Fuzzy equivalence relations and their equivalence classes

Fuzzy Sets and Systems 158 (2007) 1295 – 1313 www.elsevier.com/locate/fss

Fuzzy equivalence relations and their equivalence classes夡 ´ ca,∗ , Jelena Ignjatovi´ca , Stojan Bogdanovi´cb Miroslav Ciri´ a Faculty of Sciences and Mathematics, University of Niš, Višegradska 33. P.O. Box 224, 18000 Niš, Serbia and Montenegro b Faculty of Economics, University of Niš, Trg Kralja Aleksandra 11, 18000 Niš, Serbia and Montenegro

Received 28 February 2006; received in revised form 10 December 2006; accepted 24 January 2007 Available online 6 February 2007

Abstract In this paper we investigate various properties of equivalence classes of fuzzy equivalence relations over a complete residuated lattice. We give certain characterizations of fuzzy semi-partitions and fuzzy partitions over a complete residuated lattice, as well as over a linearly ordered complete Heyting algebra. In the latter case, for a fuzzy equivalence relation over a linearly ordered complete Heyting algebra, we construct an algorithm for calculation of a minimal family of its equivalence classes which generates it. Most of the presented results are new, but some of them are generalizations of known results given in a way which simplifies and clarifies them. © 2007 Elsevier B.V. All rights reserved. Keywords: Fuzzy equivalence relation; Fuzzy equivalence class; Fuzzy semi-partition; Fuzzy partition; Complete residuated lattice; Linearly ordered Heyting algebra

1. Introduction Fuzzy equivalence relations were introduced by Zadeh [37] as a generalization of the concept of an equivalence relation. They have been since widely studied as a way to measure the degree of indistinguishability or similarity between the objects of a given universe of discourse, and they have shown to be useful in different contexts such as fuzzy control, approximate reasoning, fuzzy cluster analysis, etc. Depending on the authors and the context in which they appeared, they have received other names such as similarity relations (original Zadeh’s name [37]), indistinguishability operators [36,23,24,5,25,14,15], T-equivalences [10,11], many-valued equivalence relations [12,13], etc. The first definition of a fuzzy partition was given by Ruspini [33], and it has played a significant role in many studies in fuzzy cluster analysis. Butnariu [6] proposed another definition which was originally based on the Lukasiewicz t-norm, and later defined for an arbitrary t-norm. But, none of these definitions of a fuzzy partition gives a bijective correspondence between fuzzy partitions and fuzzy equivalence relations. Definition of a fuzzy partition which provides such correspondence is the one by means of fuzzy equivalence classes. The concept of a fuzzy equivalence class was introduced by Zadeh [37] as a natural generalization of the concept of a crisp equivalence class, and it has been since studied in numerous papers (cf. [30,32,8,10]). Ovchinnikov [32] and De Baets et al. [8] defined a fuzzy partition as the set of all fuzzy equivalence classes of some fuzzy equivalence relation, and this natural definition has shown oneself to 夡

Research supported by Ministry of Science and Environmental Protection, Republic of Serbia, Grant No. 144011.

∗ Corresponding author. Tel.: +38 118224492; +38 118533014.

´ c). E-mail address: [email protected] (M. Ciri´ 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.01.010

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be convenient if we want to have a bijective correspondence between fuzzy partitions and fuzzy equivalence relations. De Baets et al. [8] also defined a fuzzy semi-partition as the set of (not necessarily all) fuzzy equivalence classes of some fuzzy equivalence relation. In various contexts fuzzy semi-partitions were studied in [19,27,8,10,22] and other papers. The main aim of this paper is study of certain properties of fuzzy equivalence classes. The central place in our research is held by three fuzzy equivalence relations Sf , Tf and Ef assigned to a fuzzy subset f of a set A, and three fuzzy equivalence relations SC , TC and EC assigned to a family C of fuzzy subsets of A. The relations Ef and EC are well-known, and play a crucial role in the representation theory of fuzzy equivalence relations initiated by Valverde [36], whereas the relation Sf is known as a decomposable relation [5]. After the introductory section, in Section 2 we introduce basic concepts and present elementary properties of fuzzy equivalence relations and their equivalence classes that will be used in the sequel. The main part of the paper is Section 3, where we consider fuzzy equivalence classes, fuzzy semi-partitions and fuzzy partitions. For a normalized fuzzy subset f of A, in Theorem 3.1 we prove that the set of all fuzzy equivalence relations on A having f as its fuzzy equivalence class is equal to the closed interval [Sf , Ef ] of the lattice E(A) of all fuzzy equivalence relations on A. In Theorem 3.2 we determine three necessary and sufficient conditions for a family of normalized fuzzy subsets of A to be a fuzzy semi-partition. The first of them was already determined by Klawonn and Kruse [27], in the case when truth values are taken from the real unit interval and a left-continuous t-norm on it is considered (see also a paper by De Baets and Mesiar [10]). Theorem 3.2 also shows that if any of the mentioned three conditions is fulfilled, then the set of all fuzzy equivalence relations on A having C as its fuzzy semi-partition is equal to the closed interval [SC , EC ] of E(A). In the sequel we give several interesting consequences of Theorem 3.2, and then we pass to study of fuzzy partitions. In Theorem 3.4 we give two necessary and sufficient conditions for a family of normalized fuzzy subsets of A to be a fuzzy partition, and then we present three ways for construction of the fuzzy equivalence relation corresponding to a fuzzy partition. The least theorem of the section, Theorem 3.5, gives two interesting properties of fuzzy equivalence relations that play a crucial role in the next section. In Section 4 we are interested in the case when fuzzy equivalence relations Tf and Ef coincide for every fuzzy subset f of a set A over a complete residuated lattice L. First we prove that this holds if and only if L is a linearly ordered complete Heyting algebra, that leads us to study of fuzzy equivalence relations over such algebras. Then we simplify Theorem 3.2 and obtain Theorem 4.1 that characterizes fuzzy semi-partitions over a linearly ordered complete Heyting algebra, and by Theorem 3.5 we derive Theorem 4.2. This theorem we then use to construct an algorithm for calculation of a minimal generating fuzzy semi-partition of a given fuzzy equivalence relation over a linearly ordered complete Heyting algebra. Our work concerning fuzzy semi-partitions and fuzzy partitions is closely related to that of De Baets and Mesiar [10], Klawonn and Kruse [27], Klawon [26], Höhle [22], Demirci [12] and Ovchinnikov [32], and work concerning generating families of fuzzy equivalence classes is closely related to that of Valverde [36], Jacas [23], Jacas and Recasens [25], [24], Demirci and Recasens [16], and others. 2. Preliminaries In this paper we will use complete residuated lattices as the structures of truth values. A complete residuated lattice is an algebra L = (L, ∧, ∨, ⊗, →, 0, 1) such that (L1) (L, ∧, ∨, 0, 1) is a complete lattice with the least element 0 and the greatest element 1, (L2) (L, ⊗, 1) is a commutative monoid with the unit 1, (L3) ⊗ and → form an adjoint pair, i.e., they satisfy the adjunction property: for all x, y, z ∈ L, x ⊗ y z ⇔ x y → z.

