Preservation of graded properties of fuzzy relations by aggregation functions

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1 Preservation of graded properties of fuzzy relations by aggregation functions Urszula Dudziak Institute of Mathematics, University of Rzeszów, Rzeszów, ul. Rejtana 16a, Poland. Abstract We study graded properties (α properties) of fuzzy relations, which are parameterized versions of fuzzy relation properties defined by L.A. Zadeh. Namely, we take into account fuzzy relations which are: α reflexive, α irreflexive, α symmetric, α antisymmetric, α asymmetric, α connected, α transitive, where We also pay our attention to the composed versions of these basic properties, e.g. an α equivalence, α orders. Using functions of n variables we consider aggregation fuzzy relation of given fuzzy relations. We give conditions for functions to preserve graded properties of fuzzy relations. Keywords: aggregation function, fuzzy relation properties. 1 Introduction Aggregations of fuzzy relations are important in group choice theory and multiple criteria decision making so it is interesting to know which functions preserve properties of fuzzy relations during aggregation process. There are many works devoted to this topic (e.g. [5, 8, 9]). We consider graded properties of fuzzy relations during aggregations of finite families of these relations. These graded properties of fuzzy relations are parameterized versions of the well-known properties of fuzzy relations and they are generalizations of these properties. In order to receive more general results, we concentrate on a fixed property of fuzzy relations and give conditions on operations in [0, 1] (not only aggregation functions) to preserve this property in aggregation process. Next, we put examples of aggregation functions which fulfil the respective conditions. Presented results are the continuation of the ones obtained in [3, 4]. First, in Section 2 we recall definitions which will be used in the sequel. Next, we discuss the graded properties of fuzzy relations (Section 3), and we examine properties such as: reflexivity and irreflexivity, asymmetry and connectedness, symmetry and transitivity (Section 4), composed properties (Section 5). 2 Preliminaries Now we recall some definitions which will be used in our considerations. Definition 1 ([10]). A fuzzy relation in X is an arbitrary function R : X X [0, 1]. The family of all fuzzy relations in X is denoted by FR(X). Definition 2 (cf. [1]). Let n 2. Operation F : [0, 1] n [0, 1] is called an aggregation function if it is increasing in each variable and fulfils boundary conditions F (0,..., 0) = 0, F (1,..., 1) = 1. With the use of n argument functions F we aggregate given fuzzy relations R 1,..., R n. Definition 3 (cf. [5], p. 107). Let F : [0, 1] n [0, 1], R 1,..., R n F R(X). An aggregation fuzzy relation R F F R(X) is described by the formula R F (x, y) = F (R 1 (x, y),..., R n (x, y)), x, y X. (1) A function F preserves a property of fuzzy relations if for every R 1,..., R n F R(X) having this property, R F also has this property. Example 1. Projections P k (t 1,..., t n ) = t k, t 1,..., t n [0, 1], k {1,..., n} are aggregation functions and they preserve each property of fuzzy relations because for F = P k we get R F = R k. In our further considerations we will need the following Definition 4 ([6], p. 4). Triangular norm T : [0, 1] 2 [0, 1] (triangular conorm S : [0, 1] 2 [0, 1]) is an arbitrary associative, commutative, increasing in both variables operation having a neutral element e = 1 (e = 0).

