Distributivity equations and Mayor’s aggregation operators

Knowledge-Based Systems 52 (2013) 194–200

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Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys

Distributivity equations and Mayor’s aggregation operators Dragan Jocˇic´ a, Ivana Štajner-Papuga b,⇑ a b

Higher School of Professional Business Studies, Vladimira Peric´a-Valtera 4, 21000 Novi Sad, Serbia Department of Mathematics and Informatics, University of Novi Sad, Trg D. Obradovic´a 4, 21000 Novi Sad, Serbia

a r t i c l e

i n f o

Article history: Received 21 January 2013 Received in revised form 30 July 2013 Accepted 2 August 2013 Available online 16 August 2013

a b s t r a c t The focus of this paper are distributivity equations involving the binary aggregation operators on the unit interval [0, 1] with either absorbing or neutral element from the open interval (0, 1), and the Mayor’s aggregation operators from [28]. In the second part of this paper, problem is extended to aggregation operators that have neither neutral nor absorbing element. Ó 2013 Elsevier B.V. All rights reserved.

Keywords: Aggregation operators Uninorm Nullnorm Absorbing element Neutral element Distributivity equations

1. Introduction Aggregation operators play an important role in many different theoretical and practical fields (fuzzy sets theory, theory of optimization, operations research, information theory, engineering design, game theory, voting theory, integration theory, etc.), particularly in decision making theory (see [2,15,17,21,23]). Lately, a high level of attention is directed towards characterizations of pairs of aggregation operators that are satisfying the distributivity law. Investigation of this problem has roots in [1] and, in recent years, it has been focused on t-norms and t-conorms [15], aggregation operators, quasi-arithmetic means [5], pseudo-arithmetical operations [3], fuzzy implications [30,31], uninorms and nullnorms [8,14,25, 26,33]. Additionally, many authors are considering distributivity inequalities [9,10], as well as distributivity equations on a restricted domain [4,12,13,18,20–22,29,32]. An interesting application of this restricted setting on two Borel-Cantelli lemmas and independence of events for decomposable measures is given in [7]. The aim of this paper is to extend research from [5] towards binary aggregation operators that have either an absorbing element or a neutral element from (0, 1). In [5] the previous problem is solved for one special case, i.e., when neutral elements are limited to 1 and 0 (t-norms and t-conorms). Furthermore, the presented research also extends results form [5] towards non-commutative and non-associative operators. This line of research ⇑ Corresponding author. Tel.: +381 21 485 2869; fax: +381 21 6350 458. E-mail addresses: [email protected] (D. Jocˇic´), ivana.stajner-papuga@ dmi.uns.ac.rs (I. Štajner-Papuga). 0950-7051/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.knosys.2013.08.013

presents a contemporary topic (see [20,33]) that is highly interesting since it opens some new possibilities in the utility theory (see [19]). Therefore, the main concern of this paper is how to solve functional equations

Fðx; Gðy; zÞÞ ¼ GðFðx; yÞ; Fðx; zÞÞ;

x; y; z 2 ½0; 1

and

FðGðy; zÞ; xÞ ¼ GðFðy; xÞ; Fðz; xÞÞ;

x; y; z 2 ½0; 1

where one of unknown functions is an aggregation operator defined in the sense of G. Mayor (see [28]), and another one is either a relaxed uninorm or a relaxed nullnorm [8]. The second part of this paper contains even further extension of this problem involving aggregation operators that have neither neutral nor absorbing element. Also, results presented in this paper are additionally clarifying structure of the observed GM-operators. Since paper [5] has considered only min and max as options for the GM-operators, stricture of the GM-operators for other cases was not investigated, presented results present a step forward for this investigation. This paper is organized as follows. Section 2 contains preliminary notions concerning aggregation operators defined in the sense of G. Mayor, aggregation operators with neutral and absorbing elements and distributivity equations. Results on distributivity between aggregation operator given in the sense of G. Mayor and relaxed nullnorm are given in the third section. Section 4 consists of results on distributivity between aggregation operator in the sense of G. Mayor and relaxed uninorm from the classes N max and e N min e . Topic of the fifth section is distributivity when one of the

D. Jocˇic´, I. Štajner-Papuga / Knowledge-Based Systems 52 (2013) 194–200

aggregation operators has neither neutral nor absorbing element. Some concluding remarks are given in the sixth section.