(1)

The operations ⊗ (called multiplication) and → (called residuum)  are intended formodeling the conjunction and implication of the corresponding logical calculus, and supremum ( ) and infimum ( ) are intended for modeling of the existential and general quantifier, respectively. An operation ↔ defined by x ↔ y = (x → y) ∧ (y → x),

(2)

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called biresiduum (or biimplication), is used for modeling the equivalence of truth values. Emphasizing the monoidal structure, in some sources residuated lattices are called integral, commutative, residuated l-monoids (cf. [20,21]), and in some sources the name residuated lattice was used for a more general algebraic structure (cf. [3,18]). It can be verified that with respect to , ⊗ is isotonic in both arguments, → is isotonic in the second and antitonic in the first argument, and for all x, y ∈ L and {xi }i∈I ⊆ L the following hold: x ⊗ y x ∧ y x ↔ y, x y ⇔ x → y = 1, x ↔ y = 1 ⇔ x = y, x → 1 = 1; 1 → x = x, y x → (x ⊗ y),     xi ⊗ x = (xi ⊗ x). i∈I

(3) (4) (5) (6) (7) (8)

i∈I

For other properties of complete residuated lattices we refer to [1,20]. The most studied and applied set of truth values is the real unit interval [0, 1] with x ∧ y = min(x, y), x ∨ y = max(x, y), and with three important pairs of adjoint operations: the Łukasiewicz one (x ⊗ y = max(x + y − 1, 0), x → y = min(1 − x + y, 1)), product one (x ⊗ y = x · y, x → y = 1 if x y and = y/x otherwise) and Gödel one (x ⊗ y = min(x, y), x → y = 1 if x y and = y otherwise). More generally, an algebra ([0, 1], ∧, ∨, ⊗, →, 0, 1) is a complete residuated lattice if and only if ⊗ is a left-continuous t-norm and the residuum is defined by x → y = {u ∈ [0, 1] | u ⊗ x y}. Another important set of truth values is {a0 , a1 , · · · , an }, 0 = a0 < . . . < an = 1, with ak ⊗ al = amax(k+l−n,0) and ak → al = amin(n−k+l,n) . A special case of the latter algebras is the two-element Boolean algebra of classical logic with the support {0, 1}. The only adjoint pair on the two-element Boolean algebra consist of the classical conjunction and implication operations. A residuated lattice L is called a Heyting algebra if x ⊗ y = x ∧ y, for all x, y ∈ L. If, in addition, L is a complete lattice, then it is a complete Heyting algebra, and if the partial order  in L is linear, then L is a linearly ordered Heyting algebra. The most important example of a linearly ordered complete Heyting algebra is the real unit interval [0, 1] with the Gödel pair of adjoint operations, i.e., with the standard minimum t-norm. In the further text L will be a complete residuated lattice. A fuzzy subset of a set A over L, or simply a fuzzy subset of A, is any mapping from A into L. Let f and g be two fuzzy subsets of A. The equality of f and g is defined as the usual equality of mappings, i.e., f = g if and only if f (x) = g(x), for every x ∈ A. The inclusion f g is also defined pointwise: f g if and only if f (x) g(x), for every x ∈ A. Endowed with this partialorder the set F(A) of all fuzzy  subsets of A forms a completely distributive lattice, in which the meet (intersection) i∈I fi and the join (union) i∈I fi of an arbitrary family {fi }i∈I of fuzzy subsets of A are mappings from A into L defined by 





fi (x) =

i∈I

 i∈I

 fi (x),

 i∈I

 fi (x) =



fi (x).

i∈I

The product f ⊗ g is a fuzzy subset defined by: f ⊗ g(x) = f (x) ⊗ g(x), for every x ∈ A. The crisp part of a fuzzy subset f of A, in notation c(f ), is a crisp subset of A defined by c(f ) = {x ∈ A | f (x) = 1}, the height of f, in notation f , is defined by f  =



f (x).

x∈A

Note that many authors used the name “kernel” instead of “crisp part”, and the notation “ker f ” instead of “c(f )”, but here we will use the name “kernel” in its usual meaning—for the crisp equivalence relation associated in a natural way with a mapping. Namely, the kernel of a fuzzy subset f of A, in notation ker f , is a crisp equivalence relation on A defined by ker f = {(x, y) ∈ A × A | f (x) = f (y)}. A fuzzy relation on A is any mapping from A × A into L, that is to say, any fuzzy subset of A × A, and the equality, inclusion, joins, meets and ordering of fuzzy relations are defined as for fuzzy sets. For two fuzzy relations R and S,

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their product is a fuzzy relation R ◦ S defined by  (R ◦ S)(x, y) = (R(x, a) ⊗ S(a, y)), a∈A

for all x, y ∈ A. A fuzzy relation E on A is said to be (R) reflexive (or fuzzy reflexive) if E(x, x) = 1, for every x ∈ A; (S) symmetric (or fuzzy symmetric) if E(x, y) = E(y, x), for all x, y ∈ A; (T) transitive (or fuzzy transitive) if E ◦ E E, i.e., if for all x, a, y ∈ A E(x, a) ⊗ E(a, y) E(x, y). It is not hard to check that if E is reflexive and transitive, then E ◦ E = E. A reflexive, symmetric and transitive fuzzy relation on A is called a fuzzy equivalence relation, or just a fuzzy equivalence, on A. With respect to the ordering of fuzzy relations, the set E(A) of all fuzzy equivalence relations on a set A is a complete lattice, in which the meet coincide with the ordinary intersection of fuzzy relations, but in the general case, the join in E(A) does not coincide with the ordinary union of fuzzy relations (cf. [30]). Let E be a fuzzy equivalence relation on A. For each a ∈ A we define a fuzzy subset Ea of A as follows: Ea (x) = E(a, x)

for every x ∈ A,

and we call Ea a fuzzy equivalence class, or just an equivalence class, of E determined by the element a. The set A/E = {Ea | a ∈ A} is called the factor set of A with respect to E. Let f be a fuzzy subset of a set A and let E be a fuzzy equivalence relation on A. Then f is said to be extensional (or observable) with respect to E if f (x) ⊗ E(x, y) f (y)

(9)

for all x, y ∈ A (cf. [16,29]), and to be indistinguishable (indiscernible) w.r.t. E if f (x) ⊗ f (y)E(x, y)

(10)

for all x, y ∈ A. In [32], fuzzy sets indistinguishable w.r.t. E were called pre-classes of E. A fuzzy subset f of A is normalized (or modal, in some sources) if f (x) = 1 for at least one x ∈ A. If f is normalized, extensional and indistinguishable w.r.t. a fuzzy equivalence relation E, then it is called a fuzzy point with respect to E (cf. [29]). Lemma 2.1. Let f be a normalized fuzzy subset of a set A and let E ∈ E(A). Then f is indistinguishable w.r.t. E if and only if it is contained in an equivalence class of E. Proof. Let a ∈ A such that f (a) = 1. If f is indistinguishable w.r.t. E, then for every x ∈ A we have that f (x) = f (a) ⊗ f (x)E(a, x) = Ea (x), and therefore, f Ea . Conversely, let f Ea , for some a ∈ A. Then for every x, y ∈ A we have that f (x)E(a, x) and f (y)E(a, y), which yields f (x) ⊗ f (y)E(x, a) ⊗ E(a, y) E(x, y). Hence, f is indistinguishable w.r.t. E.



