0 , y) for some x0 ∈ X. (ii) ⇒ (i) Consider A = {x0 }. Note that ( R T A)(x0 ) = sup T (R(x0 , y), A(y)) = R(x0 , x0 ) < sup R(x0 , y). y∈X

y∈X

Then we have  ( R T X ∩T A)(x0 ) = T

 sup T (R(x0 , y), 1), A(x0 ) = T y∈X



 sup R(x0 , y), 1 y∈X

= sup R(x0 , y) > ( R T A)(x0 ). y∈X

Hence R T X ∩T A
R T A. Taking A = x0 R and proceeding in the similar way (iii) ⇒ (i) can be proved.  Proposition 7. For every R ∈ R(X) the following conditions are equivalent: (i) R is irreflexive, (ii) 'R (T A ⊆ coN A for every A ∈ F (X), for some T ∈ Tlc , and for N = NT , (iii) coN A ⊆ R

T A for every A ∈ F (X), for some T ∈ Tlc , and for N = NT . Proof. The proofs of (i) ⇒ (ii) and (i) ⇒ (iii) are by an easy verification. Assume that R is not irreflexive, i.e. there is x0 ∈ X such that R(x0 , x0 ) > 0. (ii) ⇒ (i) For A = x0 R one can easily verify that ('R (T A)(x0 ) = 1. From Lemma 1(ii) it follows that N(x) = I (x, 0) < 1 iff x > 0, so N(A(x0 )) = N(R(x0 , x0 )) < 1. Then ('R (T A)(x0 ) > N(A(x0 )), which implies 'R (T A
coN A. (iii) ⇒ (i) For A ∈ F (X) such that A(y) = N(R(x0 , y)), y ∈ X, we have ( R

T A)(x0 ) = sup T (N(R(x0 , y)), N(N (R(x0 , y)))) y∈X

= sup T (N(R(x0 , y)), I (N(R(x0 , y)), 0)) = 0 y∈X

by Lemma 1(iii). By Lemma 1(xiv), (coN A)(x0 ) = N(N(R(x0 , x0 )))  R(x0 , x0 ) > 0, hence (coN A)(x0 ) > ( R

T A)(x0 ), so coN A
R

T A.  Proposition 8. For every R ∈ R(X), (i) R is weakly irreflexive iff A ∩T 'R (T A ⊆ 'R (T X for every A ∈ F (X), for some T ∈ Tlc , and for N = NT , (ii) if R is weakly irreflexive then A ∩T R

T ∅ ⊆ R

T coN A for every A ∈ F (X), for some T ∈ Tlc , and for N = NT .

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Proof. i(⇒) For every A ∈ F (X) and for every x ∈ X, we have:   (A ∩T 'R (T A)(x) = T A(x), inf I (A(y), R(x, y)) y∈X


T (A(x), I (A(x), R(x, x)))
R(x, x)

by Lemma 1(iii). By assumption, R(x, x) = inf y∈X R(x, y) for every x ∈ X. Then, using Lemma 1(i), we obtain for every x ∈ X (A ∩T 'R (T A)(x)
inf R(x, y) = inf I (1, R(x, y)) = ('R (T X)(x), y∈X

y∈X

so A ∩T 'R (T A ⊆ 'R (T X. (⇐) Assume that R is not weakly irreflexive, that is inf y∈X R(x0 , y) < R(x0 , x0 ) for some x0 ∈ X. Consider A = x0 R. It is easy to verify that (A ∩T 'R (T A)(x0 ) = R(x0 , x0 ). By Lemma 1(i) and by assumption we have ('R (T X)(x0 ) = inf I (1, R(x0 , y)) = inf R(x0 , y) < R(x0 , x0 ). y∈X

y∈X

Therefore (A ∩T 'R (T A)(x0 ) > ('R (T X)(x0 ), so A ∩T 'R (T A
'R (T X. The proof of (ii) is similar.  Proposition 9. For every R ∈ R(X) the following conditions are equivalent: (i) R is symmetric, (ii) R T [R]T A ⊆ A for every A ∈ F (X) and for some T ∈ Tlc , (iii) A ⊆ [R]T R T A for every A ∈ F (X) and for some T ∈ Tlc . Proof. For the proofs of implications (i) ⇒ (ii) and (i) ⇒ (iii), see [38]. Assume that R is not symmetric, i.e. R(x0 , y0 ) = R(y0 , x0 ) for some x0 , y0 ∈ X. (ii) ⇒ (i) Let R(x0 , y0 ) < R(y0 , x0 ) and consider A = x0 R. Then we have ( R T [R]T A)(y0 ) = sup T (R(y0 , x), inf I (R(x, z), A(z))) z∈X

x∈X

= sup T (R(y0 , x), inf I (R(x, z), R(x0 , z))) z∈X x∈X    T R(y0 , x0 ), inf I (R(x0 , z), R(x0 , z)) z∈X

= T (R(y0 , x0 ), 1) = R(y0 , x0 ) > R(x0 , y0 ) = A(y0 ).