195

1

2. Preliminaries A short overview of notions that are essential for this paper is given in this section [6,8,16,17,21,24,28,34].

s

B

A

s

2.1. Aggregation operators First, let us recall the basic definition of an aggregation operator on [0, 1].

s

Definition 1 [17]. An aggregation operator is a function A(n):[0, 1]n ? [0, 1] that is nondecreasing in each variable and that fulfills the following boundary conditions

s

1

Fig. 1. An operator from Zs.

AðnÞ ð0; . . . ; 0Þ ¼ 0 and AðnÞ ð1; . . . ; 1Þ ¼ 1: Of course, the previous definition of aggregation operators can be extended to an arbitrary real interval [a, b]. Perhaps the oldest example of an aggregation operator is the arithmetic mean defined by n 1X A ðx1 ; . . . ; xn Þ ¼ xi : n i¼1 ðnÞ

The integer n represents number of input values of the observed aggregation operator. Since the topic of this paper are the binary aggregation operators, they will be denoted simply by A instead of A(2). Many additional properties such as continuity, associativity, commutativity, idempotency, decomposability, autodistributivity, bisymmetry, and neutral and absorbing elements, etc., are often required for aggregation operators, depending on background in which the aggregation is performed (see [17]). More accurately, the focus is on the binary aggregation operators introduced by Mayor in [28] that, for the sake of simplicity, will be refereed to as the GM aggregation operators, the nullnorms and the uninorms. 2.1.1. GM aggregation operators Definition 2 [28]. A GM aggregation operator F is a commutative binary aggregation operator that satisfy the following boundary conditions for all x 2 [0, 1]:

Fð0; xÞ ¼ Fð0; 1Þx and Fðx; 1Þ ¼ ð1  Fð0; 1ÞÞx þ Fð0; 1Þ: The following properties of the GM aggregation operators are essential for the further characterizations.

Vðx; 0Þ ¼ x for x 6 s and Vðx; 1Þ ¼ x for x P s: It is clear that s from the previous definition is an absorbing element: V(x, s) = s for all x. The previous definition, as special cases, contains definitions of triangular norms and triangular conorms. For s = 0 operator V is a t-norm denoted by T, and for s = 1 operator V is a t-conorm denoted by S. The form of relaxed nullnorms that is obtained by omitting commutativity and associativity from the previous definition was introduced in [8]. Family of all such operators is denoted by Zs. The following representation theorem for this type of aggregation operators with absorbing element (annihilator) was given in [8] (see Fig. 1). Theorem 5 [8]. Let s 2 [0, 1]. G 2 Zs if and only if

8 2 > < A on½0; s ; G ¼ B on½s; 12 ; > : s otherwise;

ð1Þ

where A:[0, s]2 ? [0, s] is a binary aggregation operator with neutral element 0 and B:[s, 1]2 ? [s, 1] is a binary aggregation operator with neutral element 1. Remark 6. The only example of an idempotent operator from Zs, s 2 (0, 1), is obtained for A = max and B = min. Obviously, operator of that form is a nullnorm, not only a relaxed nullnorm. The following example presents a non-idempotent, non-associative and non-commutative operator from the class Z 1 . 2

Theorem 3 [28]. Let F be a GM aggregation operator. Then, the following holds: (i) F is associative if and only if F is a t-norm or t-conorm; (ii) F = min or F = max if and only if F(0, 1) = 0 or F(0, 1) = 1 and F(x, x) = x for all x 2 [0, 1]; (iii) F is idempotent if and only if min 6 F 6 max.

