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A generalization of the migrativity property of aggregation functions
Information Sciences 191 (2012) 76–85
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A generalization of the migrativity property of aggregation functions H. Bustince a,⇑, B. De Baets b, J. Fernandez a, R. Mesiar c,d, J. Montero e a
Departamento de Automática y Computación, Universidad Pública de Navarra, Campus Arrosadia s/n, P.O. Box 31006, Pamplona, Spain Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Coupure Links 653, 9000 Ghent, Belgium Department of Mathematics and Descriptive Geometry, Slovak University of Technology, SK-813 68 Bratislava, Slovakia d IRAFM, University of Ostrava, 30. dubna 22, 703 01 Ostrava, Czech Republic e Faculty of Mathematics, Complutense University, Madrid 28040, Spain b c
a r t i c l e
i n f o
Article history: Received 17 June 2011 Received in revised form 28 November 2011 Accepted 28 December 2011 Available online 3 January 2012 Keywords: Aggregation function Migrativity Associativity Additive generator t-Norm Uninorm
a b s t r a c t This paper brings a generalization of the migrativity property of aggregation functions, suggested in earlier work of some of the present authors by imposing the a-migrativity property of Durante and Sarkoci for all values of a instead of a single one. Replacing the algebraic product by an arbitrary aggregation function B naturally leads to the properties of a–B-migrativity and B-migrativity. This generalization establishes a link between migrativity and a particular case of Aczel’s general associativity equation, already considered by Cutello and Montero as a recursive formula for aggregation. Following a basic investigation, emphasis is put on aggregation functions that can be represented in terms of an additive generator, more specifically, strict t-norms, strict t-conorms and representable uninorms. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction The a-migrativity property (a-migrativity for short) of an aggregation function was introduced by Durante and Sarkoci [12] (see also [14]) in order to express the fact that the effect of reducing one of its arguments by a factor a is the same regardless of which argument is reduced. Explicitly, a function G: [0, 1]2 ? [0, 1] is said to be a-migrative, with a 2 [0, 1], if for any x, y 2 [0, 1] it holds that G(ax, y) = G(x, ay). Requiring this property to hold for all values of a in [0, 1], we obtain the migrativity property (migrativity for short). Note that in [7] any value a P 0 was allowed provided that ax, ay 2 [0, 1], showing among other things that the only migrative function with neutral element 1 is the algebraic product TP. Recently, several papers have dealt with a-migrativity and migrativity. A non-exhaustive list is [4,5,11,12,15,24]. From an application point of view, migrativity is particularly interesting whenever one has to aggregate partial information coming from sources with meaningful differences (information about recent events or places close to one another should in general not be treated similarly as information about events at distant moments in time or at remote locations). This is, for instance, the case in decision making (see, e.g., [27,25,26] for a general discussion). Also in image processing migrativity is very useful, since it expresses the invariance of a given property under a proportional rescaling of some part of the image (see, e.g., [6] for an analysis of conceptual overlapping). The main aim of this work is to go one step further, by subjecting the arguments of the aggregation functions not to a reduction by a factor a, but to a rescaling induced by a more sophisticated aggregation function B. This leads to a–B-migrativity and Bmigrativity, as natural generalizations of a-migrativity and migrativity. We will show that our generalization provides a con⇑ Corresponding author. E-mail address: [email protected] (H. Bustince). 0020-0255/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2011.12.019
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nection between migrativity and associativity, having the recursivity equation considered in [9] as their link. In this setting, migrativity can be recovered by considering the algebraic product TP as the scaling aggregation function, whereas associativity corresponds to the particular case in which both the scaling and the scaled aggregation function are the same. This work is structured as follows. In Section 2 we recall some basic concepts about aggregation functions needed in the subsequent sections. In Section 3, we introduce the a–B-migrativity and B-migrativity properties, generalizing the known concepts of a-migrativity and migrativity, and perform a first basic investigation. Section 4 is devoted to a–B-migrativity and B-migrativity in the context of generated aggregation functions. 2. Preliminary concepts 2.1. Aggregation functions Since migrativity was initially introduced for bivariate functions, the present work focuses on bivariate (usually called binary in this context) aggregation functions as well; we will therefore drop the adjective ‘bivariate/binary’. Recall that a function A: [0, 1]2 ? [0, 1] is called an aggregation function if it is increasing and satisfies the boundary conditions A(0, 0) = 0 and A(1, 1) = 1. Potential properties of an aggregation function A are: (i) A is called symmetric if A(x, y) = A(y, x) for any x, y 2 [0, 1]. (ii) A is called associative if A(A(x, y), z) = A(x, A(y, z)) for any x, y, z 2 [0, 1]. (iii) An element e 2 [0, 1] is called a neutral element of A if A(x, e) = A(e, x) = x for any x 2 [0, 1]. An aggregation function can have at most one neutral element. (iv) An element a 2 [0, 1] is called an absorbing element of A if A(x, a) = A(a, x) = a for any x 2 [0, 1]. An aggregation function can have at most one absorbing element. Exploiting duality is an interesting way of deriving a new aggregation function from a given one. Although here we only consider the standard negation n, defined by n(x) = 1 x, other negation functions could be considered as well (see [8,20,21]). Definition 1. The dual of an aggregation function A is the aggregation function Ad defined by
Ad ðx; yÞ ¼ 1 Að1 x; 1 yÞ:
ð1Þ
Obviously, we can use any monotone bijection to isomorphically transform aggregation functions. Proposition 1. Let A be an aggregation function. For any monotone bijection u: [0, 1] ? [0, 1], the function Au: [0, 1]2 ? [0, 1] defined by
Au ðx; yÞ ¼ u1 ðAðuðxÞ; uðyÞÞÞ;
ð2Þ
is also an aggregation function, called the u-transform of A. Note that it trivially holds that