2 Examples of triangular norms (t-norms) and triangular conorms (t-conorms) and results concerning these binary operations, which are examples of aggregation functions, are given in [6]. In our considerations we will need some more notions and properties. Definition 5 (cf. [9], Definition 2.5). Let m, n N. Operation F : [0, 1] m [0, 1] dominates operation G : [0, 1] n [0, 1] ( F G), if for arbitrary matrix [a ik ] = A [0, 1] m n we have F (G(a 11,..., a 1n ),..., G(a m1,..., a mn )) (2) G(F (a 11,..., a m1 ),..., F (a 1n,..., a mn )). Theorem 1 (cf. [9], Proposition 5.1). Increasing in each variable function F : [0, 1] n [0, 1] dominates minimum if and only if for any t 1,..., t n [0, 1] F (t 1,..., t n ) = min(f 1 (t 1 ),..., f n (t n )), (3) where functions f k : [0, 1] [0, 1] are increasing, for k = 1,..., n. Example 2. Here are examples of functions which fulfil the condition (3): if f k (t) = t, k = 1,..., n then F = min, if for some k {1,..., n}, f k (t) = t, f i (t) = 1 for i k then F = P k, (cf. Example 1), if f k (t) = max(1 v k, t), v k [0, 1], k = 1,..., n, max v k = 1 then F is the weighted minimum F (t 1,..., t n ) = min max(1 v k, t k ), (4) where t = (t 1,..., t n ) [0, 1] n. Lemma 1 (cf. [2], Lemma 1.1). a,b [0,1] a,b [0,1] (a α b α) b a, (5) α [0,1] (a α b α) a b. (6) α [0,1] Remark 1. If card X = n, X = {x 1,..., x n }, then R F R(X) may be presented by a matrix: R = [r ik ], where r ik = R(x i, x k ), i, k = 1,..., n. 3 Graded properties of fuzzy relations We will consider the following parameterized versions of fuzzy relations properties. Definition 6 (cf. [2], p. 75). Let Relation R F R(X) is: α reflexive, if α irreflexive, if x X R(x, x) α, (7) R(x, x) 1 α, (8) x X α symmetric, if R(x, y) 1 α R(y, x) R(x, y), (9) α asymmetric, if α antisymmetric, if x,y,x y X totally α connected, if α connected, if x,y,x y X α transitive, if x,y,z X min(r(x, y), R(y, x)) 1 α, (10) min(r(x, y), R(y, x)) 1 α, (11) max(r(x, y), R(y, x)) α, (12) max(r(x, y), R(y, x)) α, (13) min(r(x, y), R(y, z)) 1 α (14) R(x, z) min(r(x, y), R(y, z)). Example 3. Let card X = 2. Relation R F R(X), where [ ] R = is α reflexive for α [0, 0.3], α irreflexive for α [0, 0.6], α symmetric for α [0, 0.3), α asymmetric and α antisymmetric for α [0, 0.5], totally α connected for α [0, 0.3] and α connected for α [0, 0.7]. It is worth mentioning that in [7] a special case of α transitivity is considered. Namely, this is the 0.5 transitivity according to condition (14). However, the problem of preservation of this property during aggregation process is not discussed. Example 4. Let R c, where c [0, 1]. Relation R F R(X) is α transitive for arbitrary Remark 2. Conditions (7) - (14) for α = 1 become properties of fuzzy relations introduced by Zadeh [11]. More results describing α properties (7) - (8), (10) - (13) one can find in [3], where connections between the properties of fuzzy relations and its cuts properties are discussed. Here, we give such considerations only for α symmetry and α transitivity, which were not discussed in [3]. We recall the following Definition 7 (cf. [10]). Let R F R(X), An α cut of a relation R is a set of the form R α = {(x, y) : R(x, y) α} X X. (15)

3 Theorem 2. Let β [0, 1]. If relation R F R(X) is β symmetric (β transitive), then it is α symmetric (α transitive) for any α [0, β]. Proof. Let x, y X,α, β [0, 1] and α β. If relation R F R(X) is β-symmetric and R(x, y) 1 α, then R(x, y) 1 α 1 β and by β symmetry of R we get R(y, x) R(x, y). As a result relation R is α symmetric for arbitrary α [0, β]. The proof for α transitivity is analogous. There are the following connections between the properties of fuzzy relations and the properties of its cuts. Theorem 3. Let If relation R F R(X) is α symmetric, then R 1 α is symmetric. Proof. Let α [0, 1], R F R(X), x, y X. If R is α symmetric, then by (9) and (15) (x, y) R 1 α