Example 7 [8]. Aggregation operator G given by the formula

  8 x þ y  2xy for ðx; yÞ 2 ½12 ð1  yÞ; 12  0; 12 > > >   > < maxðx; yÞ for ðx; yÞ 2 ½0; 12 ð1  yÞ  0; 12 Gðx; yÞ ¼  2 > minðx; yÞ for ðx; yÞ 2 12 ; 1 > > > :1 otherwise 2

ð2Þ

belongs to the class Z 1 .

2.1.2. Relaxed nullnorm Another type of aggregation operators that will be used in this paper is the aggregation operator with an absorbing element, namely the relaxed nullnorm.

More general families of operators with absorbing element were studied in [27].

Definition 4 [6]. A nullnorm V is a binary aggregation operator on [0, 1] that is commutative, associative and for which there exists an element s in [0, 1] such that

2.1.3. Relaxed uninorm The following type of aggregation operators that are necessary for the presented research is an aggregation operator with a neutral element, namely the relaxed uninorm.

2

D. Jocˇic´, I. Štajner-Papuga / Knowledge-Based Systems 52 (2013) 194–200

196

where A:[0, e]2 ? [0, e] and B:[e, 1]2 ? [e, 1] are binary aggregation operators with neutral element e.

1

C

Remark 11. The first uninorms considered by Yager and Rybalov [34] are idempotent uninorms U min and U max from classes N min e e e max and N e of the following form

B

(

e

U min e

A

C e

max on ½e; 12 ;

¼

min

otherwise;

min

on ½0; e2 ;

ð6Þ

and

(

1

U max e

¼

ð7Þ

max otherwise:

Fig. 2. An operator from Ne.

An example of a non-idempotent aggregation operator from the class N max follows. e Definition 8 [34]. A uninorm U is a binary aggregation operator on [0, 1] that is commutative, associative, and for which there exists a neutral element e 2 [0, 1], i.e., U(x, e) = x for all x 2 [0, 1]. Again, triangular norms and triangular conorms can be obtained as special cases of the previous definition. Now, for e = 1 operator U is a t-norm and for e = 0 it is a t-conorm. As in the case of nullnorms, relaxed uninorms are obtained by omitting commutativity and associativity from the previous definition [8]. The family of all such operators is denoted by Ne and the representation theorem for that type of more general aggregation operators with neutral element follows (see Fig. 2).

on ½0; e2 ; on ½e; 12 ;

Fðx; yÞ ¼

8 minðx; yÞ > > > > < xy

1 3 3 for ðx; yÞ 2 ½0; 16   ½16 ; [  34  1 for ðx; yÞ 2 ½0; 16  0; 16  2 ;1 for ðx; yÞ 2 13 16

1

;3 16 4



 ½0; 34

> > 1 > > : maxðx; yÞ otherwise

ð8Þ max

belongs to the class N3 4

.

2.2. Distributivity equations Finally, let us recall the functional equations that are called left and right distributivity laws [1, p. 318].

Theorem 9 [8]. Let e 2 [0, 1]. F 2 Ne if and only if

8 > : C

Example 12 [10]. The aggregation operator F of the form

ð3Þ

otherwise;

where A and B are binary aggregation operators with neutral element e and C is an increasing operator that fulfils min 6 C 6 max. If F 2 Ne satisfy the additional condition

Definition 13. Let F, G be binary aggregation operators. F is distributive over G, if the following two laws hold: (LD) F is a left distributive over G, i.e.,

Fðx; Gðy; zÞÞ ¼ GðFðx; yÞ; Fðx; zÞÞ;

for all x; y; z 2 ½0; 1

and (RD) F is a right distributive over G, i.e.,

8x 2 ðe; 1Fð0; xÞ ¼ Fðx; 0Þ ¼ x

FðGðy; zÞ; xÞ ¼ GðFðy; xÞ; Fðz; xÞÞ;

for all x; y; z 2 ½0; 1

or Of course, for commutative F (LD) and (RD) coincide. Since results for (RD) are analogous to results for (LD), the focus will be only on (LD) case for non-commutative F.