A ¼ ðAu Þu1 ¼ ðAu1 Þu :
ð3Þ
The most common aggregation functions in fuzzy set theory are t-norms, t-conorms and uninorms (see, e.g., [19]). Recall that uninorms are symmetric and associative, and have a neutral element e 2 [0, 1]. The cases e = 0 and e = 1 correspond to tconorms and t-norms, respectively. For any uninorm U it holds that U(0, 1) = U(1, 0) = 0 (conjunctive uninorm) or U(0, 1) = U(1, 0) = 1 (disjunctive uninorm). Moreover, any uninorm comes in a conjunctive as well as a disjunctive variant, i.e., given a conjunctive uninorm U, setting U0 = U on [0, 1]2n{(0, 1), (1, 0)} and U0 (0, 1) = U0 (1, 0) = 1 yields a disjunctive uninorm (and vice versa). 2.2. Representable aggregation functions Further on, strict t-norms, strict t-conorms and representable uninorms will play an important role. Definition 2. A t-norm T is said to be a strict t-norm if it is continuous and has strictly increasing partial mappings T(x, ) = T(, x), for any x 2 ]0, 1]. The prototypical strict t-norm is the algebraic product TP. Proposition 2 [19]. A t-norm T is strict if and only if there exists a decreasing bijection t: [0, 1] ? [0, 1] such that
Tðx; yÞ ¼ t 1 ðtðxÞ þ tðyÞÞ;
ð4Þ
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for any x, y 2 [0, 1]. The function t is called an additive generator of T and is determined up to a positive multiplicative constant. Definition 3. A t-conorm S is said to be a strict t-conorm if it is continuous and has strictly increasing partial mappings S(x, ) = S(, x), for any x 2 [0, 1[. The prototypical strict t-conorm is the probabilistic sum SP. Proposition 3 [19]. A t-conorm S is strict if and only if there exists an increasing bijection s: [0, 1] ? [0, 1] such that
Sðx; yÞ ¼ s1 ðsðxÞ þ sðyÞÞ;
ð5Þ
for any x, y 2 [0, 1]. The function s is called an additive generator of S and is determined up to a positive multiplicative constant. Proposition 4 ([17,18]). Consider a uninorm U that is strictly increasing on ]0, 1[2. Then U is continuous on [0, 1]2n{(0, 1), (1, 0)} if and only if there exists an increasing bijection u: [0, 1] ? [1, 1] (called additive generator of U) such that
Uðx; yÞ ¼ u1 ðuðxÞ þ uðyÞÞ;
ð6Þ
for any x, y 2 [0, 1], with the convention that +1 + (1) = 1 if U(0, 1) = 0 and +1 + (1) = +1 if U(0, 1) = 1. A uninorm is called representable if it allows the above representation. Representable uninorms have been in use for decades (see, e.g., [10]); for a recent alternative characterization, see [13]. The prototypical representable uninorm is the 3-poperator (see Example 4). Note that for any two strict t-norms T1 and T2, it is possible to find an increasing bijection u: [0, 1] ? [0, 1] such that T1 = (T2)u. It suffices to take u ¼ t 1 t1 , with t1 an additive generator of T1 and t2 an additive generator of T2. A similar 2 observation holds for two strict t-conorms and two representable uninorms. 2.3. The a-migrativity and migrativity properties In this section, we recall the basic definitions of a-migrativity and migrativity. Definition 4. Let a 2 [0, 1]. An aggregation function A is called a-migrative if for any x, y 2 [0, 1] it holds that
Aðxa; yÞ ¼ Aðx; ayÞ:
ð7Þ
Any aggregation function A is trivially 1-migrative, while it is 0-migrative if and only if it has absorbing element 0. For continuous t-norms, the notion of a-migrativity is rather restrictive [14]. Proposition 5. Let T be continuous t-norm and a 2 ]0, 1[. If T is a-migrative, then it is a strict t-norm. Moreover, a-migrativity leads to a kind of fractal structure. Let T be an a-migrative t-norm, with a 2 ]0, 1[, then
Tðxan ; yÞ ¼ Tðx; an yÞ; for any n P 1. In particular, since T has neutral element 1, it holds that
Tðan ; yÞ ¼ Tð1 an ; yÞ ¼ Tð1; an yÞ ¼ an y: Hence, the partial mappings T(an, ) = T(, an) coincide with those of the algebraic product TP. It seems natural to narrow down Definition 4 by imposing that (7) not only holds for a specific a, but for all a 2 [0, 1]. This leads to the notion of a migrative aggregation function. Definition 5. An aggregation function A is called migrative if for any x,y 2 [0, 1] and any a 2 [0, 1] it holds that
Aðxa; yÞ ¼ Aðx; ayÞ: An in-depth study of migrative aggregation functions can be found in [7]. The main result states that migrative aggregation functions are distortions of the algebraic product. Theorem 6. An aggregation function A is migrative if and only if there exists an increasing function g: [0, 1] ? [0, 1], with g(0) = 0 and g(1) = 1, such that
Aðx; yÞ ¼ gðxyÞ; for any x, y 2 [0, 1].
ð8Þ
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3. a–B-migrativity and B-migrativity 3.1. Definition and examples Note that (7) can be written as
AðT P ðx; aÞ; yÞ ¼ Aðx; T P ða; yÞÞ; suggesting the following generalization. Definition 6. Let B be an aggregation function and a 2 [0, 1]. An aggregation function A is called a–B-migrative if for any x, y 2 [0, 1] it holds that
AðBðx; aÞ; yÞ ¼ Aðx; Bða; yÞÞ:
ð9Þ
As in the case of migrativity, we can omit the restriction to a single value of a. The case where B is a t-norm, in particular a continuous ordinal sum, was very recently considered in [16]. Definition 7. Let B be an aggregation function. An aggregation function A is called B-migrative if for any x, y 2 [0, 1] and any
a 2 [0, 1] it holds that AðBðx; aÞ; yÞ ¼ Aðx; Bða; yÞÞ: Note that neither A nor B are assumed to be symmetric, so it is important to keep the ‘correct’ order of the arguments in Eq. (9). Obviously, an aggregation function A is A-migrative if and only if it is associative. Note that the functional equation A(B(x, y), z) = A(x, B(y, z)) was considered by Cutello and Montero [9] in order to obtain a consistent recursive representation of n-ary aggregation functions in terms of binary aggregation functions. This functional equation is a special case of the general associativity equation
AðBðx; yÞ; zÞ ¼ Cðx; Dðy; zÞÞ; studied by Aczél [2], see also [23]. Assuming that all aggregation functions involved are strictly increasing, some specific solutions were characterized in [3], taking advantage of results in [22]. The following example illustrates that not every aggregation function B allows for the existence of a B-migrative aggregation function. Example 1. Let c 2 ]0, 1[ and consider the aggregation function Bc defined by Bc(x, y) = max(0, min(1, x + y c)). (i) For any aggregation function A, it holds that
AðBc ðx; cÞ; yÞ ¼ Aðx; yÞ ¼ Aðx; Bc ðc; yÞÞ; whence A is c–Bc-migrative. (ii) Suppose that A is a 0–Bc-migrative aggregation function, then it holds that
Að0; 1Þ ¼ AðBc ð0; 0Þ; 1Þ ¼ Að0; Bc ð0; 1ÞÞ ¼ Að0; 1 cÞ: Similarly, it holds that
