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Fuzzy Sets and Systems ••• (••••) •••–•••www.elsevier.com/locate/fss

Fuzzy α–C-equivalences

Urszula Bentkowska, Anna Król ∗

Interdisciplinary Centre for Computational Modelling, Department of Mathematics and Natural Sciences, University of Rzeszów, Pigonia 1, 35-310 Rzeszów, Poland

Received 24 June 2017; received in revised form 10 November 2017; accepted 7 January 2018

Abstract

In this paper new notions of fuzzy equivalences interpreted as fuzzy connectives are introduced. One of the axioms of such fuzzy equivalences is transitivity defined by the use of a fuzzy conjunction or both a fuzzy conjunction and a parameter α from the unit interval, that is why they are called fuzzy α–C-equivalences. In particular, fuzzy C-equivalences, fuzzy semi-C-equivalences as well as fuzzy weak C-equivalences are considered. Dependencies between the new types of fuzzy equivalences are presented. Moreover, the problem of preservation of axioms of such defined fuzzy equivalences by aggregation functions is examined. As a conclusion, ways of generating new fuzzy equivalences of the considered type from given ones are indicated.© 2018 Elsevier B.V. All rights reserved.

Keywords: Fuzzy connectives; Aggregation functions; Fuzzy C-equivalences; Fuzzy α–C-equivalences; Fuzzy semi-C-equivalences; Fuzzy weak C-equivalences; Dominance

1. Introduction

In this contribution new types of fuzzy equivalences, interpreted as fuzzy connectives, are introduced. The concepts come from the idea of a fuzzy relation and especially an equivalence relation. Namely, the type of the possible definition of a fuzzy equivalence, which is based on the notion of fuzzy equivalence relation which is reflexive, symmetric and transitive, is taken into account. If it comes to fuzzy relations, first of all the transitivity property is of the most interest because of the application reasons. The standard transitivity property is too strong for many relations and this is a reason why this property is modified in different ways according to demands of real-life problems [7,11,25,28,45].

In this paper both classical transitivity axiom of a fuzzy equivalence connective and its weaker versions are con-sidered. These types of transitivity for fuzzy connectives are defined by the use of a fuzzy conjunction or both a fuzzy conjunction and a parameter α from the unit interval and are called fuzzy α–C-equivalences. Diverse types of transi-tivity properties are discussed and new types of fuzzy equivalences as fuzzy connectives are obtained. In particular, fuzzy C-equivalences, fuzzy semi-C-equivalences as well as fuzzy weak C-equivalences are considered. Furthermore,

* Corresponding author.E-mail addresses: ududziak@ur.edu.pl (U. Bentkowska), annakrol@ur.edu.pl (A. Król).

https://doi.org/10.1016/j.fss.2018.01.0040165-0114/© 2018 Elsevier B.V. All rights reserved.

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in the paper diverse dependencies between obtained classes are presented. Moreover, the problem of preservation of axioms of such defined fuzzy equivalences by aggregation functions is examined. As a conclusion, ways of generat-ing new fuzzy equivalences of the considered type from given ones are indicated. Some of the considered here new concepts were presented at conferences IWIFSGN 2012 (cf. [22]) and IFSA-EUSFLAT 2015 (cf. [2]).

Aggregation functions can be useful in a variety of information fusion problems [5,18]. Fundamental properties of aggregation functions were collected in [23]. The problem of aggregation of diverse mathematical objects is rather well known. We may aggregate for example fuzzy relations and consider the problem of preservation of fuzzy relation properties during aggregation process (e.g. [12,23,34,35]) or examine fuzzy connectives [17] and preservation of their properties by aggregation functions [16]. Examination of preservation of axioms and properties of fuzzy connectives finds its applications in decision making, approximate reasoning and fuzzy control. Moreover, preservation in the aggregation process not only individual properties, but all axioms of a given kind of fuzzy connective shows the possible way of generating new fuzzy connectives from a given one. In this context, some of the fuzzy connectives (negation, conjunction, disjunction and implication) were examined in [16], a fuzzy implication was examined in [10].

This paper is structured as follows. In Section 2, basic notions useful in the paper are presented. In Section 3, diverse types of fuzzy equivalences are described and dependencies between them are shown, and in Section 4, aggregation of these types of fuzzy equivalences are discussed and new ways of generating fuzzy equivalences by aggregation process are proposed.

2. Preliminaries

Here we recall basic notions which will appear in the sequel. We consider some properties of functions, next aggregation functions, relation of dominance between operations, and fuzzy conjunctions.

Lemma 1 (cf. [13], Lemma 1.1).

∀a,b∈[0,1] ∀

α∈[0,1] (a � α ⇒ b � α) ⇔ b � a, (1)

∀a,b∈[0,1] ∀

α∈[0,1] (a � α ⇒ b � α) ⇔ a � b. (2)

Lemma 2 ([21]). Let n ∈N, n � 2. If F : [0, 1]n → [0, 1] is increasing with respect to any variable and has a neutral element e = 1, i.e. F(1, . . . , 1, t, 1, . . . , 1) = t for any t ∈ [0, 1] where t is in the k-th position, then F � min.

Definition 1. Let n ∈N, n � 2. We say that a function F : [0, 1]n → [0, 1]:• has a zero element z ∈ [0, 1] (cf. [9], Definition 10) if

∀1�k�n

∀x1,...,xk−1,xk+1,...,xn∈[0,1] F(x1, ..., xk−1, z, xk+1, ..., xn) = z, (3)

• is without zero divisors (where z is the zero element) if

∀x1,...,xn∈[0,1] (F (x1, ..., xn) = z ⇒ ( ∃

1�k�nxk = z)), (4)

• fulfils strong 1-boundary condition if

∀x1,...,xn∈[0,1] (F (x1, ..., xn) = 1 ⇔ ( ∀

1�k�nxk = 1)). (5)

Lemma 3 ([19]). Let n ∈ N, n � 2. A function F : [0, 1]n → [0, 1] has a zero element z = 0 and is without zero divisors if and only if

∀x1,...,xn∈[0,1] (F (x1, ..., xn) > 0 ⇔ ( ∀

1�k�nxk > 0)). (6)

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2.1. Aggregation functions

Now we present some properties of aggregation functions.