8x 2 ½0; eÞFð1; xÞ ¼ Fðx; 1Þ ¼ x: The family of all such operators is denoted by Nmax or Nmin and the e e following theorem holds. Theorem 10 [8]. Let e 2 [0, 1]. (i) F 2

N min e

Let us suppose that F is a GM aggregation operator and that G is a relaxed nullnorm, i.e., G 2 Zs. Two cases can be distinguished: distributivity of G over F and distributivity of F over G.

if and only if

8 on ½0; e2 ; > : min otherwise;

3.1. Case I

ð4Þ Since operator G need not be commutative, it is necessary to observe left and right distributivity. The following results are given only for the left distributivity, while the problem of the right distributivity can by observed in the analogous way.

(ii) F 2 N max if and only if e

8 A on ½0; e2 ; > > < F¼ B on ½e; 12 ; > > : max otherwise;

3. Distributivity: GM aggregation operator and relaxed nullnorm

ð5Þ

Theorem 14. Let F be a GM aggregation operator such that F(0, 1) = k and let G 2 Zs, where 0 < s < 1. G is left distributive over F if and only if F = max for k > s and F = min for k < s.

D. Jocˇic´, I. Štajner-Papuga / Knowledge-Based Systems 52 (2013) 194–200

Proof. ()) First, let us prove that k 2 {0, 1}. Since the (LD) condition insures

197

(Ü) On the other hand, it can be easily shown that F 2 {max, min} is distributiveoveranidempotentnullnorm. h

Gð0; kÞ ¼ Gð0; Fð0; 1ÞÞ ¼ FðGð0; 0Þ; Gð0; 1ÞÞ ¼ Fð0; sÞ ¼ Fð0; 1Þ  s ¼ k  s;

we have the following:  if k = s, since G(0, s) = s, it follows s = s2, i.e., s = 0 or s = 1 which is a contradiction,  if k < s it follows that G(0, k) = k, therefore k = ks, i.e., k = 0,  if k > s it follows that G(0, k) = s, therefore s = ks, i.e., k = 1. That is, k has to be either 0 or 1. Let us now show that F is indeed an idempotent operator, i.e., that F(x, x) = x for all x 2 [0, 1]:  if x 6 s, then

x ¼ Gðx; 0Þ ¼ Gðx; Fð0; 0ÞÞ ¼ FðGðx; 0Þ; Gðx; 0ÞÞ ¼ Fðx; xÞ;  if x P s, then

x ¼ Gðx; 1Þ ¼ Gðx; Fð1; 1ÞÞ ¼ FðGðx; 1Þ; Gðx; 1ÞÞ ¼ Fðx; xÞ: Therefore, F(x, x) = x for all x 2 [0, 1] and from Theorem 3 it follows that F = max or F = min. (Ü) Conversely, (LD) holds from the fact that F = max or F = min while G is an increasing operator. h Remark 15. The previous problem has been solved in [5] for G being a t-norm or a t-conorm or an idempotent nullnorm. 3.2. Case II In this case, since operator F is commutative, left and right distributivity are not separated. Theorem 16. Let F be an idempotent GM aggregation operator such that F(0, 1) = k and let G 2 Zs, with 0 < s < 1. F is distributive over G if and only if F = max or F = min and G is an idempotent nullnorm. Proof. ()) First, let us prove that k 2 {0, 1}. Since G(1, 0) = s, the assumed distributivity insures