Að0; 1 cÞ ¼ AðBc ð0; 0Þ; 1 cÞ ¼ Að0; Bc ð0; 1 cÞÞ ¼ Að0; 1 2cÞ: In the same way, we find that
Að1; 0Þ ¼ Að1; Bc ð0; 0ÞÞ ¼ AðBc ð1; 0Þ; 0Þ ¼ Að1 c; 0Þ and
Að1 c; 0Þ ¼ Að1 c; Bc ð0; 0ÞÞ ¼ AðBc ð1 c; 0Þ; 0Þ ¼ Að1 2c; 0Þ: Continuing this line of reasoning, once nc > 1, we find that A(0, 1) = A(1, 0) = A(0, 0) = 0. (iii) Analogously, if A is 1–Bc-migrative, then it holds that
Að1; 0Þ ¼ AðBc ð1; 1Þ; 0Þ ¼ Að1; Bc ð1; 0ÞÞ ¼ Að1; 1 cÞ and
Að1; 1 cÞ ¼ Að1; Bc ð1; 1 cÞÞ ¼ Að1; nð1 cÞÞ:
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Continuing this line of reasoning, once n(1 c) > 1, we find that A(1, 0) = A(0, 1) = A(1, 1) = 1. (iv) Combining (ii) and (iii), we conclude that there does not exist a binary aggregation function A that is at the same time 0–Bc-migrative and 1–Bc-migrative. Hence, there do not exist Bc-migrative aggregation functions. Example 2. Consider the smallest aggregation function
A ðx; yÞ ¼
1; if x ¼ y ¼ 1; 0; otherwise:
(i) For a 2 [0, 1[, it holds that an aggregation function A is a–A⁄-migrative if and only if 0 is the absorbing element of A. Indeed, suppose that A is a–A⁄-migrative, then for any x 2 [0, 1] it holds that
Aðx; 0Þ ¼ Aðx; A ða; 0ÞÞ ¼ AðA ðx; aÞ; 0Þ ¼ Að0; 0Þ ¼ 0; and similarly A(0, x) = 0. Hence, 0 is the absorbing element of A. Conversely, if 0 is the absorbing element of A, then for any x, y 2 [0, 1] it holds that
AðA ðx; aÞ; yÞ ¼ Að0; yÞ ¼ 0 ¼ Aðx; 0Þ ¼ Aðx; A ða; yÞÞ: (ii) For a = 1, it holds that an aggregation function A is 1–A⁄-migrative if and only if A(x, 1) = A(0, 1) and A(1, y) = A(1, 0) for any x, y 2 [0, 1[. This follows immediately from the observation that for any x 2 [0, 1[,
Aðx; 1Þ ¼ Aðx; A ð1; 1ÞÞ
and
AðA ðx; 1Þ; 1Þ ¼ Að0; 1Þ;
and similarly for A(1, y). (iii) Finally, one easily verifies that A⁄ is B-migrative for some aggregation function B if and only if B only takes value 1 in (1, 1). (iv) Note that the minimum operator TM is not 1–A⁄-migrative. Indeed, for any y 2 ]0, 1[ it holds that
minðA ð1; 1Þ; yÞ ¼ y – 0 ¼ minðx; A ð1; yÞÞ: So, contrary to the fact that any aggregation function is 1-migrative, 1–B-migrativity is not a trivially fulfilled condition. Example 3. Consider the greatest aggregation function
A ðx; yÞ ¼
0; if x ¼ y ¼ 0; 1; otherwise:
(i) For a 2 ]0, 1], it holds that an aggregation function A is a–A⁄-migrative if and only if 1 is the absorbing element of A. (ii) For a = 0, it holds that an aggregation function A is 0–A⁄-migrative if and only if A(x, 0) = A(1, 0) and A(0, y) = A(0, 1) for any x, y 2 ]0, 1]. (iii) A⁄ is B-migrative for some aggregation function B if and only if B only takes value 0 in (0, 0). 3.2. Structural consequences Aggregation functions that are a–B-migrative have some particular structure, as the following proposition shows. Proposition 7. Let A and B be two aggregation functions and a 2 [0, 1]. If A is a–B-migrative, then (i) A takes the constant value A(0, 1) on the rectangle [0, B(0, a)] [B(a, 1), 1]; (ii) A takes the constant value A(1, 0) on the rectangle [B(1, a), 1] [0, B(a, 0)]. Proof. Let us show (i). Since A is a–B-migrative, it holds that
AðBð0; aÞ; 1Þ ¼ Að0; Bða; 1ÞÞ: Since (0, B(a, 1)) 6 (B(0, a), 1), the result immediately follows from the monotonicity of A and B. h Corollary 8. Let A and B be two aggregation functions. Let I be a linearly ordered set of indices and {ai}i2I # [0, 1] be a family of real numbers such that i > j implies ai > aj. If A is ai–B-migrative for all i 2 I, then (i) A takes the constant value A(0, 1) on the rectangle
½0; Bð0; sup ai Þ ½ Bðinf ai ; 1Þ; 1: i2I
i2I
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(ii) A takes the constant value A(1, 0) on the rectangle
Bð1; inf ai Þ; 1 ½0; Bðsup ai ; 0Þ½: i2I
i2I
Proof. Let us show (i). For any (x, y) 2 [0, B(0, supi2I ai) [] B(infi2Iai, 1), 1], there exist ax, ay 2 {ai}i2I such that x 6 B(0, ax) and y P B(ay, 1). This implies that
Aðx; yÞ 6 AðBð0; ax Þ; yÞ 6 AðBð0; ax Þ; 1ÞÞ ¼ Að0; 1Þ; where the last equality follows from Proposition 7. Analogously,
Aðx; yÞ P Aðx; Bðay ; 1ÞÞ P Að0; Bðay ; 1ÞÞ ¼ Að0; 1Þ: Combining the above leads to (i). h Remark 1. If the supremum and the infimum of the family {ai}i2I are in fact a maximum and a minimum, then A is constant on the closed versions of the rectangles in Corollary 8. 3.3. Influence of neutral elements Since a-migrativity and migrativity are based on the use of the algebraic product TP, it seems natural to study what happens if A and/or B show some of the usual properties of t-norms. In particular, we start by investigating how the existence of a neutral element e either for A or B affects B-migrativity. In case B has a neutral element, the following result is easy to check. Proposition 9. Let B be an aggregation function with neutral element eB 2 [0, 1]. Then any aggregation function A is eB–B-migrative. Proof. Since eB is the neutral element of B, for any aggregation function A we can write
Aðx; yÞ ¼ AðBðx; eB Þ; yÞ
and
Aðx; yÞ ¼ Aðx; BðeB ; yÞÞ;
whence the eB–B migrativity of A is obvious. h In fact, in case of B-migrativity, the existence of a neutral element eB for B implies that any B-migrative aggregation function A is basically a distortion of B, as is stated in the following result. Proposition 10. Let B be an aggregation function with neutral element eB 2 [0, 1]. For any B-migrative aggregation function A there exists an increasing function hA: [0, 1] ? [0, 1], with h(0) = 0 and h(1) = 1, such that A(x, y) = hA(B(x, y)) for any x, y 2 [0, 1]. Proof. Consider the function hA: [0, 1] ? [0, 1] defined by
hA ðxÞ ¼ AðeB ; xÞ: It follows immediately that
Aðx; yÞ ¼ AðBðeB ; xÞ; yÞ ¼ AðeB ; Bðx; yÞÞ ¼ hA ðBðx; yÞÞ: The fact that hA is increasing follows directly from the definition of hA and the fact that A is an aggregation function. h Remark 2. Regarding the existence of a converse result for Proposition 10, note that whenever B is associative, for any function h as in Proposition 10, we have that the aggregation function A defined by A(x, y) = h(B(x, y)) is B-migrative. However, as shown in Example 1, there exists no B1/2-migrative aggregation function. Hence, whatever the choice of the function h, the aggregation function h(B1/2) cannot be B1/2-migrative. Next, we consider the case of A having a neutral element eA 2 [0, 1]. Proposition 11. Let A be an aggregation function with neutral element eA 2 [0, 1]. Let B be an aggregation function and a 2 [0, 1]. If A is a–B-migrative, then exactly one of the following three cases holds: (i) B(0, a) < eA < B(a, 1); (ii) eA 6 B(0, a) and B(a, 1) = 1; (iii) eA P B(a, 1) and B(0, a) = 0.