Definition 2 (cf. [9], pp. 6–22, [27], pp. 216–218). Let n ∈N, n � 2. A function A : [0, 1]n → [0, 1] which is increas-ing with respect to any variable is called an aggregation function if A(0, . . . , 0) = 0, A(1, . . . , 1) = 1. Moreover, an aggregation function A is called a mean if it is idempotent, i.e. A(x, . . . , x) = x for any x ∈ [0, 1].

Example 1 (cf. [9], pp. 44–56). Some examples of aggregation functions are:

• quasi-arithmetic means

A(x1, . . . , xn) = ϕ−1(1

n

n∑k=1

ϕ(xk)), (7)

• quasi-linear means

A(x1, ..., xn) = ϕ−1(

n∑k=1

wkϕ(xk)), (8)

where wk > 0, k = 1, ..., n, n∑

k=1wk = 1, x1, . . . , xn ∈ [0, 1] and ϕ : [0, 1] → [−∞, +∞] is a continuous, strictly

monotonic function.

Well known means such as weighted geometric means or weighted arithmetic means are special cases of quasi-linear means. Other families of aggregation functions can be found for example in [1,9].

Theorem 1 ([19]). Let A : [0, 1]n → [0, 1] be an aggregation function, ϕ : [0, 1] → [0, 1] be an increasing bijection. If A has a zero element z = 0 and it is an operation without zero divisors, then operation Aϕ has the same properties, where Aϕ is isomorphic to A, i.e.

Aϕ(x1, ..., xn) = ϕ−1(A(ϕ(x1), ..., ϕ(xn))), x1, ..., xn ∈ [0,1]. (9)

2.2. Fuzzy conjunctions

Fuzzy conjunctions are examples of aggregation functions (for n = 2). There are diverse approaches to define a fuzzy conjunction. We will apply the most general one (with the monotonicity condition involved).

Definition 3 ([16]). An operation C : [0, 1]2 → [0, 1] is called a fuzzy conjunction if it is increasing with respect to any variable and

C(1,1) = 1, C(0,0) = C(0,1) = C(1,0) = 0.

Corollary 1. A fuzzy conjunction has a zero element z = 0. Conversely, if a binary aggregation function has a zero element z = 0, then it is a fuzzy conjunction.

Corollary 2. If a fuzzy conjunction C : [0, 1]2 → [0, 1] has a neutral element 1, then it is a fuzzy conjunction fulfilling property C � min.

There exist many subfamilies of fuzzy conjunctions. We recall the ones used in this contribution.

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Definition 4. An operation C : [0, 1]2 → [0, 1] is called:

• an overlap function [8] if it is a commutative, continuous fuzzy conjunction without zero divisors, fulfilling condition C(x, y) = 1 if and only if x = y = 1,

• a t-seminorm [24] if it is a fuzzy conjunction and has a neutral element 1,• a t-norm [43] if it is increasing, commutative, associative, and has a neutral element 1• a quasi-copula [32, Section 6.2] if it is a 1-Lipschitz t-seminorm, i.e.

|C(x, y) − C(u, v)| � |x − u| + |y − v|, x, y,u, v ∈ [0,1].

Example 2. Consider the following family of fuzzy conjunctions for a ∈ [0, 1]

Ca(x, y) =

⎧⎪⎨⎪⎩

1, if x = y = 1

0, if x = 0 or y = 0

a otherwise

. (10)

Operations C0 and C1 are the least and the greatest fuzzy conjunction, respectively. A fuzzy conjunction Ca has no zero divisors if and only if a ∈ (0, 1]. Other examples of fuzzy conjunctions are listed below. These are well-known t-norms: minimum, product, Łukasiewicz, drastic, which are denoted in the traditional way TM , TP , TL, TD , respectively, where TP is a fuzzy conjunction without zero divisors:

TM(x, y) = min(x, y), TP (x, y) = xy,

TL(x, y) = max(x + y − 1,0), TD(x, y) =

⎧⎪⎨⎪⎩

x, if y = 1

y, if x = 1

0, otherwise

.

Example 3. The condition C � min is fulfilled by the following fuzzy conjunctions from Example 2: C0, TM , TP , TL, TD . Moreover, except for overlap functions, all families of fuzzy conjunctions from Definition 4 fulfil the inequality C � min. For overlap functions this inequality does not necessarily hold.

The following classes of t-norms will be useful in the sequel.

Definition 5. [27, p. 28] A strict t-norm T : [0, 1]2 → [0, 1] is a t-norm which is continuous and strictly increasing in (0, 1]2.

Theorem 2. [27, p. 126] Any strict t-norm T is a function isomorphic to the product t-norm TP , i.e.

T (x, y) = ϕ−1(TP (ϕ(x),ϕ(y))), x, y ∈ [0,1],where ϕ : [0, 1] → [0, 1] is an increasing bijection.

Corollary 3. Strict t-norms are conjunctions without zero divisors.

2.3. Dominance

Dominance is a dependence between operations. It will be applied in Section 4. This is why we recall or prove some useful results which are related to the concept of dominance.

Definition 6 (cf. Saminger et al. [35], Definition 2.5; cf. [44], p. 209). Let m, n ∈ N. An operation F : P m → P

dominates an operation G : P n → P (shortly F G), if for any matrix [ai,k] = A ∈ P m×n they fulfil the inequality

F(G(a1,1, . . . , a1,n), . . . ,G(am,1, . . . , am,n)) � G(F(a1,1, . . . , am,1), . . . ,F (a1,n, . . . , am,n)). (11)

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Let us consider examples of dominance between operations.

Example 4. Any two projections

Pk(x1, ..., xn) = xk, k ∈ {1, ..., n}, x1, ..., xn ∈ [0,1] (12)

dominate each other.

We may also characterize aggregation functions A which dominate C for some given fuzzy conjunction C.