ð1  kÞs þ k ¼ Fðs; 1Þ ¼ Fð1; sÞ ¼ Fð1; Gð1; 0ÞÞ ¼ GðFð1; 1Þ; Fð1; 0ÞÞ ¼ Gð1; kÞ: Now, as in the previous theorem, it holds:  if k = s it follows that s = 0 or s = 1 which is a contradiction;  if k < s it follows that k = 0;  if k > s it follows that k = 1. By applying Theorem 3, we obtain that F = max or F = min. It remains to be shown that G is an idempotent operator:  if k = 0, then for an arbitrary x 2 [0, 1], due to supposed (LD), it holds

x ¼ Fðx; 1Þ ¼ Fðx; Gð1; 1ÞÞ ¼ GðFðx; 1Þ; Fðx; 1ÞÞ ¼ Gðx; xÞ;  if k = 1, then for an arbitrary x 2 [0, 1] we have

x ¼ Fðx; 0Þ ¼ Fðx; Gð0; 0ÞÞ ¼ GðFðx; 0Þ; Fðx; 0ÞÞ ¼ Gðx; xÞ: Therefore, G is an idempotent operator from Zs, i.e., it is an idempotent nullnorm (see Remark 6).

Remark 17. In [5] previous problem has been solved with a starting assumption that G is an idempotent nullnorm. Now Theorem 16 shows that distributivity law is a very strong condition which simplifies structure of G considerably. If the assumption that F is an idempotent operator is omitted from the previous theorem, the following result that clarifies structure of the GM aggregation operator is obtained. Theorem 18. Let F be a GM aggregation operator such that F(0, 1) = k and let G 2 Zs, with 0 < s < 1. F is distributive over G if and only if G is an idempotent nullnorm and  for k > s, F is given by

8 on ½0; s2 ; > : max otherwise;

ð9Þ

where A:[0, s]2 ? [0, s] is a commutative aggregation operator with neutral element 0 and B:[s, 1]2 ? [s, 1] is a commutative aggregation operator with neutral element s,  for k < s, F is given by

8 on ½0; s2 ; > : min otherwise;

ð10Þ

where C:[0, s]2 ? [0, s] is a commutative aggregation operator with neutral element s and D:[s, 1]2 ? [s, 1] is a commutative aggregation operator with neutral element 1.

Proof. ()) As in the previous theorem we can prove that k 2 {0, 1} and that G is idempotent nullnorm. Now let us assume that k = 1. For k = 0 proof is similar and, for that reason, omitted. Since the assumed distributivity insures

Fðx; sÞ ¼ Fðx; Gð0; 1ÞÞ ¼ GðFðx; 0Þ; Fðx; 1ÞÞ ¼ Gðx; 1Þ; we have the following

 Fðx; sÞ ¼

s

for x 6 s;

x for x P s:

ð11Þ

Therefore s is an idempotent element of F and for (x, y) 2 [0, s]2 we have 0 6 F(x, y) 6 F(s, s) = s, and for (x, y) 2 [s, 1]2 holds s = F(s, s) 6 F(x, y) 6 1. Thus the restrictions A ¼ Fj½0;s2 and B ¼ Fj½s;12 are aggregation operators with the desired properties. The remaining issue is the structure of F on [0, 1]2n([0, s]2 [ [s, 1]2). For x > s and y < s < z assumed distributivity insures

x ¼ Fðx; sÞ ¼ Fðx; Gðy; zÞÞ ¼ GðFðx; yÞ; Fðx; zÞÞ: Since s = F(s, y) 6 F(x, y), s < z = F(s, z) 6 F(x, y) and G = min on [s, 1]2, we conclude that

x ¼ GðFðx; yÞ; Fðx; zÞÞ ¼ minðFðx; yÞ; Fðx; zÞÞ ¼ Fðx; yÞ: Therefore F(x, y) = x = max (x, y) for y < s < x. (Ü) Let F be given by (9) and G be an idempotent nullnorm. On the squares [0, s]2 and [s, 1]2 distributivity holds since F is increasing and G is either max or min. Otherwise G(y, z) = s for y < s < z, and L = F(x, G(y, z)) = F(x, s) is given by (11). We consider two cases for evaluation of the right side R = G(F(x, y),F(x, z)) of the distributivity law:

D. Jocˇic´, I. Štajner-Papuga / Knowledge-Based Systems 52 (2013) 194–200

198

 if x 6 s, then L = s and since F(x, y) 6 F(s, y) = s and s = F(x, s) 6 F(x, z) we obtain R = G(F(x, y), F(x, z)) = s,  if x > s, then L = x and since F(x, z) P F(x, s) = x > s we obtain R = G(max (x, y), F(x, z)) = min(x, F(x, z)) = x. In all considered cases we obtain L = D which proves that distributivity law holds. h

Remark 19. Paper [5] addressed this issue only for G being a tnorm or a t-conorm. Therefore, Theorem 18 is a continuation of this research that is providing a more precise insides of the structure of GM-operators. The following examples illustrate possible role of operations with absorbing element that fulfill distributivity equation in the utility theory.

Gðx; kÞ ¼ Gðx; Fð0; 1ÞÞ ¼ FðGðx; 0Þ; Gðx; 1ÞÞ ¼ Fð0; 1Þ ¼ k:

 If e = k, then x = G(x, e) = e for all x 6 k, which is a contradiction.  If k < e, then k = G(0, k) 6 G(0, e) = 0, i.e., k = 0.  If k > e, then F(y, 1) P F(0, 1) = k > e, for an arbitrary y. Now, by taking y 6 e, it is obtained

Fðy; 1Þ ¼ Gð0; Fðy; 1ÞÞ ¼ FðGð0; yÞ; Gð0; 1ÞÞ ¼ Fð0; 1Þ ¼ k: From the definition of a GM aggregation operator it follows F(y, 1) = (1  k)y + k = k and, therefore, (1  k)y = 0 for all y 6 e, i.e., k = 1. It remains to be shown that F(x, x) = x for all x > e. Based on (LD), for some x > e, it holds

x ¼ Gðx; 0Þ ¼ Gðx; Fð0; 0ÞÞ ¼ FðGðx; 0Þ; Gðx; 0ÞÞ ¼ Fðx; xÞ

Example 20.

(a) Paper [12] contains an example of optimistic hybrid utility function that is constructed from a continuous t-norm T and a continuous t-conorm S that fulfill conditional distributivity law:

Uðu1 ; u2 ; l1 ; l2 Þ ¼ SðTðu1 ; l1 Þ; Tðu2 ; l2 ÞÞ where u1, u2 are two utilities with values in [0, 1] and l1, l2 are two degrees of plausibility (see [12]). (b) Paper [19] contains an example of a utility function obtained from a nullnorm F with an absorbing element k 2 (0, 1). Now proposed utility function is of the following form

U F ðu1 ; u2 ; l1 ; l2 Þ ¼ SðFðu1 ; l1 Þ; Fðu2 ; l2 ÞÞ

Proof. (i) First, it will be shown that k 2 {0, 1}. For an arbitrary x 6 e, based on (LD), it holds

ð12Þ

where u1, u2 are two utilities with values in [0, 1] and l1, l2 are two degrees of plausibility (see [12]). This absorbing element allows a Decision Maker to predetermine a threshold for a certain decision process. Examination of the behavioral characteristics of (12) is given in [19]. 4. Distributivity: GM aggregation operator and relaxed uninorm

which proves (i). (ii) ()) From assumption that F is right-continuous at the point x = e easily follows F(e, e) = e. Now, by applying (LD) to an arbitrary x 2 [0, 1] and y = z = e (see [25]), it is obtained F(x, x) = x and the claim follows from Theorem 3. (Ü) The proof is analogous to the one of Theorem 14. h The next theorem is focused on relaxed uninorms from the class N min e . Theorem 22. Let F be a GM aggregation operator such that F(0, 1) = k and let G 2 N min where 0 < e < 1. e (i) If G is left distributive over F, then k 2 {0, 1} and F(x, x) = x for all x < e. (ii) If F is left-continuous at the point x = e, then G is left distributive over F if and only if F = min or F = max.

Proof. Proof is similar to the one of Theorem 21. h 4.2. Case II Now we consider distributivity of F over G.