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As well as exactly one of the following three cases: (iv) B(a, 0) < eA < B(1, a); (v) eA 6 B(a, 0) and B(1, a) = 1; (vi) eA P B(1, a) and B(a, 0) = 0. Proof. Suppose that eA 6 B(0, a), then it follows from Proposition 7(i) that A is constant on {eA} [B(a, 1), 1]. Since eA is the neutral element of A, this is only possible if B(a,1) = 1. The case eA P B(a, 1) can be treated similarly. The second group of cases follows in the same way from Proposition 7(ii). h Proposition 12. Let A be an aggregation function with neutral element eA 2 [0, 1] and let B be an aggregation function. If A is 0–B-migrative and 1–B-migrative, then both B(0, 1) and B(1, 0) belong to {0, 1}. Proof. From Corollary 8 and Remark 1, it follows that A is constant on [0, B(0, 1)] [B(0, 1), 1]. From Proposition 11, it follows that if B(0, 1) < eA, then B(0, 1) = 0, whereas if B(0, 1) > eA, then B(0, 1) = 1. Finally, if B(0, 1) = eA, then either eA = 0 or eA = 1. Summarizing, we find that B(0, 1) 2 {0, 1}. In a similar way, it follows that B(1, 0) 2 {0, 1}. h Finally, if both A and B have a neutral element, we have the following result. Proposition 13. Let A and B be two aggregation functions with neutral element eA 2 [0, 1] and eB 2 [0, 1], respectively, and let a 2 [0, 1]. If A is a–B-migrative, then (i) if eA 6 eB, then B(x, a) 6 A(x, a) and B(a, x) 6 A(a, x) for any x 2 [0, 1]; (ii) if eA P eB, then B(x, a) P A(x, a) and B(a, x) P A(a, x) for any x 2 [0, 1]. Proof. Let us show (i). Using the a–B-migrativity of A, it follows that
Bðx; aÞ ¼ AðBðx; aÞ; eA Þ 6 AðBðx; aÞ; eB Þ ¼ Aðx; Bða; eB ÞÞ ¼ Aðx; aÞ and
Bða; xÞ ¼ AðeA ; Bða; xÞÞ 6 AðeB ; Bða; xÞÞ ¼ AðBðeB ; aÞ; xÞ ¼ Bða; xÞ; and thus (i). h Corollary 14. Let A and B be two aggregation functions with the same neutral element e 2 [0, 1]. If A is B-migrative, then A = B and A is associative. 3.4. Influence of absorbing elements With respect to the existence of an absorbing element, we have the following proposition, which enlarges the number of examples of a–B-migrative aggregation functions. Proposition 15. Let A be an aggregation function such that A(a, 0) = A(0, a) = A(a, 1) = A(1, a) for some a 2 [0, 1]. For any aggregation function B with absorbing element a, it holds that A is a–B-migrative. Proof. Note that A takes a constant value, say c, on [0, a] [a, 1] [ [a, 1] [0, a]. It then easily follows that
c ¼ Aða; yÞ ¼ AðBðx; aÞ; yÞ and
c ¼ Aðx; aÞ ¼ Aðx; Bða; yÞÞ: Hence, A is a–B-migrative. h 3.5. Transformation of migrative aggregation functions Isomorphic transformations have a preserving effect on a–B-migrativity, as stated in the following theorem: Theorem 16. Consider an aggregation function B, a 2 [0, 1] and a monotone bijection u: [0, 1] ? [0, 1]. An aggregation function A is a–B-migrative if and only if Au is u1(a)–Bu-migrative.
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Proof. By definition, it holds that
Au ðBu ðx; u1 ðaÞÞ; yÞ ¼ Au ðu1 ðBðuðxÞ; aÞÞ; yÞ ¼ u1 ðAðBðuðxÞ; aÞ; uðyÞÞÞ: Now, if A is a-B-migrative, then it follows that
u1 ðAðBðuðxÞ; aÞ; uðyÞÞÞ ¼ u1 ðAðuðxÞ; Bða; uðyÞÞÞÞ ¼ Au ðx; u1 Bða; uðyÞÞÞ ¼ Au ðx; Bu ðu1 ðaÞ; yÞÞ: Hence, Au is u1(a)–Bu-migrative. The implication from right to left follows by applying the implication from left to right to A⁄ :¼ Au, u⁄ :¼ u1 and a⁄ :¼ u(a). h Corollary 17. Consider an aggregation function B and a 2 [0, 1]. An aggregation function A is a–B-migrative if and only if its dual Ad is (1 a)–Bd-migrative. 4. Migrativity and additive generators Several interesting results for a-migrative t-norms (and associative copulas) can be found in [4,14,15]. The following result is of interest for the present discussion [14]. Theorem 18. Let t be an additive generator of a strict t-norm T and a 2 ]0, 1[. Then T is a-migrative if and only if there exists a continuous, strictly decreasing function t0: [a, 1] ? [0, 1], with t0(a) < +1 and t0(1) = 0, such that
tðxÞ ¼ kt0 ðaÞ þ t0
x ; k
a
for any k 2 N and any x 2 ]ak+1, ak]. This theorem expresses that the additive generator t is fully determined by its restriction to the interval [a, 1]. Both strict t-norms and strict t-conorms are isomorphic to the algebraic product TP. Hence, due to Theorem 16, all results for a-migrativity can be translated into corresponding results for a–T-migrativity, with T a strict t-norm, and a–S-migrativity, with S is a strict t-conorm. This leads to the following corollaries: Corollary 19. Let t be an additive generator of a strict t-norm T and a 2 ]0, 1[. A continuous t-norm T1 is a–T-migrative if and only if it is generated by an additive generator t1: [0, 1] ? [0, 1] such that
t1 ðxÞ ¼ kc þ hðt1 ðtðxÞ ktðaÞÞÞ; for any k 2 N and any x 2 ]t1((k + 1)t(a), t1(kt(a))], where h is a decreasing bijection h: [a, 1] ? [0, c], with c = t(a) a positive constant. The above corollary expresses that the additive generator t1 is built up from its restriction to the interval [a, 1], which can be chosen arbitrarily. Corollary 20. Let s be an additive generator of a strict t-conorm S and a 2 ]0, 1[. A continuous t-conorm S1 is a–S-migrative if and only if it is generated by an additive generator s1: [0, 1] ? [0, 1] such that
s1 ðxÞ ¼ kc þ hðs1 ðsðxÞ ksðaÞÞÞ for any k 2 N and any x 2 [s1(ks(a)), s1((k + 1)s(a))[, where h is an increasing bijection h: [0, a] ? [0, c], with c = s(a) a positive constant. As expressed in Corollary 14, T being a t-norm, the only T-migrative t-norm is T itself; similarly, S being a t-conorm, the only S-migrative t-conorm is S itself. As Corollaries 19 and 20 concern strict t-norms and strict t-conorms, it is natural to consider representable uninorms next. Since any uninorm comes in a conjunctive and disjunctive variant, it is sufficient to consider a–U-migrativity for conjunctive representable uninorms only. Theorem 21. Let u be an additive generator of a conjunctive representable uninorm U, with neutral element e = u1(0), and a 2 ]0, 1[. A conjunctive representable uninorm U1 with the same neutral element e is a–U-migrative if and only if it is generated by an additive generator u1: [0, 1] ? [1, 1] such that: (i) u1 is arbitrary if a = e; (ii) if a > e, then
u1 ðxÞ ¼ nc þ hðn1 ðnðxÞ nuðaÞÞÞ; for any n 2 Z and any x 2 [u1(nu(a)), u1((n + 1)u(a))], where h is an increasing bijection h: [e, a] ? [0, c], with c = u(a) a positive constant. Note that h is the restriction of u1 to [e, a].