Lemma 2.2. A fuzzy subset f of a set A is a fuzzy point w.r.t. E if and only if it is an equivalence class of E.

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Proof. Let f be a fuzzy point w.r.t. E. By Lemma 2.1, f Ea , where a ∈ A such that f (a) = 1. On the other hand, for any x ∈ A, the extensionality of f w.r.t. E yields Ea (x) = E(a, x) = f (a) ⊗ E(a, x) f (x), and therefore, Ea f . Hence, f = Ea . Conversely, let f = Ea for some a ∈ A. Then f (a) = 1, so f is normalized. For arbitrary x, y ∈ A, by the ⊗-transitivity of E we have that f (x) ⊗ E(x, y) = E(a, x) ⊗ E(x, y) E(a, y) = f (y), f (x) ⊗ f (y) = E(x, a) ⊗ E(a, y) E(x, y) and hence, f is a fuzzy point.



The above lemma shows that the definition of a fuzzy point given here is equivalent to the one given by Klawonn in [26], who defined a fuzzy point as the extensional hull of a crisp point, w.r.t. a fuzzy equivalence relation E, i.e., as an equivalence class of E. The next two properties of fuzzy equivalence relations are known (cf. [8,32]), but, for the sake of completeness, we will prove them here. Lemma 2.3. Let E be a fuzzy equivalence relation on a set A. Then for every x, y ∈ A the following is true: (a) E(x, y) = Ex ⊗ Ey ; (b) Ex = Ey ⇔ E(x, y) = 1. Proof. (a) The following is true   (Ex (a) ⊗ Ey (a)) = (E(x, a) ⊗ E(a, y)) = E ◦ E(x, y) = E(x, y). Ex ⊗ Ey  = a∈A

a∈A

(b) If Ex = Ey , then E(x, y) = Ex (y) = Ey (y) = E(y, y) = 1. Conversely, let E(x, y) = 1. Then for any a ∈ A we have Ex (a) = E(x, a) E(x, y) ⊗ E(y, a) = E(y, a) = Ey (a) and similarly, Ey (a)Ex (a). Therefore, Ex = Ey .



3. Fuzzy equivalence classes, fuzzy semi-partitions and fuzzy partitions We start this section defining three kinds of fuzzy equivalence relations induced, in a natural way, by fuzzy sets and families of fuzzy sets. Let f be a fuzzy subset of a set A. Define fuzzy relations Sf , Tf and Ef on A as follows:   1 if x = y, 1 if f (x) = f (y), , Tf (x, y) = Sf (x, y) = f (x) ⊗ f (y) if x = y, f (x) ⊗ f (y) if f (x) = f (y), Ef (x, y) = f (x) ↔ f (y). It can be easily shown that Sf , Tf and Ef are fuzzy equivalence relations on A, and by (3) we have that Sf Tf Ef , for every fuzzy subset f of A. The relation Ef plays a crucial role in the representation theory of fuzzy equivalence relations, initiated by Valverde in [36], whereas the relation Sf is known as a decomposable relation [5]. Next, let C be a family of fuzzy subsets of a set A. Then fuzzy equivalence relations TC and EC on A are defined by   Tf (x, y), EC (x, y) = Ef (x, y), TC (x, y) = f ∈C

f ∈C

for all x, y ∈ A, whereas SC is defined as the least fuzzy equivalence relation on A such that f SC , for every f ∈ C. For an explicit expression of SC in terms of the join operation in L we refer to [30].

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The first theorem of this section characterizes all fuzzy equivalence relations having a given fuzzy set as an equivalence class. Theorem 3.1. A fuzzy subset f of a set A is an equivalence class of some fuzzy equivalence relation on A if and only if it is normalized. In this case, the set of all E ∈ E(A) such that f is an equivalence class of E is equal to the closed interval [Sf , Ef ] of E(A). Proof. It is clear that any equivalence class of a fuzzy equivalence relation is normalized. Conversely, if f is normalized, then it can be easily verified that f is an equivalence class of any of the fuzzy equivalence relations Sf , Tf and Ef . Further, for any E ∈ E(A) and x, y ∈ A, by the adjunction property (1) and the definition of biresiduum (2) it follows that E(x, y) Ef (x, y) ⇔ E(x, y)f (x) ↔ f (y) ⇔ E(x, y)f (x) → f (y) & E(x, y)f (y) → f (x) ⇔ f (x) ⊗ E(x, y)f (y) & f (y) ⊗ E(x, y)f (x) and we conclude that f is extensional w.r.t. E if and only if E Ef . Therefore, the set of all E ∈ E(A) such that f is extensional w.r.t. E is equal to the principal ideal (Ef ] of E(A) generated by Ef . Moreover, for any E ∈ E(A) and x, y ∈ A, x = y, by (10) and the definition of Sf we conclude f is indistinguishable w.r.t. E if and only if Sf E, and hence, the set of all E ∈ E(A) such that f is indistinguishable w.r.t. E is equal to the principal dual ideal (principal filter) [Sf ) of E(A) generated by Sf . Now, according to Lemma 2.2, the set of all E ∈ E(A) such that f is an equivalence class of E is the intersection of the principal ideal (Ef ] and the principal dual ideal [Sf ) of E(A), i.e., it is equal to the closed interval [Sf , Ef ] of E(A).  Let A = {Ai }i∈I be a family of crisp subsets of a set A. If Ai = ∅, for every i ∈ I , and Ai ∩ Aj = ∅, for any two different i, j ∈ I , then A is called a semi-partition of A, if i∈I Ai = A, then A is called a cover of A, and if A is both a semi-partition and a cover of A, then it is a partition of A. It is well-known that A is a partition of A if and only if it is the set of all equivalence classes of some equivalence relation on A, and that it is a semi-partition of A if and only if it is a subset of the set of equivalence classes of some equivalence relation on A. As we have noted in the introduction, there were various definitions of fuzzy partitions. Since we are particularly interested in relationships between fuzzy partitions and fuzzy equivalence relations, fuzzy partitions and a fuzzy semipartitions will be defined here in a way used by De Baets et al. [8] and Ovchinnikov [32]. Namely, a family C of normalized fuzzy subsets of a set A will be called a fuzzy partition of A if it is the set of all distinct equivalence classes of some fuzzy equivalence relation on A. We also say that C is a fuzzy partition of this fuzzy equivalence relation, or that it is induced by it. A family C is said to be a fuzzy semi-partition of A if it is the set of equivalence classes (not necessarilly all) of some fuzzy equivalence relation on A (i.e., a subset of some fuzzy partition). We also say that C is a fuzzy semi-partition of this equivalence relation. The next theorem gives several characterizations of fuzzy semi-partitions. Theorem 3.2. Let C be a family of normalized fuzzy subsets of a set A. Then the following conditions are equivalent: (i) C is a fuzzy semi-partition of A; (ii) for all f, g ∈ C   f (x) ⊗ g(x) f (y) ↔ g(y); x∈A

(iii) for all x, y ∈ A   f (x) ⊗ f (y)  g(x) ↔ g(y); f ∈C

(iv) SC EC .