Theorem 3 (cf. [35], Proposition 5.2). Let A : [0, 1]n → [0, 1] be an aggregation function. Then A TD if and only if there exists a non-empty subset I = {k1, ..., km} ⊂ {1, ..., n}, k1 < ... < km, and an increasing mapping B :[0, 1]m → [0, 1] satisfying the following conditions: B(0, ..., 0) = 0 and B(u1, ..., um) = 1 ⇔ u1 = ... = um = 1, such that A(x1, ..., xn) = B(xk1 , ..., xkm).

A function B in Theorem 3 is an aggregation function and for a t-norm T we have T (x1, x2) = 1 if and only if x1 = x2 = 1 and thus I = {1, 2}, so B = T and T TD ([35], p. 32).

Example 5. Any quasi-arithmetic mean dominates TD and any t-seminorm dominates TD (both aggregation functions fulfil strong 1-boundary condition (5)).

Below we recall a characterization of aggregation functions which dominate minimum.

Theorem 4 (cf. [35], Proposition 5.1). Let A be an aggregation function. Thus A min if and only if

A(x1, ..., xn) = min(f1(x1), ..., fn(xn)), x1, ..., xn ∈ [0,1], (13)

where fk : [0, 1] → [0, 1] are increasing mappings, k = 1, ..., n.

Let us notice some simple examples of functions that fulfil (13).

Example 6. Let us consider an aggregation function A and increasing functions fk : [0, 1] → [0, 1] for k = 1, ..., nthat fulfil (13). If fk(x) = x, k = 1, ..., n, then A = min.

If for some k ∈ {1, ..., n}, fk(x) = x, fi(x) = 1 for i �= k, then A = Pk .If fk(x) = max(1 − vk, x), vk ∈ [0, 1], k = 1, ..., n, max

1�k�nvk = 1, then A is the weighted minimum

A(x1, ..., xn) = min1�k�n

max(1 − vk, xk), v = (v1, ..., vn) ∈ [0,1]n, max1�k�n

vk = 1. (14)

Here we present some other examples of dominance between functions.

Example 7 (cf. [15,35]). A weighted geometric mean dominates t-norm TP . A weighted arithmetic mean dominates t-norm TL. An aggregation function

A(t1, . . . , tn) = p

n

n∑k=1

tk + (1 − p) min1�k�n

tk (15)

dominates TL, where p ∈ (0, 1). A weighted minimum dominates any t-norm (see [15]). Moreover, A Pk and Pk A for any aggregation function A (see [14]).

Corollary 4 ([3]). Minimum dominates any fuzzy conjunction. A weighted minimum dominates any t-seminorm.

We also recall some very known results on dominance in the family of t-norms and t-conorms and some properties of dominance for arbitrary operations.

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Theorem 5. (cf. [35, p. 16]) Let T be a t-norm. Then

T T , TM T TD, TD T ⇔ T = TD, T TM ⇔ T = TM.

Proposition 1 (cf. [35]). Let us consider increasing binary operations F, G : [0, 1]2 → [0, 1], a bijection ϕ : [0, 1] →[0, 1] and

Fϕ(x, y) = ϕ−1(F (ϕ(x),ϕ(y))), x, y ∈ [0,1]. (16)

If ϕ is increasing, then F G ⇔ Fϕ Gϕ .If ϕ is decreasing, then F G ⇔ Fϕ Gϕ .

Proposition 2 (cf. [44], p. 209). Let F, G : [0, 1]2 → [0, 1] have a common neutral element e ∈ [0, 1]. If F G, then F � G.

By Propositions 1 and 2 we obtain the results for fuzzy conjunctions.

Corollary 5. For any two t-seminorms C1, C2 and any order isomorphism ϕ : [0, 1] → [0, 1], it holds that C1 C2if and only if (C1)ϕ (C2)ϕ . Moreover, for any t-seminorms C1, C2 if C1 C2, then C1 � C2.

Results for dominance of a fuzzy conjunction with a neutral element 1 are listed below.

Theorem 6 ([36]). If a t-seminorm dominates TL then it is a quasi-copula.If a t-seminorm dominates TP then it is a quasi-copula.If a t-seminorm C1 dominates a t-seminorm C2, then C1 = TM whenever C2 = TM .

Other results on dominance may be found in [27, pp. 152–156], [3,4,26,30,37–42].

3. Dependencies between new types of fuzzy equivalences

There exist various definitions of a fuzzy equivalence. In many contributions, for example those concerning gener-alized logical laws, it is used an equality, that is the function E : [0, 1]2 → [0, 1] given by the formula

E(x, y) ={

1, if x = y

0, if x �= y. (17)

It is natural to expect that such notion of a fuzzy equivalence is a generalization of the equivalence of classical propositional calculus, that is the function E : [0, 1]2 → [0, 1] that fulfils conditions E(0, 1) = E(1, 0) = 0, E(0, 0) =E(1, 1) = 1. We will apply the approach in which definition of a fuzzy equivalence follows from the notion of a fuzzy equivalence relation, namely relation which is reflexive, symmetric and transitive (cf. [31,46]). There are also many other approaches of defining fuzzy equivalences (cf. [6,7,23,29]).

Definition 7 (cf. [33], p. 33). Let C be a fuzzy conjunction. A fuzzy C-equivalence is a function E : [0, 1]2 → [0, 1]fulfilling the following conditions

E(0,1) = 0 (boundary property), (18)

E(x, x) = 1, x ∈ [0,1] (reflexivity), (19)

E(x, y) = E(y, x), x, y ∈ [0,1] (symmetry), (20)

C(E(x, y),E(y, z)) � E(x, z) x, y, z ∈ [0,1] (C-transitivity). (21)

In the cited monograph [33] property (18) is omitted. However, in this case the constant function E(x, y) = 1, x, y ∈ [0, 1] fulfils the definition of a fuzzy equivalence although it is not a generalization of crisp equivalence. This is why we add this assumption to the definition.

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In the transitivity condition (21) an operation C is involved. We assume that C is a fuzzy conjunction, in particular it can be a triangular norm widely used in the literature. Now, we give example of a fuzzy C-equivalence for some family of fuzzy conjunctions.

Proposition 3. Let C be a fuzzy conjunction. The function

E(x, y) ={

1, if x = y

min(x, y), otherwise(22)

is a fuzzy C-equivalence if and only if C � min.