R(x, y) 1 α R(y, x) R(x, y) R(y, x) 1 α (y, x) R 1 α. Thus the cut R 1 α is symmetric. The converse theorem to Theorem 3 does not hold. Example 5. Let card X = 2, α [0, 1], R F R(X) be the one from Example 3. The cuts R β are symmetric for β [0, 0.5] (0.7, 1], so the cuts R 1 α have this property for α 0.5 and α < 0.3. The fuzzy relation R is α symmetric for α [0, 0.3), so for α = 0.5, the cut R 0.5 is symmetric, but R is not 0.5 symmetric. Now we will consider the property of an α transitivity. Theorem 4. Let If relation R F R(X) is α transitive, then R 1 α is transitive. Proof. Let α [0, 1], R F R(X), x, y, z X. If R is α transitive, then by (15) and (14) we get ((x, y), (y, z) R 1 α ) (R(x, y) 1 α, R(y, z) 1 α) min(r(x, y), R(y, z)) 1 α R(x, z) min(r(x, y), R(y, z)) R(x, z) 1 α (x, z) R 1 α, so relation R 1 α is transitive. The converse theorem to Theorem 4 does not hold. Example 6. Let R F R(X), card X = 3, R = The cuts R β are transitive for β [0, 0.6] (0.8, 1], so the cuts R 1 α have this property for α [0, 0.2) [0.4, 1]. We see that 0.8 = min(r 32, r 21 ) 1 α, for α [0.4, 1] and min(r 32, r 21 ) = 0.8 > 0.6 = r 31, so relation R is not α transitive for α [0.4, 1]. 4 Preservation of graded properties Results concerning preservation of an α reflexivity, an α irreflexivity, an α asymmetry, an α antisymmetry and both types of an α connectedness are given in [4]. Now symmetry and transitivity will be discussed. Presented conditions will be stated, in general, for operations F in [0, 1] and next adequate examples of aggregation functions will be indicated. Let us notice that condition (9) may be written in a more convenient way. Lemma 2. Let Relation R F R(X) is α symmetric iff R(x, y) 1 α R(y, x) = R(x, y). (16) Proof. Let α [0, 1], x, y X. If R fulfils (16) and R(x, y) 1 α, then by (16) we get R(y, x) = R(x, y), which implies (9). If relation R fulfils (9) and R(x, y) 1 α, then R(y, x) R(x, y), and as a result R(y, x) 1 α. Applying once again (9) to the pair (y, x) we see that R(x, y) R(y, x). Thus R(y, x) = R(x, y) and relation R fulfils (16). Lemma 3. Let α [0, 1], t = (t 1,..., t n ) [0, 1] n. F : [0, 1] n [0, 1] fulfils if and only if F [0,1] n \[1 α,1] n < 1 α (17) F (t 1,..., t n ) 1 α min t k 1 α. (18) t [0,1] n Proof. Let We see that the condition (18) is equivalent to the following one ( t k < 1 α) F (t 1,..., t n ) < 1 α. t [0,1] n The above condition is equivalent to (17) which finishes the proof. Theorem 5. Let If F : [0, 1] n [0, 1] fulfils (17), then it preserves an α symmetry of relations R 1,..., R n F R(X).

4 Proof. Let α [0, 1], x, y X and relations R 1,..., R n F R(X) be α symmetric. If F fulfils (17) then using notations and Lemma 3 we get t k = R k (x, y), k = 1, 2,..., n, R F (x, y) 1 α F (t) 1 α min t k 1 α t k 1 α R k(x, y) 1 α. By α symmetry of relations R 1,..., R n and Lemma 2 we obtain R k (x, y) = R k (y, x), k = 1,..., n. Thus R F (x, y) = R F (y, x) and by Lemma 2 relation R F is α symmetric. Theorem 6. If F : [0, 1] n [0, 1] fulfils condition F min, then it preserves an α symmetry of fuzzy relations for arbitrary Proof. By Lemma 3 a function F fulfils (17) for arbitrary α [0, 1] iff α [0,1] F (t 1,..., t n ) 1 α min t k 1 α. t [0,1] n So by (6) this is equivalent to the fact that F min. By Theorem 5 a function F preserves an α symmetry of fuzzy relations for any For our further considerations we need the following statement Lemma 4. If F : [0, 1] n [0, 1] is increasing in each variable and has a neutral element e = 1, i.e. t [0,1] F (1,..., 1, t, 1,..., 1) = t, (19) where t is in the k-th position, then F min. Proof. Let t 1,..., t n [0, 1]. By assumption on F we get for any k = 1,..., n F (t 1,..., t k 1, t k, t k+1,..., t n ) F (1,..., 1, t k, 1,..., 1) = t k. As a result so F min. F (t 1,..., t n ) min t k, Corollary 1. Let n = 2. Every t-norm preserves an α symmetry of fuzzy relations for