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(iii) if a < e, then
u1 ðxÞ ¼ nd þ hðn1 ðnðxÞ nuðaÞÞÞ; for any n 2 Z and any x 2 [u1((n + 1)u(a)), u1(nu(a))], where h is an increasing bijection h: [a, e] ? [d, 0], with d = d(a) a negative constant. Note that h is the restriction if u1 to [a, e]. Proof. Sufficiency is immediate. To check necessity, let U1 be an a-U-migrative conjunctive representable uninorm with neutral element e = u1(0). Evidently, if a = e, any aggregation function is e–U-migrative. Suppose that a 2 ]e, 1[ (the case a 2 ]0, e[ is similar) and let u1: [0, 1] ? [1, 1] be an additive generator of U1. Since U1 also has neutral element e, it holds that u1(e) = 0, and the a–U-migrativity of U1 implies that for any x, y 2 [0, 1] it holds that
u1 ðu1 ðuðxÞ þ uðaÞÞÞ þ u1 ðyÞ ¼ u1 ðxÞ þ u1 ðu1 ðuðaÞ þ uðyÞÞÞ: 1
Let us consider the increasing bijection g = u1 u then (10) can be rewritten as
ð10Þ
: [1, 1] ? [1, 1], with g(0) = 0. If we denote p = u(x) and r = u(y),
gðp þ uðaÞÞ þ gðrÞ ¼ gðpÞ þ gðr þ uðaÞÞ
ð11Þ
for any p, r 2 [1, 1]. Following [1], this type of Cauchy equation has the following solutions. For any increasing bijection k: [0, u(a)] ? [0, c], with c a positive constant, it holds that the function g: [1, 1] ? [1, 1] defined by
gðpÞ ¼ nc þ kðp nuðaÞÞ; for any n 2 Z and any p 2 [nu(a), (n + 1)u(a)] is a solution. Now, it suffices to take into account that u1 = g u, h = k uj[e, a], and the result follows. h Remark 3. It is also worthwhile to point out the following observations: (i) The functional Eq. (11) can be seen as u(a)- + -migrativity on R of a function G : R2 ! R defined by G(x, y) = g(x) + g(y). (ii) Under the same assumptions of Theorem 21, it holds that
ðnÞ ðnÞ U 1 x; aU ¼ U x; aU ; for any x 2 [0, 1] and any n 2 Z, where: ðnÞ (a) if n > 0, then aU ¼ Uða; . . . ; aÞ (the argument a being repeated n times); ð0Þ (b) aU ¼ e; ðnÞ ðnÞ ðnÞ (c) if n < 0, then aU is the unique solution of the equation Uðx; aU Þ ¼ e, i.e., aU ¼ u1 ðnuðaÞÞ. (iii) For a representable uninorm U with neutral element e, the functions T, S: [0, 1]2 ? [0, 1] defined by
Tðx; yÞ ¼
Uðex; eyÞ e
Sðx; yÞ ¼
Uðe þ ð1 eÞx; e þ ð1 eÞyÞ e 1e
and
are a strict t-norm and a strict t-conorm, respectively (see [17–19]). If we denote the underlying strict t-norm and strict tconorm of another representable uninorm U1 with neutral element e by T1 and S1, respectively, then the above results imply that e (a) if U1 is a–U-migrative with a > e, then S1 is b–S-migrative with b ¼ a1e ; (b) if U1 is a–U-migrative with a < e, then T1 is l–T-migrative with l ¼ ae .
Example 4. Let us consider the best known representable uninorm, the so-called the 3-p operator U3p, introduced by [28] and defined by
U 3p ðx; yÞ ¼
xy ; xy þ ð1 xÞð1 yÞ
1 with the convention 00 ¼ 0. The 3-p operator is a conjunctive representable uninorm with neutral element e¼ 2 and additive x generator u: [0, 1] ? [1, 1] defined by uðxÞ ¼ log 1x . Let a ¼ 34 and consider the function h : 12 ; 34 ! 0; 14 defined by hðxÞ ¼ x 12. Following Theorem 21(ii), we construct a function u1: [0, 1] ? [1, 1] such that u1 ð0Þ ¼ 1; u1 ð1Þ ¼ 1; u1 ðxÞ ¼ hðxÞ ¼ x 12 if x 2 12 ; 34 , and
H. Bustince et al. / Information Sciences 191 (2012) 76–85
u1 ðxÞ ¼
85
n2 x þ n ; 4 3 ð1 xÞ þ x
for any n 2 Z and any x 2
h
3n 3nþ1
i 3nþ1 . Then u1 generates a representable conjunctive uninorm U1 that is 3/4–U-migrative. ; 3nþ1 þ1
Moreover, the underlying strict t-conorm S of U is the so-called Einstein sum given by
Sðx; yÞ ¼
xþy : 1 þ xy
As for the uninorm U1, the underlying strict t-conorm S1 has an additive generator s1: [0, 1] ? [0, 1] given by s1 ðxÞ ¼ u1 ðe þ ð1 eÞxÞ ¼ u1 xþ1 and is 1/2–S-migrative. 2 5. Conclusion In this paper, we have introduced a generalization of the concepts of a-migrativity and migrativity. Generalized migrativity provides a link between Aczél’s general associativity equation [1] and the recursive equation studied in [3,9]. We have focused here on some specific properties only, but it should be clear that this approach deserves further theoretical investigations. Acknowledgements Authors were partially supported by Grants TIN2010-15055 and TIN2009-07901 from the Government of Spain, and by Grants APVV-0073-10 and MSM VZ 6198898701. References [1] J. Aczél, Lectures on Functional Equations and their Applications, Academic Press Inc., New York, 1966. [2] J. Aczél, V.D. Belousov, M. Hosszú, Generalized associativity and bisymmetry on quasigroups, Acta Mathematica Academiae Scientiarum Hungaricae 11 (1960) 127–136. [3] A. Amo, J. Montero, E. Molina, Representation of consistent recursive rules, European Journal of Operational Research 130 (2001) 29–53. [4] G. Beliakov, T. Calvo, On migrative means and copulas, in: Proceedings of Fifth International Summer School on Aggregation Operators, Palma de Mallorca, 2009, pp. 107–110. [5] M. Budincˇevic´, M. Kurilic´, A family of strict and discontinuous triangular norms, Fuzzy Sets and Systems 95 (1998) 381–384. [6] H. Bustince, J. Fernández, R. Mesiar, J. Montero, R. Orduna, Overlap functions, Nonlinear Analysis 72 (2010) 1488–1499. [7] H. Bustince, J. Montero, R. Mesiar, Migrativity of aggregation functions, Fuzzy Sets and Systems 160 (2009) 766–777. [8] T. Calvo, A. Kolesárová, M. Komorníková, R. Mesiar, Aggregation operators, properties, classes and construction methods, in: T. Calvo, G. Mayor, R. Mesiar (Eds.), Aggregation Operators New Trends and Applications, Physica-Verlag, Heidelberg, 2002, pp. 3–104. [9] V. Cutello, J. Montero, Recursive connective rules, International Journal of Intelligent Systems 14 (1999) 3–20. [10] B. De Baets, J. Fodor, van Melle’s combining function in MYCIN is a representable uninorm: an alternative proof, Fuzzy Sets and Systems 104 (1999) 133–136. [11] F. Durante, R. Ghiselli Ricci, Supermigrative semi-copulas and triangular norms, Information Sciences 179 (2009) 2689–2694. [12] F. Durante, P. Sarkoci, A note on the convex combinations of triangular norms, Fuzzy Sets and Systems 159 (2008) 77–80. [13] J. Fodor, B. De Baets, A single-point characterization of representable uninorms, Fuzzy Sets and Systems, submitted for publication, doi:10.1016/ j.fss.2011.12.001. [14] J. Fodor, I.J. Rudas, On continuous triangular norms that are migrative, Fuzzy Sets and Systems 158 (2007) 1692–1697. [15] J. Fodor, I.J. Rudas, On some classes of aggregation functions that are migrative, in: Proceedings of IFSA-EUSFLAT 2009, Lisbon, Portugal, 2009, pp. 653– 656. [16] J. Fodor, I.J. Rudas, Migrative t-norms with respect continuous ordinal sums, Information Sciences 181 (2011) 4860–4866. [17] J. Fodor, R. Yager, A. Rybalov, Structure of uninorms, International Journal of Uncertainty, Fuzziness and Knowledge-based Systems 5 (1997) 411–427. [18] E.P. Klement, R. Mesiar, E. Pap, On the relationship of associative compensatory operators to triangular norms and conorms, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 4 (1996) 129–144. [19] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000. [20] K.C. Maes, B. De Baets, Negation and affirmation: the role of involutive negators, Soft Computing 11 (2007) 647–654. [21] K.C. Maes, B. De Baets, Commutativity and self-duality: two tales of one equation, International Journal of Approximate Reasoning 50 (2009) 189–199. [22] K. Mak, Coherent continuous systems and the generalized functional equation of associativity, Mathematics of Operations Research 12 (1987) 597– 625. [23] Gy. Maksa, Quasisums and generalized associativity, Aequationes Mathematicae 69 (2005) 6–27. [24] R. Mesiar, H. Bustince, J. Fernandez, On the a-migrativity of semicopulas, quasi-copulas, and copulas, Information Sciences 180 (2010) 1967–1976. [25] J. Montero, V. López, D. Gómez, The role of fuzziness in decision making, in: D. Ruan et al. (Eds.), Fuzzy Logic: A Spectrum of Applied and Theoretical Issues, Springer, 2007, pp. 337–349. [26] J. Montero, D. Gómez, S. Muñoz, Fuzzy information representation for decision aiding, in: Proceedings of the IPMU Conference, Málaga, Spain, 2008, pp. 1425–1430. [27] B. Roy, Decision sciences or decision aid sciences, European Journal of Operational Research 66 (1993) 184–203. [28] R. Yager, A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems 80 (1996) 111–120.