(11)

y∈A

g∈C

(12)

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Moreover, if any of the above four conditions is satisfied, then the set of all fuzzy equivalence relations on A having C as its fuzzy semi-partition is equal to the interval [SC , EC ] of E(A). Proof. (i) ⇒ (ii). Let C be a set of equivalence classes of a fuzzy equivalence relation E on A. Consider arbitrary f, g ∈ C. Then f = Ea and g = Eb , for some a, b ∈ A, and by Lemma 2.3 we have that the left-hand side of (11) can be written as follows:  f (x) ⊗ g(x) = f ⊗ g = Ea ⊗ Eb  = E(a, b). x∈A

On the other hand, consider an arbitrary y ∈ A and set Ey = h. Then by Theorem 3.1 we have that E(a, b)Eh (a, b) = h(a) ↔ h(b) = Ey (a) ↔ Ey (b) = Ea (y) ↔ Eb (y) = f (y) ↔ g(y). Therefore, E(a, b)



f (y) ↔ g(y)

y∈A

and hence, we have proved (ii). (ii) ⇒ (iii). Let (ii) hold. Let us observe that (ii) can be stated as f (x) ⊗ g(x)f (y) ↔ g(y),

(13)

for all f, g ∈ C and x, y ∈ A. Consider arbitrary f, g ∈ C and x, y ∈ A. Then (13) yields f (x) ⊗ g(x) f (y) → g(y) and

f (y) ⊗ g(y)f (x) → g(x)

and by the adjunction property it follows that: f (x) ⊗ g(x)f (y) → g(y) ⇔ f (x) ⊗ f (y) g(x) → g(y), f (y) ⊗ g(y)f (x) → g(x) ⇔ f (x) ⊗ f (y) g(y) → g(x). Therefore, we have that f (x) ⊗ f (y)g(x) ↔ g(y) for all f, g ∈ C and x, y ∈ A, and hence, we have proved that (iii) holds. (iii) ⇒ (iv). Let (iii) hold. For arbitrary f ∈ C and x, y ∈ A such that x = y, by (12) it follows that:  g(x) ↔ g(y) = EC (x, y), f (x) ⊗ f (y) g∈C

which yields Sf EC , for every f ∈ C, and hence SC EC . (iv) ⇒ (i). Let SC EC and let E ∈ [SC , EC ]. Then E ∈ [Sf , Ef ], for every f ∈ C, and by Theorem 3.1, any f ∈ C is an equivalence class of E. Therefore, C is a fuzzy semi-partition of A. Finally, if C is a fuzzy semi-partition of A, i.e., if SC EC , then the interval [SC , EC ] of E(A) is the intersection of the intervals [Sf , Ef ], f ∈ C, and by Theorem 3.1 we have that [SC , EC ] is the set of all fuzzy equivalence relations on A having any f ∈ C as its equivalence class.  The inequality (11) can be interpreted as “the degree of overlapping of f and g (the left-hand side) is equal to the degree of equality of f and g (the right-hand side)”, which generalizes the properties of crisp equivalence classes: “if f and g are not disjoint (i.e., have degree of overlapping 1), then f = g (i.e., have degree of equality 1)”. Note that De Baets and Mesiar [10] and Demirci [12] used the condition (ii) in the definition of fuzzy semi-partitions. In the case of fuzzy semi-partitions defined by means of a left-continuous t-norm on the real unit interval, equivalence of the conditions (i) and (ii) was proved by Klawonn and Kruse in [27] (see also papers by De Baets [7], De Baets and Mesiar [10] and Klawonn [26]). Klawonn and Kruse also proved that SC and EC are, respectively, the smallest and the greatest fuzzy equivalence relations having C as its fuzzy semi-partition (see also [26]).

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The next two corollaries describe some properties of fuzzy semi-partitions that will be used in the further text. The first of them was proved by De Baets and Mesiar [10] for T-semi-partitions (fuzzy semi-partitions defined by means of a t-norm on the real unit interval), but here we give a slightly simplified proof. Corollary 3.1. Let C be a fuzzy semi-partition of a set A. Then for all f, g ∈ C 

f (x) ⊗ g(x) =

x∈A



f (y) ↔ g(y).

(14)

y∈A

Proof. For arbitrary f, g ∈ C and a ∈ c(f ), by (6) it follows: 

f (y) ↔ g(y)f (a) ↔ g(a) = g(a) = f (a) ⊗ g(a)



f (x) ⊗ g(x).

x∈A

y∈A

By this and by Theorem 3.2 we conclude that (14) holds.



Corollary 3.2. Let C be a fuzzy semi-partition of a set A. Then for any two different f, g ∈ C the following two assertions hold: (A1) f ⊗ g < 1, (A2) g(x) = f ⊗ g, for every x ∈ c(f ). Proof. The assertion (A1) is a consequence of (14) and (5), whereas (A2) follows immediately by the proof of Corollary 3.1.  Let us observe that the condition (A1) is equivalent to (A1) {c(f )}f ∈C is a semi-partition of A and it ensures that C does not contain two different members with the same crisp part. On the other hand, (A2) asserts that any member of C assumes a constant value on the crisp part of any other member of C, what was pointed out by De Baets and Mesiar in [10]. We will prove later that the condition (A2) and a stronger version of (A1) , which says that {c(f )}f ∈C is a partition of A, are necessary and sufficient for characterization of fuzzy partitions. The following theorem gives a nice representation of a fuzzy equivalence relation SC , in the case when C is a fuzzy semi-partition. It was proved by Klawonn and Kruse in [27], for fuzzy semi-partitions defined by means of a left continuous t-norm on the real unit interval (see also [26,34,35]), but, for the sake of completeness, we will give a proof here. Theorem 3.3. Let C be a fuzzy semi-partition of a set A. Then the relation SC can be represented by: SC (x, x) = 1, for each x ∈ A, and SC (x, y) =



f (x) ⊗ f (y),

(15)

f ∈C

for any two different x, y ∈ A. Proof. Let S be a fuzzy relation on A defined by: S(x, x) = 1, for any x ∈ A, and S(x, y) equals the join on the right-hand side of the equality (15), for different x, y ∈ A. We have that Sf SC , for every f ∈ C, and by the definitions of fuzzy relations Sf and S it follows S SC . On the other hand, we also have that Sf S, for every f ∈ C, and to prove inequality SC S, it is enough to prove transitivity of S.

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Since C is a fuzzy semi-partition of A, we can write C = {Ea }a∈I , for some fuzzy equivalence relation E on A and I ⊆ A, and by transitivity of E and (8) we obtain that       E(x, a) ⊗ E(a, y) ⊗ E(y, b) ⊗ E(b, z) S(x, y) ⊗ S(y, z) = 

a∈I 







E(x, a) ⊗ E(a, y) ⊗ E(y, z) =

a∈I



b∈I





 E(x, a) ⊗ E(a, y) ⊗ E(y, z)

a∈I

 E(x, a) ⊗ E(a, z) = S(y, z),

a∈I

for arbitrary x, y, z ∈ A. This completes the proof of the theorem.