The logical next step is investigation of aggregation operations with neutral element, i.e., of relaxed uninorms. Therefore, in this section F is a GM aggregation operator and G is a relaxed uninorm from N min [ N max . Again, two cases can be distinguished: distribue e tivity of G over F and distributivity of F over G.

Theorem 23. Let F be a GM aggregation operator such that F(0, 1) = k and let G 2 N max where 0 < e < 1. If F is distributive over G, then e k 2 {0, 1} and G ¼ U max . e

4.1. Case I

Proof. Let us prove that k 2 {0, 1}. From

Operator G need not be commutative, therefore it is again necessary to observe left and right distributivity. Results are given only for the left distributivity, while results for the right distributivity can be obtained in the analogous way. Theorem 21. Let F be a GM aggregation operator such that F(0, 1) = k and let G 2 N max with 0 < e < 1. e

k ¼ Fð0; 1Þ ¼ Fð0; Gð0; 1ÞÞ ¼ GðFð0; 0Þ; Fð0; 1ÞÞ ¼ Gð0; kÞ the following is obtained.  If e = k, then e = G(0, e) = 0 which is a contradiction.  If k < e, then k = G(0, k) 6 G(0, e) = 0, i.e., k = 0.  If k > e, then F(1, z) P F(1, 0) = k > e for an arbitrary z. Now, if z 6 e, the distributivity condition insures

k ¼ Fð1; 0Þ ¼ Fð1; Gð0; zÞÞ ¼ GðFð1; 0Þ; Fð1; zÞÞ (i) If G is left distributive over F, then k 2 {0, 1} and F(x, x) = x for all x > e. (ii) If F is right-continuous at the point x = e, then G is left distributive over F if and only if F = min or F = max.

¼ Gðk; Fð1; zÞÞ P Gðe; Fð1; zÞÞ ¼ Fð1; zÞ P k: Therefore, F(1, z) = k for all z 6 e. Now, as in Theorem 21, it can be proved that k = 1.

D. Jocˇic´, I. Štajner-Papuga / Knowledge-Based Systems 52 (2013) 194–200

Proof that G ¼ U max is analogous to the one of Theorem 16. h e The previous theorem gives only a necessary condition. That the assumption of distributivity is not sufficient is illustrated by the following example. Example 24. Let F be a GM aggregation operator, k = 0 and G ¼ U max where 0 < e < 1. For an arbitrary 0 < x < e, y = 0 and z = 1 e distributivity of F over G leads to

x ¼ Fðx; 1Þ ¼ Fðx; Gð0; 1ÞÞ ¼ GðFðx; 0Þ; Fðx; 1ÞÞ ¼ Gð0; xÞ ¼ 0; which is a contradiction. Unfortunately, as a contrast to the Theorem 16, the previous theorem provides only the necessary condition even if we suppose that F is an idempotent GM aggregation operator. As in the previous example, it can be shown that min is not distributive over U max . e The analogous claim for operators from N min is the following. e Theorem 25. Let F be a GM aggregation operator such that F(0, 1) = k and let G 2 N min where 0 < e < 1. If F is distributive over G, then e k 2 {0, 1} and G ¼ U min e .

5. Distributivity: aggregation operators without neutral and absorbing element In this section we study (LD) law where unknown function F is an aggregation operator that has neither absorbing nor neutral element. First, let us recall the well known result (see [5,15,17,21]).