Hence R T [R]T A
A. Let R(x0 , y0 ) > R(y0 , x0 ) and consider A = y0 R. Proceeding in the analogous way we obtain the same conclusion.

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The proof of (iii) ⇒ (i) is similar.  Proposition 10. Let T ∈ Tlc and let N = NT . Then for every R ∈ R(X) the following conditions are equivalent: (i) R is right N-asymmetric, (ii) A ⊆ [R]T R

T coN A for every A ∈ F (X), (iii) R T 'R (T A ⊆ coN A for every A ∈ F (X). Proof. (i) ⇒ (ii) For every A ∈ F (X) and for every x ∈ X, we have ([R]T R

T coN A)(x) 



= inf I R(x, y), sup T (N(R(y, z)), N(N(A(z)))) y∈X

z∈X

 inf I (R(x, y), T (N(R(y, x))), N(N(A(x)))). y∈X

By assumption, N(R(y, x))  R(x, y) for all x, y ∈ X. Next, by Lemma 1(xiv), it holds N(N(A(x)))  A(x) for every x ∈ X. Then T (N(R(y, x)), N(N(A(x))))  T (R(x, y), A(x)), so by Lemma 1(xi), inf I (R(x, y), T (N(R(y, x)), N(N(A(x)))))

y∈X

 inf I (R(x, y), T (R(x, y), A(x))) y∈X

 A(x).

Hence ([R]T R

T coN A)(x)  A(x) for every x ∈ X, so A ⊆ [R]T R

T coN A. The proof of (i) ⇒ (iii) is similar. Assume R is not right N-asymmetric, i.e. R(x0 , y0 ) > N(R(y0 , x0 )) for some x0 , y0 ∈ X. (ii) ⇒ (i) For A = {x0 } by simple calculations we get ([R]T R

T coN A)(x0 ) < 1 = A(x0 ), whence A
[R]T R

T coN A. (iii) ⇒ (i) For A = y0 R, in the similar way one can show that R T 'R (T A
coN A.  Proposition 11. Let T ∈ Tlc and let N = NT . Then for every R ∈ R(X) the following conditions are equivalent: (i) R is left N-asymmetric, (ii) A ⊆ 'R (T [R]T coN A for every A ∈ F (X).

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243

Proof. (i) ⇒ (ii) For every A ∈ F (X) and for every y ∈ X, ('R (T [R]T coN A)(y)   = inf I inf I (R(x, z), N(A(z))), R(y, x) x∈X

z∈X

 inf I (I (R(x, y), N(A(y))), R(y, x)) x∈X

 inf I (I (R(x, y), N(A(y))), N(R(x, y))), x∈X

since, by assumption, R(y, x)  N(R(x, y)) for all x, y ∈ X. Next, by Lemma 1(xiii), I (R(x, y), N(A(y)))
I (N(N(A(y))), N(R(x, y))), so I (I (R(x, y), N(A(y))), N(R(x, y)))  I (I (N(N(A(y))), N(R(x, y))), N(R(x, y))). Then we have: ('R (T [R]T coN A)(y)  inf I (I (N(N(A(y))), N(R(x, y))), N(R(x, y))) x∈X

 N(N(A(y)))  A(y)

by Lemma 1(x) and 1(xiv). Hence we finally obtain ('R (T [R]T coN A)(y)  A(y) for every y ∈ X. Therefore, A ⊆ 'R (T [R]T coN A. (ii) ⇒ (i) Assume that R is not left N-asymmetric and let A ∈ F (X) be such that A(z) = N(R(x0 , z)) for any z ∈ X. By simple calculations we get ('R (T [R]T coN A)(y0 ) < A(y0 ), which implies A
'R (T [R]T coN A.  Proposition 12. For every T ∈ Tlc and for every R ∈ R(X) the following conditions are equivalent: (i) R is T-Euclidean, (ii) R T [R]T A ⊆ [R]T A for every A ∈ F (X), (iii) R T A ⊆ [R]T R T A for every A ∈ F (X). Proof. (i) ⇒ (ii) Note first that for every A ∈ F (X) and for all x, y ∈ X, ([R]T A)(y) = inf I (R(y, z), A(z)) z∈X


inf I (T (R(x, y), (R(x, z))A(z)))

by assumption

= inf I (R(x, y), I (R(x, z), A(z))) z∈ X  = I R(x, y), inf I (R(x, z), A(x))

by Lemma 1(ix)

z∈X

z∈X

= I (R(x, y), ([R]T A)(x)).

by Lemma 1(iv)

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Then for every A ∈ F (X) and for every x ∈ X, ( R T [R]T A)(x) = sup T (R(x, y), ([R]T A)(y)) y∈X


sup T (R(x, y), I (R(x, y), ([R]T A)(x)))
([R]T A)(x) x∈X

by Lemma 1(iii), so R T [R]T A ⊆ [R]T A. The proof of (i) ⇒ (iii) is similar. Assume now that R is not T-Euclidean, that is T (R(x0 , y0 ), R(x0 , z0 )) > R(y0 , z0 ) for some x0 , y0 , z0 ∈ X. (ii) ⇒ (i) Let A = y0 R. Note that ( R T [R]T A)(x  0)