As we have seen in Theorem 3.2, a fuzzy equivalence relation corresponding to a fuzzy semi-partition is not necessarily uniquely determined. But, the next corollary shows that in some cases such fuzzy equivalence relation can be unique. Corollary 3.3. Let C be a family of normalized fuzzy subsets of a set A. Then the following conditions are equivalent: (i) there exists a unique fuzzy equivalence relation having C as its fuzzy semi-partition; (ii) for any two different x, y ∈ A   f (x) ⊗ f (y) = g(x) ↔ g(y); f ∈C

(16)

g∈C

(iii) SC = EC . Proof. This follows by Theorems 3.2 and 3.3.



Example 3.1. (a) Let L = ([0, 1], ∧, ∨, ⊗, →, 0, 1), with the product pair of adjoined operations, and let a fuzzy equivalence relation E on A = {1, 2, 3, 4} be given by the following table: 1

2

3

4

1

1.00

0.12

0.41

0.13

2

0.12

1.00

0.12

0.23

3

0.41

0.12

1.00

0.27

4

0.13

0.23

0.27

1.00

E

For each i ∈ A let Ei denote the equivalence class determined by i. Then C = {E1 , E2 , E3 } is a fuzzy semi-partition which satisfies the conditions of Corollary 3.3, i.e., SC = EC = E. By Theorem 3.4 proved below, C is not a fuzzy partition. (b) Let L = ([0, 1], ∧, ∨, ⊗, →, 0, 1), with the Gödel pair of adjoined operations, and let C = {f, g}, where f and g are fuzzy subsets of A = {1, 2, 3, 4} given by

f

1

2

3

4

1.0

0.1

0.4

0.2

g

1

2

3

4

0.4

0.1

1.0

0.3

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It can be easily verified that Ef = Sf < Sg = Eg , so SC = Sg > Ef = EC . Now, according to Theorem 3.2, we have that C is not a fuzzy semi-partition, but, in spite of that, SC can be represented as in Theorem 3.3, that is, the relation S defined as in the proof of Theorem 3.3 is transitive. Now we will aim our attention to fuzzy partitions, which are characterized by the next theorem: Theorem 3.4. Let C be a family of normalized fuzzy subsets of a set A. Then the following conditions are equivalent: (i) C is a fuzzy partition of A; (ii) C is a fuzzy semi-partition of A and satisfies the condition: (A3) for every x ∈ A there exists f ∈ C such that f (x) = 1. (iii) C satisfies the following conditions: (A4) for every x ∈ A there exists a unique fx ∈ C such that fx (x) = 1; (A5) fx (y) = fx ⊗ fy , for all x, y ∈ A; Moreover, if any of the above five conditions is satisfied, then SC = EC and it is the unique fuzzy equivalence relation on A such that C is its fuzzy partition. Proof. (i) ⇒ (ii). If C is a fuzzy partition of a fuzzy equivalence relation E on A, then for every x ∈ A we have that Ex ∈ C and Ex (x) = 1. Therefore, (ii) holds. (ii) ⇒ (iii). Let (ii) hold. Then by (A3) we have that for every x ∈ A there exists f ∈ C such that f (x) = 1, and if there exist another g ∈ C such that g(x) = 1, then f ⊗ g f (x) ⊗ g(x) = 1, which contradicts (A1). Therefore, we conclude that (A4) holds. The condition (A5) is just another formulation of (A2). (iii) ⇒ (i). Let (iii) hold. In notation from (A4) and (A5), let us define a fuzzy relation E on A as follows: for arbitrary x, y ∈ A let E(x, y) = fx ⊗ fy .

(17)

It is evident that E is reflexive and symmetric. To prove the transitivity, consider arbitrary x, y, z ∈ A. Then by (A5) it follows that: E(x, y) ⊗ E(y, z) = fx ⊗ fy  ⊗ fy ⊗ fz  = fx (y) ⊗ fz (y) fx ⊗ fz  = E(x, z). Hence, E is a fuzzy equivalence relation on A. Since by (A5) we have that E(x, y) = fx (y), for all x, y ∈ A, we conclude that C is the set of all equivalence classes of E, i.e., a fuzzy partition of E. Further, suppose that any of the conditions (i)–(iii) is satisfied. In notations from (iii), by (A4), (A5) and Corollary 3.1 we have that    f (x) ⊗ f (y) = fa (x) ⊗ fa (y) = fx (a) ⊗ fy (a) a∈A

f ∈C

=



a∈A

fx (a) ↔ fy (a) =

a∈A



a∈A

fa (x) ↔ fa (y) =



g(x) ↔ g(y)

g∈C

and by Corollary 3.3 we conclude that SC = EC , and it is the unique fuzzy equivalence relation on A such that C is its fuzzy partition.  Let us note that the condition (A3) and (A4) are, respectively, equivalent to (A3) {c(f )}f ∈C is a cover of A; (A4) {c(f )}f ∈C is a partition of A; and that the fuzzy equivalence relation corresponding to a fuzzy partition C of a set A can be represented in three ways, i.e., for all x, y ∈ A   EC (x, y) = f (x) ↔ f (y) = f (x) ⊗ f (y) = fx ⊗ fy . (18) f ∈C

f ∈C

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If E is a fuzzy equivalence relation on A and x, y ∈ A, then the first equality in (18) yields E(x, y) =



Ex (z) ↔ Ey (z)

z∈A

what means that the degree of relationship of elements x and y is equal to the degree of equality of fuzzy equivalence classes Ex and Ey . Note also that De Baets and Mesiar [10] defined a T-partition (where T is a t-norm) as a T-semi-partition satisfying the condition (A4) , and Ovchinnikov [32] characterized fuzzy partitions by the condition (A4) and a condition that can be stated in our notation as (A5) fx (y) ⊗ fy (x) = fx ⊗ fy , for all x, y ∈ A. It is not hard to verify that (A5) is equivalent to (A5). Some related characterizations of fuzzy partitions can be found in the above mentioned paper by De Baets and Mesiar [10]. Also, for examples of fuzzy partitions of infinite sets, in particular of T-partitions of the set of reals, we refer to the paper by De Baets et al. [9]. Thiele and Schmechel [35] and Demirci [12] studied very close notions to the fuzzy semi-partition and the fuzzy partition in this paper. Namely, a family C of normalized fuzzy subsets of a set A is called a fuzzy semi-partition of A in the sense of Thiele–Schmechel if for any f, g ∈ C and any x, y ∈ A we have g(x) = 1 ⇒ f (x) ⊗ f (y) g(y).