Proof. Obviously, conditions (18)–(20) are fulfilled. Let us examine the property of a C-transitivity (21). (⇒) Let x, z ∈ [0, 1]. If x �= z then for y = 1 one has

C(x, z) = C(E(x,1),E(1, z)) � E(x, z) = min(x, z).

On the other hand if x = z then for x = 1 one has C(x, x) = C(1, 1) = 1 � 1 = x, and for x �= 1 there exists t � x and then C(x, x) � C(x, t) � min(x, t) = x.

(⇐) Let C � min, x, y, z ∈ [0, 1]. If x �= y and y �= z then

C(E(x, y),E(y, z)) = C(min(x, y),min(y, z)) � min(min(x, y),min(y, z)) � min(x, z) � E(x, z).

For x = y one obtains

C(E(x, y),E(y, z)) = C(1,min(y, z)) � min(y, z) = min(x, z) � E(x, z).

Similarly, for y = z one has

C(E(x, y),E(y, z)) = C(min(x, y),1) � min(x, y) = min(x, z) � E(x, z). �We may weaken conditions given in the previous definition by replacing transitivity property with the appropriate

weaker transitivity conditions.

Definition 8. Let α ∈ [0, 1], C be a fuzzy conjunction. A fuzzy weak C-equivalence is a function E : [0, 1]2 → [0, 1]fulfilling conditions (18)–(20) and

∀x,y,z∈X

C(E(x, y),E(y, z)) > 0 ⇒ E(x, z) > 0. (23)

A fuzzy semi-C-equivalence is a function E : [0, 1]2 → [0, 1] fulfilling conditions (18)–(20) and

∀x,y,z∈X

C(E(x, y),E(y, z)) = 1 ⇒ E(x, z) = 1. (24)

A fuzzy α–C-equivalence is a function E : [0, 1]2 → [0, 1] fulfilling conditions (18)–(20) and

∀x,y,z∈X

C(E(x, y),E(y, z)) � 1 − α ⇒ C(E(x, y),E(y, z)) � E(x, z). (25)

The conditions (23)–(25) will be called a weak C-transitivity, a semi-C-transitivity and an α–C-transitivity, respec-tively.

Example 8. For any fuzzy conjunction C and α ∈ [0, 1], the function (17) is a fuzzy α–C-equivalence and a fuzzy C-equivalence.

The following statement indicates further examples of fuzzy α–C-equivalences.

Proposition 4. Let a ∈ (0, 1), α ∈ [0, 1). The function E given by (22) is a fuzzy α–Ca-equivalence if and only if a < 1 − α.

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Fig. 1. Dependencies between classes of fuzzy equivalences.

Proof. Obviously, conditions (18)–(20) are fulfilled. Let us examine the property of an α–Ca-transitivity (25).(⇒) For the proof by contradiction let us assume that a � 1 − α and let us consider x ∈ (0, 1) such that x < a and

y = z = 1. We have

Ca(E(x, y),E(y, z)) = Ca(E(x,1),E(1,1)) = Ca(x,1) = a � 1 − α

and E(x, z) = E(x, 1) = x < a. Hence Ca(E(x, y), E(y, z)) > E(x, z) and the function E is not α–Ca-transitive.(⇐) Let us assume that for some x, y, z ∈ [0, 1] we have Ca(E(x, y), E(y, z)) � 1 − α. By the assumption that

a < 1 − α and the formula of a fuzzy conjunction Ca (10) we obtain that Ca(E(x, y), E(y, z)) = 1. Again by the definition of Ca and E we have E(x, y) = E(y, z) = 1 and x = y = z. Thus, we obtain E(x, z) = 1, what shows that inequality Ca(E(x, y), E(y, z)) � E(x, z) holds and the function E is α–Ca-transitive. �

Examples of other types of fuzzy equivalences will be provided in the sequel. Now, in the consecutive statements and examples, we will present dependencies both inside one class of fuzzy connectives (with respect to a fuzzy conjunction and with respect to a parameter α) and between the considered classes. These dependencies, with respect to a fixed fuzzy conjunction C, are also presented in Fig. 1, where SC means the class of fuzzy semi-C-equivalences, WC means the class of fuzzy weak-C-equivalences, C means the class of fuzzy C-equivalences and α–C means the class of all fuzzy α–C-equivalences. First we present the figure and then provide the appropriate statements to justify the picture.

If it comes to dependencies between the considered classes, directly from definitions of a fuzzy C-equivalence, fuzzy weak C-equivalence and fuzzy semi-C-equivalence we get the following properties.

Corollary 6. Let C be a fuzzy conjunction. If E : [0, 1]2 → [0, 1] is a fuzzy C-equivalence, then it is both a fuzzy weak C-equivalence and a fuzzy semi-C-equivalence.

Corollary 7. Let C : [0, 1]2 → [0, 1] be a fuzzy conjunction. Function E : [0, 1]2 → [0, 1] is a fuzzy 1–C-equivalence if and only if E is a fuzzy C-equivalence. Function E is a fuzzy 0–C-equivalence if and only if E is a fuzzy semi-C-equivalence.

The following example shows that the notions of a fuzzy weak C-equivalence and a fuzzy semi-C-equivalence are not equivalent to the notion of a fuzzy C-equivalence.

Example 9. Let us consider a function

E(x, y) ={

1, if x = ymin(x,y)max(x,y)

, if x �= y(26)

and the fuzzy conjunction C = min. Obviously E fulfils conditions (18)–(20). We shall show that it does not fulfil (21).

For x = 0.2, y = 0.4, z = 0.8 one has

C(E(x, y),E(y, z)) =min

(min(x, y)

max(x, y),

min(y, z)

max(y, z)

)= min

(0.2

0.4,

0.4

0.8

)= 0.5.

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Moreover, E(x, z) = min(x,z)max(x,z)

= 0.20.8 = 0.25. Thus, C(E(x, y), E(y, z)) > E(x, z) and the function (26) is not a fuzzy

min-equivalence.Now, let us observe that from the assumption min(E(x, y), E(y, z)) > 0 it follows that both E(x, y) > 0 and

E(y, z) > 0. This holds if x = y = z or x, y, z > 0. In both cases we have E(x, y) > 0. This is why the function (26)is a fuzzy weak min-equivalence.