Contents lists available at SciVerse ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
A generalization of the migrativity property of aggregation functions H. Bustince a,⇑, B. De Baets b, J. Fernandez a, R. Mesiar c,d, J. Montero e a
Departamento de Automática y Computación, Universidad Pública de Navarra, Campus Arrosadia s/n, P.O. Box 31006, Pamplona, Spain Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Coupure Links 653, 9000 Ghent, Belgium Department of Mathematics and Descriptive Geometry, Slovak University of Technology, SK-813 68 Bratislava, Slovakia d IRAFM, University of Ostrava, 30. dubna 22, 703 01 Ostrava, Czech Republic e Faculty of Mathematics, Complutense University, Madrid 28040, Spain b c
a r t i c l e
i n f o
Article history: Received 17 June 2011 Received in revised form 28 November 2011 Accepted 28 December 2011 Available online 3 January 2012 Keywords: Aggregation function Migrativity Associativity Additive generator t-Norm Uninorm
a b s t r a c t This paper brings a generalization of the migrativity property of aggregation functions, suggested in earlier work of some of the present authors by imposing the a-migrativity property of Durante and Sarkoci for all values of a instead of a single one. Replacing the algebraic product by an arbitrary aggregation function B naturally leads to the properties of a–B-migrativity and B-migrativity. This generalization establishes a link between migrativity and a particular case of Aczel’s general associativity equation, already considered by Cutello and Montero as a recursive formula for aggregation. Following a basic investigation, emphasis is put on aggregation functions that can be represented in terms of an additive generator, more specifically, strict t-norms, strict t-conorms and representable uninorms. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction The a-migrativity property (a-migrativity for short) of an aggregation function was introduced by Durante and Sarkoci [12] (see also [14]) in order to express the fact that the effect of reducing one of its arguments by a factor a is the same regardless of which argument is reduced. Explicitly, a function G: [0, 1]2 ? [0, 1] is said to be a-migrative, with a 2 [0, 1], if for any x, y 2 [0, 1] it holds that G(ax, y) = G(x, ay). Requiring this property to hold for all values of a in [0, 1], we obtain the migrativity property (migrativity for short). Note that in [7] any value a P 0 was allowed provided that ax, ay 2 [0, 1], showing among other things that the only migrative function with neutral element 1 is the algebraic product TP. Recently, several papers have dealt with a-migrativity and migrativity. A non-exhaustive list is [4,5,11,12,15,24]. From an application point of view, migrativity is particularly interesting whenever one has to aggregate partial information coming from sources with meaningful differences (information about recent events or places close to one another should in general not be treated similarly as information about events at distant moments in time or at remote locations). This is, for instance, the case in decision making (see, e.g., [27,25,26] for a general discussion). Also in image processing migrativity is very useful, since it expresses the invariance of a given property under a proportional rescaling of some part of the image (see, e.g., [6] for an analysis of conceptual overlapping). The main aim of this work is to go one step further, by subjecting the arguments of the aggregation functions not to a reduction by a factor a, but to a rescaling induced by a more sophisticated aggregation function B. This leads to a–B-migrativity and Bmigrativity, as natural generalizations of a-migrativity and migrativity. We will show that our generalization provides a con⇑ Corresponding author. E-mail address: [email protected] (H. Bustince). 0020-0255/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2011.12.019
H. Bustince et al. / Information Sciences 191 (2012) 76–85
77
nection between migrativity and associativity, having the recursivity equation considered in [9] as their link. In this setting, migrativity can be recovered by considering the algebraic product TP as the scaling aggregation function, whereas associativity corresponds to the particular case in which both the scaling and the scaled aggregation function are the same. This work is structured as follows. In Section 2 we recall some basic concepts about aggregation functions needed in the subsequent sections. In Section 3, we introduce the a–B-migrativity and B-migrativity properties, generalizing the known concepts of a-migrativity and migrativity, and perform a first basic investigation. Section 4 is devoted to a–B-migrativity and B-migrativity in the context of generated aggregation functions. 2. Preliminary concepts 2.1. Aggregation functions Since migrativity was initially introduced for bivariate functions, the present work focuses on bivariate (usually called binary in this context) aggregation functions as well; we will therefore drop the adjective ‘bivariate/binary’. Recall that a function A: [0, 1]2 ? [0, 1] is called an aggregation function if it is increasing and satisfies the boundary conditions A(0, 0) = 0 and A(1, 1) = 1. Potential properties of an aggregation function A are: (i) A is called symmetric if A(x, y) = A(y, x) for any x, y 2 [0, 1]. (ii) A is called associative if A(A(x, y), z) = A(x, A(y, z)) for any x, y, z 2 [0, 1]. (iii) An element e 2 [0, 1] is called a neutral element of A if A(x, e) = A(e, x) = x for any x 2 [0, 1]. An aggregation function can have at most one neutral element. (iv) An element a 2 [0, 1] is called an absorbing element of A if A(x, a) = A(a, x) = a for any x 2 [0, 1]. An aggregation function can have at most one absorbing element. Exploiting duality is an interesting way of deriving a new aggregation function from a given one. Although here we only consider the standard negation n, defined by n(x) = 1 x, other negation functions could be considered as well (see [8,20,21]). Definition 1. The dual of an aggregation function A is the aggregation function Ad defined by
Ad ðx; yÞ ¼ 1 Að1 x; 1 yÞ:
ð1Þ
Obviously, we can use any monotone bijection to isomorphically transform aggregation functions. Proposition 1. Let A be an aggregation function. For any monotone bijection u: [0, 1] ? [0, 1], the function Au: [0, 1]2 ? [0, 1] defined by
Au ðx; yÞ ¼ u1 ðAðuðxÞ; uðyÞÞÞ;
ð2Þ