199

Now, for an arbitrary x 2 [0, 1], from similar arguments, follows that A(x, 1) is an idempotent element of S. Again, based on continuity of A(, 1), the conclusion is that all elements from [k, 1] are idempotents of S. Since [0, 1] = [0, k] [ [k, 1], it is obtained that all elements from [0, 1] are idempotent elements of S, i.e., S = max. (

) Conversely can be proved as in Theorem 14. h

The previous result can be extended to the GM aggregation operators. The first theorem gives us only the necessary condition (see [5]). Theorem 29. Let A be an aggregation operator such that A(0, 1) = A(1, 0) = k and let partial mappings A(, 0) and A(, 1) be continuous. Let G be a GM aggregation operator. If A is left distributive over G, then G is an idempotent operator. The next theorem again extends the starting result to the GM aggregation operators and, under an additional assumption, provides the necessary and sufficient condition (see [5]). Theorem 30. Let A be a GM aggregation operator such that A(0, 1) = A(1, 0) = k and let partial mapping A(, 0) and A(, 1) be continuous. Let G be a GM aggregation operator such that G(0, 1) 2 {0, 1}. A is left distributive over G if and only if G = min or G = max.

Remark 31. It is easy to conclude that theorems from this section hold if the assumption A(0, 1) = A(1, 0) = k is replaced by A(1, 0) P A(0, 1).

Theorem 26. Let T be a t-norm and S be a t-conorm, then 6. Conclusion (i) T is distributive over S if and only if S = max, (ii) S is distributive over T if and only if T = min. The focus of the presented results is on the case (i) since the results for the case (ii) can be obtained analogously. It is easy to show that Theorem 26 holds when the assumptions of commutativity and associativity from the definition of a t-norm T are omitted. Now, the remaining question is if the properties of the operator T can be relaxed even more, i.e., what is happening if we omit the existence of both neutral and absorbing element. As it is shown in the next theorem, a certain type of continuity for the operator T is required. Remark 27. Although continuity is not one of the basic properties for aggregation operators, it is very often required in applications. This property insures that a small error that might appear at the input level remains small at the output level.

Theorem 28. Let A be an aggregation operator such that A(0, 1) = A(1, 0) = k and let partial mappings A(, 0) and A(, 1) be continuous. Let S be a t-conorm. A is left distributive over S if and only if S = max. Proof. ()) Let us assume that A is left distributive over S. Based on (LD), the following holds for an arbitrary x 2 [0, 1]

Aðx; 0Þ ¼ Aðx; Sð0; 0ÞÞ ¼ SðAðx; 0Þ; Aðx; 0ÞÞ ,i.e, A(x, 0), for all x 2 [0, 1], is an idempotent element of S. From the assumed continuity of A(, 0) follows that all elements from [0, k] are idempotents of S.

In this paper we have considered distributivity equations where one of unknown functions is the GM aggregation operator [28]. Results in Sections 3 and 4, where another function is relaxed nullnorm or relaxed uninorm [8], extend corresponding ones from [5]. Theorem 18 significantly clarifies structure of the GM-operators. Also, we have seen that distributivity law is a strong condition since it simplifies the structure of involved operators significantly. In Section 5 the same problem is studied when one of operators has neither neutral nor absorbing element. Since pairs of aggregation operators that are satisfying distributivity law play an important role in utility theory (see [11–13,19,23]), further investigations will go to this direction. Acknowledgment This paper has been supported by the Ministry of Science and Technological Development of Republic of Serbia 174009 and by the Provincial Secretariat for Science and Technological Development of Vojvodina. References [1] J. Aczél, Lectures on Functional Equations and their Applications, Academic Press, New York, 1966. [2] G. Beliakov, A. Pradera, T. Calvo, Aggregation Functions: A Guide for Practitioners, Springer, Heidelberg, 2007. [3] P. Benvenuti, R. Mesiar, Pseudo-arithmetical operations as a basis for the general measure and integration theory, Information Sciences 160 (2004) 1– 11. [4] C. Bertoluzza, V. Doldi, On the distributivity between t-norms and t-conorms, Fuzzy Sets and Systems 142 (2004) 85–104. [5] T. Calvo, On some solutions of the distributivity equations, Fuzzy Sets and Systems 104 (1999) 85–96. [6] T. Calvo, B. De Baets, J. Fodor, The functional equations of Frank and Alsina for uninorms and nullnorms, Fuzzy Sets and Systems 120 (2001) 385–394.

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