(19)

If, in addition, C satisfies the condition (A3) (or equivalently (A3)), then it is a fuzzy partition of A in the sense of Thiele–Schmechel. Any fuzzy semi-partition, defined as in this paper, is a fuzzy semi-partition in the sense of Thiele– Schmechel (this is an immediate consequence of Theorem 3.2(iii)), but the converse does not hold. For example, a family C = {f, g} from Example 3.1(b) is not a fuzzy semi-partition, but it is a fuzzy semi-partition in the sense of Thiele–Schmechel. However, C is a fuzzy partition, defined as in this paper, if and only if it is a fuzzy partition in the sense of Thiele–Schmechel (cf. [35,12]), and the bijective correspondence between fuzzy equivalence relations and fuzzy partitions in the sense of Thiele–Schmechel was first established by Thiele and Schmechel in [35]. Let E be a fuzzy equivalence relation on a set A and let C be a family of fuzzy subsets of A. If E = EC , then C is said to generate E, and it is called a generating family of E, and the members of C are called generators of E. The well-known Valverde’s Representation Theorem, proved in [36], says that every fuzzy equivalence relation is generated by some family of fuzzy sets. In particular, Valverde proved that any fuzzy equivalence relation is generated by the family of all its equivalence classes (what also follows by Theorem 3.4), but it is often important to find a generating family of minimal cardinality. A family of minimal cardinality in the collection of all generating families of a fuzzy equivalence relation E is called a minimal generating family of E. All minimal generating families of E have the same cardinality, and this cardinality is called the dimension of E. Fuzzy equivalence relations of dimension 1 are called one-dimensional. The concept of the dimension of a fuzzy relation was introduced in [31] (where first representation theorems for fuzzy equivalence relations were also established) and developed further in [17]. A family of minimal cardinality in the collection of all generating families of E consisting of equivalence classes of E will be called a minimal generating fuzzy semi-partition of E. In the next section we will give an algorithm for calculation of the minimal generating fuzzy semi-partition in the case of fuzzy sets and relations assuming membership degrees in a linearly ordered complete Heyting algebra. This algorithm is based on the following theorem: Theorem 3.5. Let C be a family of fuzzy subsets of a set A and E a fuzzy equivalence relation on A. Then the following assertions hold: (a) If C is a fuzzy semi-partition of E, then TC E ⇔

f ∈C

ker f ⊆ c(E)

(20)

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(b) If C is a generating set of E, then

ker f = c(E).

(21)

f ∈C

Proof. (a) Let TC E. Suppose that (x, y) ∈ ker f , for every f ∈ C. Then f (x) = f (y), for every f ∈ C, so TC (x, y) = 1, and now TC E yields E(x, y) = 1, i.e., (x, y) ∈ c(E). Conversely, suppose that

ker f ⊆ c(E), f ∈C

and consider arbitrary x, y ∈ A. If TC (x, y) = 1, then Tf (x, y) = 1, for every f ∈ C, what means that f (x) = f (y), for every f ∈ C, so by the starting hypothesis we conclude that E(x, y) = 1, and therefore, TC (x, y) E(x, y). On the other hand, if TC (x, y) < 1, then there exists f ∈ C such that f (x) = f (y) and Tf (x, y) = f (x) ⊗ f (y), and now, by the indistinguishability of f w.r.t. E, we have that TC (x, y)Tf (x, y) = f (x) ⊗ f (y) E(x, y). Therefore, we have proved that TC E. (b) Let C be a generating set of E, i.e., E = EC . Since c(Eg ) = ker g, for every fuzzy subset g of A, we have that



c(Ef ) = ker f, c(E) = c(EC ) = f ∈C

f ∈C

and hence, (21) holds.  4. Fuzzy equivalence relations over linearly ordered complete Heyting algebras In this section we will consider fuzzy equivalence relations over a complete residuated lattice L having a property that Tf = Ef , for every fuzzy set f over L. Clearly, this is true if and only if L satisfies the condition x⊗y =x ↔y

for all x, y ∈ L such that x = y

(22)

and we first show that this condition holds if and only if L is a linearly ordered complete Heyting algebra. Lemma 4.1. A complete residuated lattice L = (L, ∧, ∨, ⊗, →, 0, 1) satisfies the condition (22) if and only if L is a linearly ordered Heyting algebra. Proof. Let (22) hold. To prove that L is a linearly ordered Heyting algebra, it is enough to prove that for all x, y ∈ L the following is true: x⊗y =x

or

x ⊗ y = y.

(23)

Indeed, suppose that x ⊗ y = x. Then by (3) it follows that x ⊗ y x ∧ y x, y, so (4), (22) and (7) yield   x ⊗ y = (x ⊗ y) ∧ x = (x ⊗ y) ↔ x = (x ⊗ y) → x ∧ x → (x ⊗ y) = x → (x ⊗ y) y, and hence, x ⊗ y = y. Conversely, let L be a linearly ordered Heyting algebra and let x, y ∈ L be any two different elements. If x < y, then x ⊗ y = x ∧ y = x and by (4) and (7) it follows that: x ↔ y = (x → y) ∧ (y → x) = y → x = y → (y ⊗ x) x = x ⊗ y, so by (3) we conclude that x ↔ y = x ⊗ y. In the same way we prove that y < x implies x ↔ y = x ⊗ y, what completes the proof of the lemma. 

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In view of the above lemma, in the further text L will be a linearly ordered complete Heyting algebra. In this case, for any fuzzy subset f of a set A, fuzzy equivalence relations Sf , Tf and Ef are represented as follows:  1 if x = y, Sf (x, y) = f (x) ∧ f (y) if x = y,  1 if f (x) = f (y), Tf (x, y) = Ef (x, y) = f (x) ∧ f (y) if f (x) = f (y). Now we can observe that all fuzzy equivalence relations from the interval [Sf , Ef ] (that is, the ones having f as an equivalence class) have the same value f (x) ∧ f (y) on all pairs (x, y) such that f (x) = f (y), that is, on all pairs (x, y) out of ker f . If for any s ∈ L we set f [s] = {x ∈ A | f (x) = s}, then this situation can be explained by Fig. 1. Theorem 4.1. Let C be a family of normalized fuzzy subsets of a set A. Then the following conditions are equivalent: (i) C is a fuzzy semi-partition of A; (ii) for any two different f, g ∈ C and every y ∈ A f (y) = g(y) ⇒ f (y) ∧ g(y) = f ∧ g; (iii) for any two different f, g ∈ C and all x, y ∈ A g(x) = g(y) ⇒ f (x) ∧ f (y) g(x) ∧ g(y); (iv) for any two different f, g ∈ C the following conditions hold: (B1) g(x) = g(y), for all x, y ∈ c(f ); (B2) for all x, y ∈ A, f (x) = f (y) and g(x) = g(y) implies f (x) ∧ f (y) = g(x) ∧ g(y). Proof. (i) ⇔ (ii). Let us first observe that in the case of fuzzy sets over a linearly ordered complete Heyting algebra the condition (ii) of Theorem 3.2 can be stated as f ∧ gf (y) ↔ g(y)

(24)

for any two different f, g ∈ C and y ∈ A. Since the right-hand side of the inequality (24) is equal to 1 whenever f (y) = g(y), then (24) is equivalent to f (y) = g(y) ⇒ f ∧ g f (y) ∧ g(y)