is also an aggregation function, called the u-transform of A. Note that it trivially holds that
A ¼ ðAu Þu1 ¼ ðAu1 Þu :
ð3Þ
The most common aggregation functions in fuzzy set theory are t-norms, t-conorms and uninorms (see, e.g., [19]). Recall that uninorms are symmetric and associative, and have a neutral element e 2 [0, 1]. The cases e = 0 and e = 1 correspond to tconorms and t-norms, respectively. For any uninorm U it holds that U(0, 1) = U(1, 0) = 0 (conjunctive uninorm) or U(0, 1) = U(1, 0) = 1 (disjunctive uninorm). Moreover, any uninorm comes in a conjunctive as well as a disjunctive variant, i.e., given a conjunctive uninorm U, setting U0 = U on [0, 1]2n{(0, 1), (1, 0)} and U0 (0, 1) = U0 (1, 0) = 1 yields a disjunctive uninorm (and vice versa). 2.2. Representable aggregation functions Further on, strict t-norms, strict t-conorms and representable uninorms will play an important role. Definition 2. A t-norm T is said to be a strict t-norm if it is continuous and has strictly increasing partial mappings T(x, ) = T(, x), for any x 2 ]0, 1]. The prototypical strict t-norm is the algebraic product TP. Proposition 2 [19]. A t-norm T is strict if and only if there exists a decreasing bijection t: [0, 1] ? [0, 1] such that
Tðx; yÞ ¼ t 1 ðtðxÞ þ tðyÞÞ;
ð4Þ
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for any x, y 2 [0, 1]. The function t is called an additive generator of T and is determined up to a positive multiplicative constant. Definition 3. A t-conorm S is said to be a strict t-conorm if it is continuous and has strictly increasing partial mappings S(x, ) = S(, x), for any x 2 [0, 1[. The prototypical strict t-conorm is the probabilistic sum SP. Proposition 3 [19]. A t-conorm S is strict if and only if there exists an increasing bijection s: [0, 1] ? [0, 1] such that
Sðx; yÞ ¼ s1 ðsðxÞ þ sðyÞÞ;
ð5Þ
for any x, y 2 [0, 1]. The function s is called an additive generator of S and is determined up to a positive multiplicative constant. Proposition 4 ([17,18]). Consider a uninorm U that is strictly increasing on ]0, 1[2. Then U is continuous on [0, 1]2n{(0, 1), (1, 0)} if and only if there exists an increasing bijection u: [0, 1] ? [1, 1] (called additive generator of U) such that
Uðx; yÞ ¼ u1 ðuðxÞ þ uðyÞÞ;
ð6Þ
for any x, y 2 [0, 1], with the convention that +1 + (1) = 1 if U(0, 1) = 0 and +1 + (1) = +1 if U(0, 1) = 1. A uninorm is called representable if it allows the above representation. Representable uninorms have been in use for decades (see, e.g., [10]); for a recent alternative characterization, see [13]. The prototypical representable uninorm is the 3-poperator (see Example 4). Note that for any two strict t-norms T1 and T2, it is possible to find an increasing bijection u: [0, 1] ? [0, 1] such that T1 = (T2)u. It suffices to take u ¼ t 1 t1 , with t1 an additive generator of T1 and t2 an additive generator of T2. A similar 2 observation holds for two strict t-conorms and two representable uninorms. 2.3. The a-migrativity and migrativity properties In this section, we recall the basic definitions of a-migrativity and migrativity. Definition 4. Let a 2 [0, 1]. An aggregation function A is called a-migrative if for any x, y 2 [0, 1] it holds that
Aðxa; yÞ ¼ Aðx; ayÞ:
ð7Þ
Any aggregation function A is trivially 1-migrative, while it is 0-migrative if and only if it has absorbing element 0. For continuous t-norms, the notion of a-migrativity is rather restrictive [14]. Proposition 5. Let T be continuous t-norm and a 2 ]0, 1[. If T is a-migrative, then it is a strict t-norm. Moreover, a-migrativity leads to a kind of fractal structure. Let T be an a-migrative t-norm, with a 2 ]0, 1[, then
Tðxan ; yÞ ¼ Tðx; an yÞ; for any n P 1. In particular, since T has neutral element 1, it holds that
Tðan ; yÞ ¼ Tð1 an ; yÞ ¼ Tð1; an yÞ ¼ an y: Hence, the partial mappings T(an, ) = T(, an) coincide with those of the algebraic product TP. It seems natural to narrow down Definition 4 by imposing that (7) not only holds for a specific a, but for all a 2 [0, 1]. This leads to the notion of a migrative aggregation function. Definition 5. An aggregation function A is called migrative if for any x,y 2 [0, 1] and any a 2 [0, 1] it holds that
Aðxa; yÞ ¼ Aðx; ayÞ: An in-depth study of migrative aggregation functions can be found in [7]. The main result states that migrative aggregation functions are distortions of the algebraic product. Theorem 6. An aggregation function A is migrative if and only if there exists an increasing function g: [0, 1] ? [0, 1], with g(0) = 0 and g(1) = 1, such that
Aðx; yÞ ¼ gðxyÞ; for any x, y 2 [0, 1].
ð8Þ
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3. a–B-migrativity and B-migrativity 3.1. Definition and examples Note that (7) can be written as
AðT P ðx; aÞ; yÞ ¼ Aðx; T P ða; yÞÞ; suggesting the following generalization. Definition 6. Let B be an aggregation function and a 2 [0, 1]. An aggregation function A is called a–B-migrative if for any x, y 2 [0, 1] it holds that
AðBðx; aÞ; yÞ ¼ Aðx; Bða; yÞÞ:
ð9Þ
As in the case of migrativity, we can omit the restriction to a single value of a. The case where B is a t-norm, in particular a continuous ordinal sum, was very recently considered in [16]. Definition 7. Let B be an aggregation function. An aggregation function A is called B-migrative if for any x, y 2 [0, 1] and any
a 2 [0, 1] it holds that AðBðx; aÞ; yÞ ¼ Aðx; Bða; yÞÞ: Note that neither A nor B are assumed to be symmetric, so it is important to keep the ‘correct’ order of the arguments in Eq. (9). Obviously, an aggregation function A is A-migrative if and only if it is associative. Note that the functional equation A(B(x, y), z) = A(x, B(y, z)) was considered by Cutello and Montero [9] in order to obtain a consistent recursive representation of n-ary aggregation functions in terms of binary aggregation functions. This functional equation is a special case of the general associativity equation
AðBðx; yÞ; zÞ ¼ Cðx; Dðy; zÞÞ; studied by Aczél [2], see also [23]. Assuming that all aggregation functions involved are strictly increasing, some specific solutions were characterized in [3], taking advantage of results in [22]. The following example illustrates that not every aggregation function B allows for the existence of a B-migrative aggregation function. Example 1. Let c 2 ]0, 1[ and consider the aggregation function Bc defined by Bc(x, y) = max(0, min(1, x + y c)). (i) For any aggregation function A, it holds that