Let us consider a condition min(E(x, y), E(y, z)) = 1. From this it follows that both E(x, y) = 1 and E(y, z) = 1and it means that x = y and y = z. Thus, x = z and E(x, z) = 1. This means that the function (26) is a fuzzy semi-min-equivalence.

There is no correspondence between properties (23) and (24) what is shown in the next examples.

Example 10. Let us consider an arbitrary fuzzy conjunction C and the function

E(x, y) =

⎧⎪⎨⎪⎩

1, if x = y

0.5, if x �= y, (x, y) ∈ (0,1)2

0, otherwise

. (27)

Directly by the formula of E it follows that it fulfils conditions (18)–(20). Let us consider a weak C-transitivity condition in its contrapositive form:

∀x,y,z∈X

E(x, z) = 0 ⇒ C(E(x, y),E(y, z)) = 0.

By the definition of the function E it follows that E(x, z) = 0 if:1) x = 0 and z > 0 or2) x > 0 and z = 0 or3) x = 1 and z < 1 or4) x < 1 and z = 1For an arbitrary y ∈ [0, 1] let us consider the value Vy := C(E(x, y), E(y, z)).1) If y > 0, then E(x, y) = E(0, y) = 0 and because of a zero element of C (cf. Corollary 1) Vy = 0. If y = 0, then for z > 0 we have E(y, z) = E(0, z) = 0 and Vy = 0.2) If y > 0, then E(y, z) = E(y, 0) = 0, so Vy = 0. If y = 0, then for x > 0 we have E(x, y) = E(x, 0) = 0 and Vy = 0.3) If y < 1, then E(x, y) = E(1, y) = 0, so Vy = 0. If y = 1, then for z < 1 we have E(y, z) = E(1, z) = 0 and Vy = 0.2) If y < 1, then E(y, z) = E(y, 1) = 0, so Vy = 0. If y = 1, then for x < 1 we have E(x, y) = E(x, 1) = 0 and Vy = 0.This is why the function (27) is weakly C-transitive for any fuzzy conjunction C. However, this function may not be semi-C-transitive for some fuzzy conjunction C. Let us consider a fuzzy conjunction

C(x, y) ={

min(x, y), if y � 1 − x

1, otherwise. (28)

For the values x = y = 0.6, z = 0.8 we have C(E(x, y), E(y, z)) = C(E(0.6, 0.6), E(0.6, 0.8)) = C(1, 0.5) = 1, but E(x, z) = E(0.6, 0.8) = 0.5. This is why the function (27) is not semi-C-transitive for the given fuzzy conjunction C.

Example 11. Now, let us assume that C is an arbitrary fuzzy conjunction fulfilling strong 1-boundary condition, where a ∈ (0, 1) is an idempotent element of C.

E(x, y) =

⎧⎪⎨⎪⎩

1, if x = y

0, if {x, y} = {0,1}a, otherwise

. (29)

Directly by the formula of E it follows that it fulfils conditions (18)–(20). Let us consider a weak C-transitivity. For x = 0, y = 0.5, z = 1 we have C(E(x, y), E(y, z)) = C(E(0, 0.5), E(0.5, 1)) = C(a, a) = a > 0, but C(x, z) =

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C(0, 1) = 0. Thus, the function (29) is not weakly C-transitive. Now, let us observe that for the given operations Cand E we have

C(E(x, y),E(y, z)) = 1 ⇔ E(x, y) = 1 ∧ E(y, z) = 1 ⇔ x = y ∧ y = z ⇒ x = z ⇒ E(x, z) = 1.

Thus, the function (29) is semi-C-transitive.

Now we will analyze dependencies inside the class of fuzzy α–C-equivalences with respect to the parameter α. As a consequence we will also get the dependencies between this class of fuzzy equivalences and the remaining classes.

Proposition 5. Let β ∈ [0, 1], C : [0, 1]2 → [0, 1] be a fuzzy conjunction. If E : [0, 1]2 → [0, 1] is a fuzzy β–C-equivalence, then it is a fuzzy α–C-equivalence for any α ∈ [0, β].

Proof. Let x, y ∈ [0, 1], α, β ∈ [0, 1] and α � β , C : [0, 1]2 → [0, 1] be a fuzzy conjunction. If E is a fuzzy β–C-equivalence and C(E(x, y), E(y, z)) � 1 − α, then C(E(x, y), E(y, z)) � 1 − α � 1 − β and by assumption that E is a fuzzy β–C-equivalence we get E(x, z) � C(E(x, y), E(y, z)). As a result E is a fuzzy α–C-equivalence for arbitrary α ∈ [0, β]. �

By Proposition 5 and Corollary 7 we get

Corollary 8. Let C : [0, 1]2 → [0, 1] be a fuzzy conjunction. If E : [0, 1]2 → [0, 1] is a fuzzy C-equivalence, then Eis a fuzzy α–C-equivalence for any α ∈ [0, 1].

Corollary 9. Let C : [0, 1]2 → [0, 1] be a fuzzy conjunction, α ∈ [0, 1]. If E : [0, 1]2 → [0, 1] is a fuzzy α–C-equivalence, then E is a fuzzy semi-C-equivalence.

The following example shows that a fuzzy α–C-equivalence may not be a fuzzy C-equivalence and simultaneously presents examples of fuzzy α–C-equivalences.

Example 12. Let us consider the function E given by (22) and a fuzzy conjunction Ca for a ∈ (0, 1). As Ca does not fulfil the condition Ca � min thus, by Proposition 3, E is not a fuzzy Ca-equivalence. On the other hand, by Proposition 4, it is a fuzzy α–Ca-equivalence for any α < 1 − a.

Let us recall that by Corollary 6 and Corollary 7, if E is a fuzzy α–C-equivalence for α = 1, then E is a fuzzy weak C-equivalence. However, if E is a fuzzy weak C-equivalence, then it need not be a fuzzy α–C-equivalence for any α ∈ [0, 1] or it may be a fuzzy α–C-equivalence for some α ∈ [0, 1]. As a result there is no clear dependencebetween a fuzzy α–C-equivalence and a fuzzy weak C-equivalence, what will be shown in the sequel (Examples 13, 14).