arbitrary Now we pay our attention to an α transitivity. Theorem 7. Let If increasing in each variable F : [0, 1] n [0, 1] fulfils (17) and F min, i.e. for any (s 1,..., s n ), (t 1,..., t n ) [0, 1] n F (min(s 1, t 1 ),..., min(s n, t n )) (20) min(f (s 1,..., s n ), F (t 1,..., t n )), then it preserves an α transitivity of fuzzy relations. Proof. Let α [0, 1], x, y, z X. If relations R 1,..., R n F R(X) are α transitive and F fulfils (17) then using notations s k = R k (x, y), t k = R k (y, z), k = 1,..., n, (21) applying Lemma 3 and by the properties of the minimum we have min(r F (x, y), R F (y, z)) 1 α min(f (s 1,..., s n ), F (t 1,..., t n )) 1 α (F (s 1,..., s n ) 1 α, F (t 1,..., t n ) 1 α) ( min s k 1 α, min t k 1 α) min(s k, t k ) 1 α min(r k(x, y), R k (y, z)) 1 α. So by α transitivity of relations R k we see that R k (x, z) min(r k (x, y), R k (y, z)), k = 1,..., n. Thus by (20), monotonicity of F and by notations (21) we have min(r F (x, y), R F (y, z)) = min(f (R 1 (x, y),..., R n (x, y)), F (R 1 (y, z),..., R n (y, z))) = min(f (s 1,..., s n ), F (t 1,..., t n )) F (min(s 1, t 1 ),..., min(s n, t n )) = F (min(r 1 (x, y), R 1 (y, z)),..., min(r n (x, y), R n (y, z))) F (R 1 (x, z),..., R n (x, z)) = R F (x, z) which proves an α transitivity of R F. Using Lemma 3, similarly like for α symmetry we receive Theorem 8. If increasing in each variable function F : [0, 1] n [0, 1] fulfils (20) and F min, then it preserves an α transitivity of fuzzy relations for any Looking for functions which fulfil (20) (cf. Theorem 1, Example 2) and F min (cf. Lemma 4) we see that the minimum, which is an example of an aggregation function, fulfils these conditions. Corollary 2. The minimum preserves an α transitivity of fuzzy relations for any

5 5 Preservation of composed graded properties Now we will consider composed α properties. These are such properties which are conjunctions of properties discussed in the previous sections. Definition 8 (cf. [2], p. 77). Let Relation R F R(X) is: an α tolerance, if it is α reflexive and α symmetric, an α tournament, if it is α asymmetric and α connected, an α equivalence, if it is α reflexive, α symmetric and α transitive, a quasi α order, if it is α reflexive and α transitive, a partial α order, if it is α reflexive, α antisymmetric and α transitive, a linear quasi α order, if it is an α connected quasi α order, a strict α order, if it is α irreflexive and α transitive, a linear strict α order, if it is α asymmetric, α transitive and α connected, a linear α order, if it is an α connected partial α order. Example 7. Let card X = 2. Relation R F R(X), where [ ] R = is 0.5 asymmetric (0.5 antisymmetric) and totally 0.5 connected (0.5 connected), so it is 0.5 tournament. Applying some results of [4] we will present functions F : [0, 1] n [0, 1] preserving composed properties of fuzzy relations. We will concentrate on the results concerning preservation of composed graded properties of fuzzy relations for any Although in [4] are given characterizations, the next theorems are, in main cases, only sufficient conditions, because statements for an α symmetry and an α transitivity are only sufficient conditions. Moreover, it will turn out that, in many cases, the described function is an n argument minimum in [0, 1]. Theorem 9. If F = min, then it preserves an α tolerance for any Proof. By [4] (Theorem 2) F preserves an α reflexivity for any α [0, 1] if and only if F min. By Theorem 6, if F min, then it preserves an α symmetry for any Taking into account the definition of an α tolerance we see that F = min preserves an α tolerance for any Theorem 10. Let card X 2. F : [0, 1] n [0, 1] preserves an α tournament for any α [0, 1] if and only if it fulfils s,t [0,1] n s,t [0,1] n min(f (s), F (t)) max min(s k, t k ), (22) max(f (s), F (t)) min max(s k, t k ). (23) Proof. As a direct consequence of [4] (Theorems 6, 8) and definition of an α tournament we see that F preserves this property for any α [0, 1] iff it fulfils (22) (preservation of an α asymmetry for any α [0, 1]) and (23) (preservation of an α connectedness for any α [0, 1]). Theorem 11. If F = min, then it preserves an α equivalence (a quasi α order, a partial α order) for any Proof. An α equivalence is an α transitive relation of an α tolerance, so by Theorem 9 and Corollary 2 we see that F = min preserves an α equivalence for any By [4] (Theorem 2), F preserves an α reflexivity for any α [0, 1] if and only if F min. By Theorem 8, Corollary 2 and by the definition of a quasi α order we see that F = min preserves this property for any A partial α order is an α antisymmteric relation of a quasi α order, so by the above results and by the fact that F = min fulfils (22), which by [4] (Theorem 6) means that it preserves an α antisymmetry for any α [0, 1], we see that F = min preserves a partial α order for any Theorem 12. If an increasing in each variable function F : [0, 1] n [0, 1] fulfils conditions F min and F min, then it preserves a strict α order for any Proof. By [4] (Theorem 2) a function F preserves an α irreflexivity for any α [0, 1] if and only if F max, so taking into account Theorem 8 and definition of a strict α order we get the required statement. Example 8. F = min preserves a strict α order for any α [0, 1], because the minimum is increasing and F min (cf. Theorem 1). In virtue of Theorems 11, 12 and by definitions of a linear quasi α order, a linear α order and a linear strict α order and by the fact that F preserves an α connectedness (a total α connectedness) if and only if it fulfils (23) (cf. [4], Theorem 8) we see that F = min does not preserve these linear orders because the minimum does not fulfil (23) (see Example 9).

6 Example 9. Let n = 2. If F = min, then condition (23) is of the form max(min(s 1, s 2 ), min(t 1, t 2 )) min(max(s 1, t 1 ), max(s 2, t 2 )). where s 1, s 2, t 1, t 2 [0, 1]. Let us take s 1 = 0, s 2 = 0.5, t 1 = 0.5, t 2 = 0. We see that for these numbers the converse inequality is fulfilled, because 0 = max(min(0, 0.5), min(0.5, 0)) < min(max(0, 0.5), max(0.5, 0)) = 0.5. Thus minimum does not fulfil condition (23). References [1] T. Calvo, A. Kolesárová, M. Komorniková, R. Mesiar, Aggregation Operators: Properties, Classes and Construction Methods, in: T. Calvo et al. (Eds.), Aggregation Operators. Physica- Verlag, Heildelberg, 2002, pp [2] J. Drewniak, Fuzzy Relation Calculus, Silesian University, Katowice, [3] U. Dudziak, Aggregations preserving graded properties of fuzzy relations, in: K. T. Atanassov et al. (Eds.), Soft Computing Foundations and Theoretical Aspects. EXIT, Warszawa, 2004, pp [4] U. Dudziak, Graded properties of fuzzy relations in aggregation process, Journal of Electrical Engineering 56 (2005) (12/s) [5] J. Fodor, M. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer Acad. Publ., Dordrecht, [6] E. P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Acad. Publ., Dordrecht, [7] V. Peneva, I. Popchev, Aggregation of fuzzy relations, C. R. Acad. Bulgare Sci. 51(9-10) (1998) [8] V. Peneva, I. Popchev, Properties of the aggregation operators related with fuzzy relations, Fuzzy Sets Syst. 139 (3) (2003) [9] S. Saminger, R. Mesiar, U. Bodenhofer, Domination of aggregation operators and preservation of transitivity, Int. J. Uncertain., Fuzziness, Knowl.- Based Syst. 10 Suppl. (2002) [10] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) [11] L.A. Zadeh, Similarity relations and fuzzy orderings, Inform. Sci. 3 (1971)

Preservation of t-norm and t-conorm based properties of fuzzy relations during aggregation process

8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2013) Preservation of t-norm and t-conorm based properties of fuzzy relations during aggregation process Urszula Dudziak Institute