(25)

and since the inequality f (y) ∧ g(y)f ∧ g holds by the definition of the height of a fuzzy set, we conclude that (25) is equivalent to f (y) = g(y) ⇒ f ∧ g = f (y) ∧ g(y). By this and by Theorem 3.2 we conclude that the conditions (i) and (ii) are equivalent. (i) ⇔ (iii). This equivalence can be proved in the same way as (i) ⇔ (ii). (i) ⇒ (iv). Let C be a fuzzy semi-partition of A. By Corollary 3.2 we have that (B1) holds, and since we have proved that (i)⇔(iii), then (B2) follows by (iii). (iv) ⇒ (i). Let any two different f, g ∈ C satisfy the conditions (B1) and (B2). We will prove that any f ∈ C is an equivalence class of E = EC . Consider arbitrary f ∈ C, x ∈ c(f ) and y ∈ A, and set Cx,y = {g ∈ C | g(x) = g(y)}. Assume first that f ∈ Cx,y . Then Eg (x, y) = 1, for every g ∈ C\Cx,y , and by (B2) we have that Eg (x, y) = Ef (x, y), for every g ∈ Cx,y , so   Ex (y) = E(x, y) = Eg (x, y) = Eg (x, y) = Ef (x, y) = f (y). g∈C

g∈Cx,y

1308

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Fig. 1. Fuzzy equivalence relations from the interval [Sf , Ef ] may differ from Sf and Ef , and each other, only in values in the gray area of the figure (here 0  t  s  1).

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Further, suppose that f ∈ / Cx,y , i.e., f (x) = f (y). Then x, y ∈ c(f ), and by (B1) it follows that g(x) = g(y), for every g ∈ C \ {f }, so we conclude that Eg (x, y) = 1, for every g ∈ C, and  Ex (y) = E(x, y) = Eg (x, y) = 1 = f (y). g∈C

Therefore, we have proved that f = Ex . This completes the proof of the theorem.



In other words, (B3) means that Ef and Eg must coincide on all pairs (x, y) from the crisp set (relation) (ker f )c ∩ (ker g)c = (ker f ∪ ker g)c . Let C be a fuzzy semi-partition of a set A such that for any two different x, y ∈ A there exists f ∈ C such that f (x) = f (y). Then for any two different x, y ∈ A and f ∈ C such that f (x) = f (y), by Lemma 4.1 we have that f (x) ∧ f (y) = f (x) ↔ f (y), so   h(x) ∧ h(y). g(x) ↔ g(y) f (x) ↔ f (y) = f (x) ∧ f (y)  g∈C

h∈C

By this and by Theorems 3.2 and 3.3 it follows that there exists a unique fuzzy equivalence relation having C as its fuzzy semi-partition. This fact was pointed out by Klawonn [26] and Demirci [12] (see Corollary 3.2 in [26] or the comment just after Theorem 3.5 in [12]). Theorem 4.2. Let C be a fuzzy semi-partition of a fuzzy equivalence relation E on a set A. Then C is a generating family of E if and only if

ker f ⊆ c(E). (26) f ∈C

Proof. If C is a generating family of E, then (26) follows by Theorem 3.5(b). Conversely, if (26) holds, then by Theorem 3.5(a) we have that EC = TC E EC , and hence, E = EC .



Corollary 4.1. Let E be a fuzzy equivalence relation on a set A. Then E is one-dimensional if and only if there exists an equivalence class f of E such that ker f ⊆ c(E). Let E be a fuzzy equivalence relation on a set A and let P be the fuzzy partition induced by E, i.e., the family of all its equivalence classes. For any two different elements x, y ∈ A, an equivalence class f ∈ P is said to separate elements x and y if f (x) = f (y). The collection of all equivalence classes of E which separate the elements x and y will be denoted by Px,y , i.e. Px,y = {f ∈ P | f (x) = f (y)}. It is easy to check that for all x, y ∈ A the following hold: (C1) Px,y = Py,x ; (C2) Px,y = ∅ if and only if E(x, y) < 1; (C3) if E(x, y) = 1, i.e., Ex = Ey , then Px,z = Py,z , for every z ∈ A. In particular, if E(x, y) < 1, then both Ex and Ey separate x and y, i.e., Ex , Ey ∈ Px,y . Further, let S(E) = {Px,y | x, y ∈ A, E(x, y) < 1}. and let Smin (E) be the set of all minimal elements of the partially ordered set (S(E), ⊆). Theorem 4.3. Let E be a fuzzy equivalence relation on a set A, let P be the fuzzy partition induced by E, and let C ⊆ P. Then the following conditions are equivalent: (i) C is a generating family of E;

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(ii) C intersects all members of S(E); (iii) C intersects all members of Smin (E). Proof. The set C intersects all members of S(E) if and only if for all x, y ∈ A such that E(x, y) < 1 there exists f ∈ C such that f (x) = f (y), what is evidently equivalent to

ker f ⊆ c(E) (27) f ∈C

and according to Theorem 4.2 we conclude that (i) ⇔ (ii). The implication (ii) ⇒ (iii) is evident, whereas (iii) ⇒ (ii) is an immediate consequence of the fact that any member of S(E) contains some member of Smin (E).  Let A¯ be a subset of A having a property that for any f ∈ P there exists exactly one a ∈ A¯ such that f = Ea , which is called a representative of this equivalence class. In other words, A¯ is a set containing exactly one element from every equivalence class of a crisp equivalence relation c(E), so for any two different a, b ∈ A¯ we have that E(a, b) < 1. By (C3) it follows that for any pair x, y ∈ A such that E(x, y) < 1 there exist two different a, b ∈ A¯ such that Px,y = Pa,b , so S(E) can be also expressed as ¯ a = b}. S(E) = {Pa,b | a, b ∈ A, Therefore, when compute the collections S(E) and Smin (E), it is enough to consider any set A¯ of representatives of equivalence classes of E instead of the whole set A. It is also possible to consider representatives of equivalence classes instead of these classes as members of S(E) and Smin (E), and in this case S(E) and Smin (E) are considered as ¯ subsets of A. 5. Algorithm for calculation of a minimal generating fuzzy semi-partition In the sequel we state an algorithm for calculation of minimal generating fuzzy semi-partitions. Algorithm 5.1. Calculation of a minimal generating fuzzy semi-partition of a given fuzzy equivalence relation E on a finite set A = {1, 2, . . . , n}: 1. 2. 3. 4. 5.

Form the family P of all equivalence classes of E; Choose any set A¯ of representatives of equivalence classes; ¯ i < j }; For any pair i, j ∈ A¯ such that i < j compute the set Pi,j , and form the collection S(E) = {Pi,j | i, j ∈ A, Eliminate all non-minimal elements of S(E) and form the collection Smin (E) of all minimal elements of S(E); Find a subset C ⊆ P with minimal number of elements, which intersects all members of Smin (E).

The resulting family C is a minimal generating fuzzy semi-partition of E. If a fuzzy equivalence relation E on A is represented by a n × n matrix, then the equivalence classes of E are represented by the rows of this matrix. In Step 2, for any i ∈ A and j > i we check if the jth row is equal to the ith one, and if it is true, we eliminate j from A. The elements which remained in A after this procedure form the set A¯ of representatives of equivalence classes of E. ¯ we check if the kth positions in the ith Then in Step 3, for any pair i, j ∈ A¯ such that i < j , and any k ∈ A, ¯ and the jth columns differ, and we form the set Pi,j from all k ∈ A having this property. In this way we form the collection S(E). In Step 4, for any P ∈ S(E) we check if it is a minimal member of S(E). Namely, for any Q ∈ S(E) we check if Q ⊂ P . If we find such Q, we conclude that P is not a minimal member of S(E), and we check the next member of S(E). Otherwise, if Q ⊂ P for every Q ∈ S(E), we conclude that P is a minimal member of S(E), and we include it into Smin (E). In this way we form the collection Smin (E).