Example 13. Let us consider the function E given by (27) and a fuzzy conjunction C given by (28). According to Example 10, E is a fuzzy weak C-equivalence, however it is not a fuzzy semi-C-equivalence. By Corollary 9, the latter denotes that E is not a fuzzy α–C-equivalence for any α ∈ [0, 1].

Example 14. Let us consider the function E given by (22), a ∈ [0, 1], and a fuzzy conjunction Ca given by (10). By Example 12, E is a fuzzy α–Ca-equivalence for any α < 1 − a and it is easy to show that it is a fuzzy weak Ca-equivalence.

There are also other dependencies for considered here classes of fuzzy equivalences. Namely, all considered here families are descending ones with respect to a fuzzy conjunction (this is an example of dependence inside each class).

Proposition 6. Let α ∈ [0, 1], C1, C2 be fuzzy conjunctions, C1 � C2. If E : [0, 1]2 → [0, 1] is a fuzzy α–C2-equiva-lence (respectively fuzzy C2-equivalence, fuzzy weak C2-equivalence, fuzzy semi-C2-equivalence), then it is also a fuzzy α–C1-equivalence (respectively fuzzy C1-equivalence, fuzzy weak C1-equivalence, fuzzy semi-C1-equivalence).

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Proof. Let α ∈ [0, 1], C1, C2 : [0, 1]2 → [0, 1] be fuzzy conjunctions, C1 � C2. If E is a fuzzy α–C2-equivalence, then

C1(E(x, y),E(y, z)) � 1 − α ⇒ C2(E(x, y),E(y, z)) � 1 − α ⇒E(x, z) � C2(E(x, y),E(y, z)) ⇒ E(x, z) � C1(E(x, y),E(y, z)),

so E is a fuzzy α–C1-equivalence. Justification for other properties is analogous. �Directly by Proposition 6 we see that an α–C-transitivity, based on the parameter α, is a weaker property than a

C-transitivity.

Corollary 10. Let α ∈ [0, 1], C : [0, 1]2 → [0, 1] be a fuzzy conjunction, C � min. If E : [0, 1]2 → [0, 1] is a fuzzy min-equivalence, then it is a fuzzy α–C-equivalence.

4. Preservation of diverse types of fuzzy C-equivalences in aggregation process

We may aggregate diverse objects, for example fuzzy relations or fuzzy connectives (cf. [23]). Here it will be considered aggregation of fuzzy equivalences defined in the previous section.

Definition 9 (cf. [23], p. 14). Let A : [0, 1]n → [0, 1] be an aggregation function. For given fuzzy equivalences E1, . . . , En, we consider an aggregated connective

E(x, y) = A(E1(x, y), . . . ,En(x, y)), x, y ∈ [0,1]. (30)

An aggregation function A preserves a property of the given fuzzy equivalences if the operation E described by (30)has such a property for arbitrary E1, . . . , En fulfilling this property. A class of fuzzy equivalences is closed under an aggregation function A if the result of the aggregation belongs to this class for arbitrary fuzzy equivalences from the class.

Preservation of the fuzzy equivalence types in aggregation process is a way of generating new examples of fuzzy equivalence of the considered type.

Example 15. Any projection preserves a fuzzy C-equivalence and a fuzzy α–C-equivalence (for any α ∈ [0, 1]) because in the formula (30) with A = Pk we get E = Ek for k ∈ {1, . . . , n}.

By the boundary conditions of aggregation functions we get

Lemma 4 (cf. [16]). Any aggregation function preserves binary truth tables and symmetry of the aggregated fuzzy equivalences.

Lemma 5. Any aggregation function preserves the property (19) of the aggregated operations.

By Lemmas 4 and 5 it follows that any aggregation function preserves boundary property, symmetry and reflexivity of any fuzzy equivalence considered in this paper. Thus, for aggregation of these fuzzy equivalences it is enough to consider only transitivity conditions. We will do it in the sequel starting with the strongest type of it, namely C-transitivity. Below we present a result from [15] written in other terminology.

Theorem 7. Let C be a fuzzy conjunction. An aggregation function A : [0, 1]n → [0, 1] preserves C-transitivity of the aggregated fuzzy equivalences E1, ..., En if and only if A dominates C.

By Theorem 7 and Proposition 6 we get

Corollary 11. Let C1, C2 be fuzzy conjunctions, C1 � C2. If an aggregation function A dominates C2 and E1, ..., En

are fuzzy C2-equivalences, then A(E1, ..., En) is a fuzzy C1-equivalence.

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By Theorem 7, Lemmas 4 and 5 one obtains

Theorem 8. The family of all fuzzy C-equivalences is closed under an aggregation function A that dominates C.

By Corollary 4 we get

Corollary 12. Minimum preserves a fuzzy C-equivalence for any fuzzy conjunction C. A weighted minimum preserves a fuzzy C-equivalence for any t-seminorm C.

By Examples 5–7 and Theorem 5 we get examples of aggregation functions preserving the given type of fuzzy equivalence.

Example 16. Quasi-arithmetic means A preserve a fuzzy TD-equivalence and t-seminorms A preserve a fuzzy TD-equivalence. A weighted minimum (14) preserves a fuzzy C-equivalence for C = min. Weighted arithmetic means preserve a fuzzy TL-equivalence. Geometric mean preserves a fuzzy TP -equivalence. An aggregation function A de-scribed by the formula (15) preserves a fuzzy TL-equivalence. Any t-norm preserves a fuzzy C-equivalence for Cbeing a t-norm. The only t-norm that preserves a fuzzy C-equivalence for C = min is min.

Now let us pay attention to preservation of a fuzzy α–C-equivalence. The special case for α = 1 (C-transitivity) has been just presented.