 = sup T R(x0 , y), inf I (R(y, z), R(y0 , z)) z∈X y∈ X   T R(x0 , y0 ), inf I (R(y0 , z), R(y0 , z)) z∈X

= T (R(x0 , y0 ), 1) R(x0 , y0 ).

by Lemma 1(ii)

Then we have I (( R  T [R]T A)(x0 ), ([R]T A)(x0 ))


I R(x0 , y0 ), inf I (R(x0 , z), R(y0 , z)) z∈X


I (R(x0 , y0 ), I (R(x0 , z0 ), R(y0 , z0 )))

= I (T (R(x0 , y0 ), R(x0 , z0 )), R(y0 , z0 )) <1

by Lemma 1(ix) by assumption and Lemma 1(ii).

Again by Lemma 1(ii), ( R T [R]T A)(x0 ) > ([R]T A)(x0 ), so R T [R]T A
[R]T A. (iii) ⇒ (i) For A = {y0 } one can easily check that I (( R T A)(z0 ), ([R]T R T A)(z0 )) < 1. Then, by Lemma 1(ii), ( R T A)(z0 ) > ([R]T R T A)(z0 ), so R T A
[R]T R T A.  Proposition 13. For every T ∈ Tlc and for every R ∈ R(X) the following conditions are equivalent: (i) R is T -transitive, (ii) [R]T A ⊆ [R]T [R]T A for every A ∈ F (X), (iii) R T R T A ⊆ R T A for every A ∈ F (X). Proof. Proofs of implications (i) ⇒ (ii) and (i) ⇒ (iii) are given in [38]. Assume that R is not T -transitive, that is T (R(x0 , y0 ), R(y0 , z0 )) > R(x0 , z0 ) for some x0 , y0 , z0 ∈ X.

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245

(ii) ⇒ (i) Let A = x0 R. Then we have: ([R]T [R]T A)(x  0)

 = inf I R(x0 , y), inf I (R(y, z), R(x0 , z)) y∈X

z∈X

= inf inf I (R(x0 , y), I (R(y, z), R(x0 , z)))

by Lemma 1(iv)


I (R(x0 , y0 ), I (R(y0 , z0 ), R(x0 , z0 ))) = I (T (R(x0 , y0 ), R(y0 , z0 )), R(x0 , z0 )) <1 = inf I (R(x0 , y), R(x0 , y))

by Lemma 1(ix) by assumption and Lemma 1(ii) by Lemma 1(ii)

y∈X z∈X

y∈X

= ([R]T A)(x0 ). Hence [R]T A
[R]T [R]T A. (iii) ⇒ (i) Taking A = {z0 } and proceeding as above we get the conclusion.  Proposition 14. Let T ∈ Tlc and let N = NT . Then for every R ∈ R(X) the following conditions are equivalent: (i) R is T -cotransitive, (ii) 'R]T A ⊆ 'R]T 'R]T A for every A ∈ F (X), (iii) R T R T A ⊆ R T A for every A ∈ F (X). Proof. (i) ⇒ (ii) For every A ∈ F (X) and for every x ∈ X, we have ('R]T 'R]T A)(x) = inf I (N(R(x, y)), inf I (N(R(y, z)), A(z))) y∈X

z∈X

= inf inf I (N(R(x, y)), I (N(R(y, z)), A(z)))

by Lemma 1(iv)

= inf inf I (T (N(R(x, y)), N(R(y, z))), A(z)) y∈X z∈ X 

by Lemma 1(ix)

= inf I sup T (N(R(x, y)), N(R(y, z))), A(z)

by Lemma 1(v)

 inf I (N(R(x, z)), A(z))

by assumption

y∈X z∈X

z∈X

y∈X

z∈X

= ('R]T A)(x). Then 'R]T A ⊆ 'R]T 'R]T A. The proof of (i) ⇒ (iii) is similar. Assume that R is not T-cotransitive, i.e. for some x0 , y0 , z0 ∈ X, T (N (R(x0 , y0 )), N(R(y0 , z0 ))) > N(R(x0 , z0 )).

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(ii) ⇒ (i) Put A = coN x0 R, i.e. A(y) = N(R(x0 , y)), y ∈ X. Proceeding as above we obtain: ('R]T 'R]T A)(x0 ) = inf inf I (T (N (R(x0 , y)), N(R(y, z)), N(R(x0 , z)))) y∈X z∈X


I (T (N(R(x0 , y0 )), N(R(y0 , z0 )), N (R(x0 , z0 ))))

<1 = inf I (N (R(x0 , y)), N(R(x0 , y))) y∈X

= ('R]T A)(x0 ). by assumption and Lemma 1(ii). Hence 'R]T A
'R]T 'R]T A. (iii) ⇒ (i) For A = {z0 }, one can easily verify that ( R T R T A)(x0 ) > ( R T A)(x0 ), so clearly R T R T A
R T A. 

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