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Finally, in Step 5 we determine all cross-sections of members of Smin (E), taking one element from any member of Smin (E), and then we find the cross-sections with minimal number of elements. They are minimal generating fuzzy semi-partitions of E. In what follows we give two examples which demonstrate application of this algorithm. Example 5.1. Let L = ([0, 1], ∧, ∨, ⊗, →, 0, 1), with the Gödel pair of adjoined operations, and let fuzzy equivalence relations E and F on A = {1, 2, . . . , 8} be given by 1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

1

1.0

0.2

0.2

0.2

0.2

0.2

0.0

0.0

1

1.0

1.0

1.0

0.4

0.4

0.3

0.0

0.0

2

0.2

1.0

0.3

0.4

0.3

0.3

0.0

0.0

2

1.0

1.0

1.0

0.4

0.4

0.3

0.0

0.0

3

0.2

0.3

1.0

0.3

0.3

0.3

0.0

0.0

3

1.0

1.0

1.0

0.4

0.4

0.3

0.0

0.0

4

0.2

0.4

0.3

1.0

0.3

0.3

0.0

0.0

4

0.4

0.4

0.4

1.0

0.4

0.3

0.0

0.0

5

0.2

0.3

0.3

0.3

1.0

0.8

0.0

0.0

5

0.4

0.4

0.4

0.4

1.0

0.3

0.0

0.0

6

0.2

0.3

0.3

0.3

0.8

1.0

0.0

0.0

6

0.3

0.3

0.3

0.3

0.3

1.0

0.0

0.0

7

0.0

0.0

0.0

0.0

0.0

0.0

1.0

0.0

7

0.0

0.0

0.0

0.0

0.0

0.0

1.0

1.0

8

0.0

0.0

0.0

0.0

0.0

0.0

0.0

1.0

8

0.0

0.0

0.0

0.0

0.0

0.0

1.0

1.0

E

F

(a) We have that c(E) is the equality relation on A, so E has 8 different classes, i.e., P = {E1 , E2 , . . . , E8 } and A¯ = A. In the sequel, instead of the classes we will write their representatives, and then we have the following: P1,i = {1, . . . , 6} for 2 i 6; Pi,7 = {1, . . . , 6, 7} for 1i 6; Pi,8 = {1, . . . , 6, 8} for 1i 6; P2,3 = P3,4 = {2, 3, 4}; P2,4 = {2, 4}; P2,5 = P2,6 = P4,5 = P4,6 = {2, 4, 5, 6}; P3,5 = P3,6 = {3, 5, 6}; P5,6 = {5, 6}; P7,8 = {7, 8}. Eliminating the sets which are not minimal, we obtain that Smin (E) consists of the sets {2, 4}, {5, 6} and {7, 8}, and E has 8 minimal generating fuzzy semi-partitions given by {Ei , Ej , Ek }, for any choice of i ∈ {2, 4}, j ∈ {5, 6} and k ∈ {7, 8}. Note that neither of these generating families is a minimal generating family in the collection of all generating families of E. Namely, E has a two-element generating family {f, g}, where f and g are given by

f

1

2

3

4

5

6

7

8

1.0

1.0

1.0

0.4

0.3

0.3

0.1

0.0

g

1

2

3

4

5

6

7

8

0.2

0.3

0.5

0.3

0.8

1.0

0.0

0.0

This is a minimal generating family, because if E would have a one-element generating family, then its single member would be an equivalence class of E.

´ c et al. / Fuzzy Sets and Systems 158 (2007) 1295 – 1313 M. Ciri´

1312

(b) It is evident that F has 5 different classes, and we can assume that A¯ = {1, 4, 5, 6, 7} and P = {F1 , F4 , F5 , F6 , F7 }. Now, P1,4 P1,5 P1,6 P1,7 P4,5

= {1, 4}; = {1, 5}; = P4,6 = P5,6 = {1, 4, 5, 6}; ¯ = P4,7 = P5,7 = P6,7 = A; = {4, 5}

and we have that Smin (F ) consists of the sets {1, 4}, {1, 5} and {4, 5}. Clearly, sets with minimal number of elements intersecting all members of Smin (F ) are the same sets {1, 4}, {1, 5} and {4, 5}, so F has 3 minimal generating fuzzy semi-partitions: {F1 , F4 }, {F1 , F5 } and {F4 , F5 }. The next example shows that Theorem 4.2, as well as Theorem 4.3 and Algorithm 5.1, hold only in the case when the complete residuated lattice L of truth values is a linearly ordered Heyting algebra, that is, if it satisfies the condition (22). Example 5.2. Suppose that L is any complete residuated lattice which does not satisfy the condition (22), i.e., there exist ,  ∈ L such that  =  and  ⊗  <  ↔ . Since  = 1 or  = 1 implies  ⊗  =  ↔ , we conclude that  < 1 and  < 1. Also, by  =  it follows that  ↔  < 1. Set  =  ↔ . Now, let f and g be fuzzy subsets of a set A = {1, 2, 3} defined as in the figure given below, and let E be a fuzzy equivalence relation on A generated by the family {f, g}.

f

g

1

2

3

1





1

2

1

1





1

2

3

1

1





1

1

3

1

2

3

Λ Λ

1

2

3

2



1

γ

2



1

γ

2

Λ

1

γ

1





3



γ

1

3



γ

1

3

Λ

γ

1

Ef

Ef

E = Ef Λ Eg

Then ker f = ker g = A = c(E) (where A is the equality relation on A), and the tables representing Ef , Eg and E demonstrate that families {f } and {g} do not generate E. This example also shows that the condition (22) is necessary to Tf = Ef be satisfied for every fuzzy set f over L, because here we have that Tf (2, 3) =  ⊗  <  ↔  = Ef (2, 3), and hence, Tf < Ef . 6. Concluding remarks In this paper we investigated various properties of equivalence classes of fuzzy equivalence relations over a complete residuated lattice. We gave certain characterizations of fuzzy semi-partitions and fuzzy partitions over a complete residuated lattice, as well as over a linearly ordered complete Heyting algebra. In the latter case, for a fuzzy equivalence relation over a linearly ordered complete Heyting algebra, we constructed an algorithm for calculation of a minimal family of its equivalence classes which generates it. Most of the presented results are new, but some of them are generalizations of known results given in a way which simplifies and clarifies them. In the follow-up papers the obtained results will be applied to study of fuzzy relations between two sets, fuzzy mappings and fuzzy homomorphisms, as well as in study of fuzzy languages and their recognition by fuzzy automata.

´ c et al. / Fuzzy Sets and Systems 158 (2007) 1295 – 1313 M. Ciri´

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Acknowledgements The author are grateful to the referees for valuable remarks which helped to improve the paper. References [1] [3] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [29] [30] [31] [32] [33] [34] [35]

[36] [37]

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