Lemma 6 ([20]). Let α ∈ [0, 1], t = (t1, ..., tn) ∈ [0, 1]n. F : [0, 1]n → [0, 1] fulfils

F |[0,1]n\[1−α,1]n < 1 − α (31)

if and only if

∀t∈[0,1]n F (t1, ..., tn) � 1 − α ⇒ min

1�k�ntk � 1 − α. (32)

Theorem 9. Let α ∈ [0, 1], C : [0, 1]2 → [0, 1] be a fuzzy conjunction. If an aggregation function A : [0, 1]n → [0, 1]fulfils (31) and A C, i.e. for any (s1, ..., sn), (t1, ..., tn) ∈ [0, 1]n

A(C(s1, t1), ...,C(sn, tn)) � C(A(s1, . . . , sn),A(t1, . . . , tn)), (33)

then it preserves a fuzzy α–C-equivalence.

Proof. Let α ∈ [0, 1], C : [0, 1]2 → [0, 1] be a fuzzy conjunction, x, y, z ∈ [0, 1]. If E1, ..., En are fuzzy α–C-equivalences, A fulfils (31) and (33) then using notations

sk = Ek(x, y), tk = Ek(y, z), k = 1, ..., n (34)

and

E(x, y) = A(E1(x, y), ...,En(x, y)), x, y ∈ [0,1], (35)

and applying Lemma 6 one obtains

C(E(x, y),E(y, z)) � 1 − α ⇔ C(A(E1(x, y), ...,En(x, y)),A(E1(y, z), ...,En(y, z))) � 1 − α ⇔C(A(s1, ..., sn),A(t1, ..., tn)) � 1 − α ⇒ A(C(s1, t1), ...,C(sn, tn)) � 1 − α ⇒min

1�k�nC(sk, tk)� 1 − α ⇒ ∀

1�k�nC(sk, tk) � 1 − α ⇔ ∀

1�k�nC(Ek(x, y),Ek(y, z)) � 1 − α.

So if Ek are fuzzy α–C-equivalences we see that Ek(x, z) � C(Ek(x, y), Ek(y, z)), for each k = 1, ..., n. Thus by the monotonicity of A and (35), (34), (33) one has

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C(E(x, y),E(y, z)) = C(A(E1(x, y), ...,En(x, y)),A(E1(y, z), ...,En(y, z))) =C(A(s1, ..., sn),A(t1, ..., tn)) � A(C(s1, t1), ...,C(sn, tn)) =A(C(E1(x, y),E1(y, z)), ...,C(En(x, y),En(y, z))) � A(E1(x, z), ...,En(x, z)) = E(x, z)

which proves that E is a fuzzy α–C-equivalence. �Theorem 10. Let C : [0, 1]2 → [0, 1] be a fuzzy conjunction. If an aggregation function A : [0, 1]n → [0, 1] fulfils A C and A � min, then it preserves a fuzzy α–C-equivalence for any α ∈ [0, 1].

Proof. Let A fulfil (33). By Lemma 6, A fulfils (31) for arbitrary α ∈ [0, 1] if and only if

∀α∈[0,1] ∀

t∈[0,1]n A(t1, ..., tn) � 1 − α ⇒ min1�k�n

tk � 1 − α.

As a result, by (2), this is equivalent to the fact that A � min. By Theorem 9 an aggregation function A preserves a fuzzy α–C-equivalence for any α ∈ [0, 1]. �Example 17. Theorem 10 gives only sufficient conditions for preservation, by an aggregation function, a fuzzy α–C-equivalence for any α ∈ [0, 1] and fuzzy conjunction C : [0, 1]2 → [0, 1]. Let us consider a projection Pk , k ∈N. We know that Pk preserves a fuzzy α–C-equivalence for any α ∈ [0, 1] (see Example 15), it dominates any operation C : [0, 1]2 → [0, 1] ([14], Corollary 1), so also a fuzzy conjunction C, but it is not true that Pk � min.

Let us observe that Lemma 2 shows a way of finding aggregation functions A : [0, 1]n → [0, 1] which fulfil the condition A � min. Typical examples of such binary operations are t-norms and fuzzy conjunctions fulfilling condition C � min (Corollary 2, Example 3). Moreover, we have many results concerning relation of dominance in the family of t-norms (Theorem 5, Corollary 4). Applying these facts the following results hold true.

Corollary 13. Let T be an arbitrary t-norm. Minimum preserves a fuzzy α–T -equivalence for any α ∈ [0, 1]. Tpreserves a fuzzy α–TD-equivalence for any α ∈ [0, 1]. T preserves a fuzzy α–T -equivalence for any α ∈ [0, 1]. Minimum preserves a fuzzy α–C-equivalence for any fuzzy conjunction C and any α ∈ [0, 1].

By Theorem 6 we obtain

Corollary 14. If a quasi-copula dominates TL (TP ), then it preserves a fuzzy α–TL-equivalence (a fuzzy α–TP -equiva-lence) for any α ∈ [0, 1].

Now we will turn to the problem of preservation of a fuzzy semi-C-equivalence which is a special case of α–C-equivalence (α = 0) which has been just considered. However, there are additional results for this special case of α–C-equivalence. Firstly, let us notice that there exists an aggregation function which preserves a fuzzy semi-C-equivalence for any fuzzy conjunction C.

Proposition 7. Let C be a fuzzy conjunction. Minimum preserves a fuzzy semi-C-equivalence.

Proof. Let E1, . . . , En be a fuzzy semi-C-equivalences, E = min(E1, . . . , En), and x, y, z ∈ [0, 1]. If C(E(x, y),

E(y, z)) = 1, then by monotonicity of C

1 = C( min1�k�n

Ek(x, y), min1�k�n

Ek(y, z)) � C(Ei(x, y),Ei(y, z)), i = 1, ..., n,

so C(Ei(x, y), Ei(y, z)) = 1 and since E1, . . . , En are fuzzy semi-C-equivalences we get Ei(x, z) = 1, for i =1, ..., n. Therefore

E(x, z) = min(E1(x, z), . . . ,En(x, z)) = 1

and E is a fuzzy semi-C-equivalence. �

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There is another sufficient condition for preservation of semi-C-transitivity by aggregation functions (cf. Theo-rem 9).

Theorem 11. If an aggregation function A : [0, 1]n → [0, 1] and a fuzzy conjunction C : [0, 1]2 → [0, 1] fulfil strong 1-boundary condition (5), then A preserves a fuzzy semi-C-equivalence.

Proof. Let x, y, z ∈ [0, 1], E1, ..., En be fuzzy semi-C-equivalences and both A and C have strong 1-boundary property. If C(E(x, y), E(y, z)) = 1, then we get E(x, y) = 1, E(y, z) = 1, so A(E1(x, y), ..., En(x, y)) = 1 and A(E1(y, z), ..., En(y, z)) = 1. In virtue of (5), we obtain E1(x, y) = 1, ..., En(x, y) = 1, E1(y, z) = 1, ..., En(y, z) =1. By condition (5) for conjunction C we have C(Ek(x, y), Ek(y, z)) = 1, k = 1, ..., n and since Ek , k = 1, ..., nare fuzzy semi-C-equivalences we obtain Ek(x, z) = 1, k = 1, ..., n. Using (5) we have A(E1(x, z), ..., En(x, z)) =E(x, z) = 1, so E is a fuzzy semi-C-equivalence. �Example 18. Theorem 11 gives only a sufficient condition for a preservation of a fuzzy semi-C-equivalence. We see that minimum preserves a fuzzy semi-C-equivalence for C = C1 (the greatest conjunction), where

C1(x, y) ={

0, if x = 0 or y = 0

1 otherwise.

Let x, y, z ∈ [0, 1], E1, ..., En be fuzzy semi-C-equivalences and C(E(x, y), E(y, z)) = 1. Thus we have E(x, y) >0 and E(y, z) > 0. So A(E1(x, y), ..., En(x, y)) > 0 and A(E1(y, z), ..., En(y, z)) > 0, where A(x1, . . . , xn) =min

1�k�nxk . As a result Ek(x, y) > 0 for k = 1, ..., n and Ek(y, z) > 0 for k = 1, ..., n, thus C(Ek(x, y), Ek(y, z)) = 1

for k = 1, ..., n. By assumption E1, ..., En are fuzzy semi-C-equivalences, so Ek(x, z) = 1 for k = 1, ..., n. Finally, E(x, z) = A(E1(x, z), ..., En(x, z)) = min(1, ..., 1) = 1. This proves that E is a fuzzy semi-C-equivalence for the given aggregation function A and conjunction C. However, C = C1 does not fulfil condition (5).

Corollary 15. Weighted arithmetic means preserve fuzzy semi-C-equivalence for a t-seminorm (or a t-norm or an overlap function) C.

Proof. The property follows from the fact that t-norms, t-seminorms, overlap functions and weighted arithmetic means have strong 1-boundary property. �

Analogously to the way presented in Theorem 1 we may generate new families of aggregation functions preserving semi-C-equivalence.

Proposition 8. Let ϕ : [0, 1] → [0, 1] be an increasing bijection. If an aggregation function A : [0, 1]n → [0, 1] fulfils (5), then Aϕ also fulfils (5).

Since quasi-linear means have the strong 1-boundary property, by Corollary 15, Lemma 2 and Proposition 8 we get

Corollary 16. Let C be a t-seminorm (or a t-norm or an overlap function). The quasi-linear means preserve a fuzzy semi-C-equivalence.

Now we want to pay attention to preservation of a fuzzy weak C-equivalence by aggregation functions. There exists an aggregation function which preserves a fuzzy weak C-equivalence for any fuzzy conjunction C. Similarly to the proof of Proposition 7, applying monotonicity of C, we obtain

Proposition 9. Let C be a fuzzy conjunction. Minimum preserves a fuzzy weak C-equivalence.

Theorem 12. If an aggregation function A : [0, 1]n → [0, 1] has a zero element z = 0 and both A and a fuzzy con-junction C are operations without zero divisors, then A preserves a fuzzy weak C-equivalence.

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Proof. Let E1, ..., En be fuzzy weak C-equivalences. If C(E(x, y), E(y, z)) > 0, then because C as a conjunction has a zero element z = 0 and by Lemma 3 we get E(x, y) > 0 and E(y, z) > 0. As a result from (30) we get

A(E1(x, y), ...,En(x, y)) > 0, A(E1(y, z), ...,En(y, z)) > 0.

By Lemma 3 and assumptions about function A we have

E1(x, y) > 0, ...,En(x, y) > 0,E1(y, z) > 0, ...,En(y, z) > 0.

Again, in virtue of Lemma 3 and by assumptions about A and C we see that C(Ek(x, y), Ek(y, z)) > 0, k = 1, ..., n. Since Ek for k = 1, ..., n are fuzzy weak C-equivalences we get Ek(x, z) > 0, k = 1, ..., n and A(E1(x, z), ..., En(x, z)) > 0, so E(x, z) > 0 which proves that E is a fuzzy weak C-equivalence. �Example 19. Theorem 12 gives only a sufficient condition for preservation of a fuzzy weak C-equivalence. Weighted arithmetic means preserve a fuzzy weak TL-equivalence but weighted arithmetic means do not have a zero element z = 0.

By Theorem 1 and Theorem 12 we get

Proposition 10. Let A : [0, 1]n → [0, 1] be an aggregation function and ϕ : [0, 1] → [0, 1] be an increasing bijection. If A preserves a fuzzy weak C-equivalence, then Aϕ also preserves this property (cf. (9)).

By Example 2, Theorem 1, Theorem 12, and Lemma 2 we get

Corollary 17. Functions TP , TM , any strict t-norm or overlap function preserve a fuzzy weak C-equivalence for C = min. A weighted geometric mean preserves a fuzzy weak C-equivalence for C being a strict t-norm or an overlap function.

5. Conclusion

In this contribution, new definitions of fuzzy equivalences as fuzzy connectives were proposed. The presented defi-nitions differ in the (generalized) transitivity property that is the one that involves a fuzzy conjunction C and optionally a parameter α ∈ [0, 1]. Dependencies between them were indicated. Preservation of these fuzzy equivalences by ag-gregation functions were examined and as a result new ways of generating fuzzy equivalences, from the given ones, by the use of admissible aggregation functions were considered. For the future work it will be interesting to discuss other equivalence notions (e.g. [23], p. 33) important from the application point of view and examine dependencies between them.

Acknowledgements

This work was partially supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge in Rzeszów, through Project Number RPPK.01.03.00-18-001/10.

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