AðBc ðx; cÞ; yÞ ¼ Aðx; yÞ ¼ Aðx; Bc ðc; yÞÞ; whence A is c–Bc-migrative. (ii) Suppose that A is a 0–Bc-migrative aggregation function, then it holds that
Að0; 1Þ ¼ AðBc ð0; 0Þ; 1Þ ¼ Að0; Bc ð0; 1ÞÞ ¼ Að0; 1 cÞ: Similarly, it holds that
Að0; 1 cÞ ¼ AðBc ð0; 0Þ; 1 cÞ ¼ Að0; Bc ð0; 1 cÞÞ ¼ Að0; 1 2cÞ: In the same way, we find that
Að1; 0Þ ¼ Að1; Bc ð0; 0ÞÞ ¼ AðBc ð1; 0Þ; 0Þ ¼ Að1 c; 0Þ and
Að1 c; 0Þ ¼ Að1 c; Bc ð0; 0ÞÞ ¼ AðBc ð1 c; 0Þ; 0Þ ¼ Að1 2c; 0Þ: Continuing this line of reasoning, once nc > 1, we find that A(0, 1) = A(1, 0) = A(0, 0) = 0. (iii) Analogously, if A is 1–Bc-migrative, then it holds that
Að1; 0Þ ¼ AðBc ð1; 1Þ; 0Þ ¼ Að1; Bc ð1; 0ÞÞ ¼ Að1; 1 cÞ and
Að1; 1 cÞ ¼ Að1; Bc ð1; 1 cÞÞ ¼ Að1; nð1 cÞÞ:
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Continuing this line of reasoning, once n(1 c) > 1, we find that A(1, 0) = A(0, 1) = A(1, 1) = 1. (iv) Combining (ii) and (iii), we conclude that there does not exist a binary aggregation function A that is at the same time 0–Bc-migrative and 1–Bc-migrative. Hence, there do not exist Bc-migrative aggregation functions. Example 2. Consider the smallest aggregation function
A ðx; yÞ ¼
1; if x ¼ y ¼ 1; 0; otherwise:
(i) For a 2 [0, 1[, it holds that an aggregation function A is a–A⁄-migrative if and only if 0 is the absorbing element of A. Indeed, suppose that A is a–A⁄-migrative, then for any x 2 [0, 1] it holds that
Aðx; 0Þ ¼ Aðx; A ða; 0ÞÞ ¼ AðA ðx; aÞ; 0Þ ¼ Að0; 0Þ ¼ 0; and similarly A(0, x) = 0. Hence, 0 is the absorbing element of A. Conversely, if 0 is the absorbing element of A, then for any x, y 2 [0, 1] it holds that
AðA ðx; aÞ; yÞ ¼ Að0; yÞ ¼ 0 ¼ Aðx; 0Þ ¼ Aðx; A ða; yÞÞ: (ii) For a = 1, it holds that an aggregation function A is 1–A⁄-migrative if and only if A(x, 1) = A(0, 1) and A(1, y) = A(1, 0) for any x, y 2 [0, 1[. This follows immediately from the observation that for any x 2 [0, 1[,
Aðx; 1Þ ¼ Aðx; A ð1; 1ÞÞ
and
AðA ðx; 1Þ; 1Þ ¼ Að0; 1Þ;
and similarly for A(1, y). (iii) Finally, one easily verifies that A⁄ is B-migrative for some aggregation function B if and only if B only takes value 1 in (1, 1). (iv) Note that the minimum operator TM is not 1–A⁄-migrative. Indeed, for any y 2 ]0, 1[ it holds that
minðA ð1; 1Þ; yÞ ¼ y – 0 ¼ minðx; A ð1; yÞÞ: So, contrary to the fact that any aggregation function is 1-migrative, 1–B-migrativity is not a trivially fulfilled condition. Example 3. Consider the greatest aggregation function
A ðx; yÞ ¼
0; if x ¼ y ¼ 0; 1; otherwise:
(i) For a 2 ]0, 1], it holds that an aggregation function A is a–A⁄-migrative if and only if 1 is the absorbing element of A. (ii) For a = 0, it holds that an aggregation function A is 0–A⁄-migrative if and only if A(x, 0) = A(1, 0) and A(0, y) = A(0, 1) for any x, y 2 ]0, 1]. (iii) A⁄ is B-migrative for some aggregation function B if and only if B only takes value 0 in (0, 0). 3.2. Structural consequences Aggregation functions that are a–B-migrative have some particular structure, as the following proposition shows. Proposition 7. Let A and B be two aggregation functions and a 2 [0, 1]. If A is a–B-migrative, then (i) A takes the constant value A(0, 1) on the rectangle [0, B(0, a)] [B(a, 1), 1]; (ii) A takes the constant value A(1, 0) on the rectangle [B(1, a), 1] [0, B(a, 0)]. Proof. Let us show (i). Since A is a–B-migrative, it holds that
AðBð0; aÞ; 1Þ ¼ Að0; Bða; 1ÞÞ: Since (0, B(a, 1)) 6 (B(0, a), 1), the result immediately follows from the monotonicity of A and B. h Corollary 8. Let A and B be two aggregation functions. Let I be a linearly ordered set of indices and {ai}i2I # [0, 1] be a family of real numbers such that i > j implies ai > aj. If A is ai–B-migrative for all i 2 I, then (i) A takes the constant value A(0, 1) on the rectangle
½0; Bð0; sup ai Þ ½ Bðinf ai ; 1Þ; 1: i2I
i2I
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(ii) A takes the constant value A(1, 0) on the rectangle
Bð1; inf ai Þ; 1 ½0; Bðsup ai ; 0Þ½: i2I
i2I
Proof. Let us show (i). For any (x, y) 2 [0, B(0, supi2I ai) [] B(infi2Iai, 1), 1], there exist ax, ay 2 {ai}i2I such that x 6 B(0, ax) and y P B(ay, 1). This implies that
Aðx; yÞ 6 AðBð0; ax Þ; yÞ 6 AðBð0; ax Þ; 1ÞÞ ¼ Að0; 1Þ; where the last equality follows from Proposition 7. Analogously,
Aðx; yÞ P Aðx; Bðay ; 1ÞÞ P Að0; Bðay ; 1ÞÞ ¼ Að0; 1Þ: Combining the above leads to (i). h Remark 1. If the supremum and the infimum of the family {ai}i2I are in fact a maximum and a minimum, then A is constant on the closed versions of the rectangles in Corollary 8. 3.3. Influence of neutral elements Since a-migrativity and migrativity are based on the use of the algebraic product TP, it seems natural to study what happens if A and/or B show some of the usual properties of t-norms. In particular, we start by investigating how the existence of a neutral element e either for A or B affects B-migrativity. In case B has a neutral element, the following result is easy to check. Proposition 9. Let B be an aggregation function with neutral element eB 2 [0, 1]. Then any aggregation function A is eB–B-migrative. Proof. Since eB is the neutral element of B, for any aggregation function A we can write
Aðx; yÞ ¼ AðBðx; eB Þ; yÞ
and
Aðx; yÞ ¼ Aðx; BðeB ; yÞÞ;
whence the eB–B migrativity of A is obvious. h In fact, in case of B-migrativity, the existence of a neutral element eB for B implies that any B-migrative aggregation function A is basically a distortion of B, as is stated in the following result. Proposition 10. Let B be an aggregation function with neutral element eB 2 [0, 1]. For any B-migrative aggregation function A there exists an increasing function hA: [0, 1] ? [0, 1], with h(0) = 0 and h(1) = 1, such that A(x, y) = hA(B(x, y)) for any x, y 2 [0, 1]. Proof. Consider the function hA: [0, 1] ? [0, 1] defined by
hA ðxÞ ¼ AðeB ; xÞ: It follows immediately that
Aðx; yÞ ¼ AðBðeB ; xÞ; yÞ ¼ AðeB ; Bðx; yÞÞ ¼ hA ðBðx; yÞÞ: The fact that hA is increasing follows directly from the definition of hA and the fact that A is an aggregation function. h Remark 2. Regarding the existence of a converse result for Proposition 10, note that whenever B is associative, for any function h as in Proposition 10, we have that the aggregation function A defined by A(x, y) = h(B(x, y)) is B-migrative. However, as shown in Example 1, there exists no B1/2-migrative aggregation function. Hence, whatever the choice of the function h, the aggregation function h(B1/2) cannot be B1/2-migrative. Next, we consider the case of A having a neutral element eA 2 [0, 1]. Proposition 11. Let A be an aggregation function with neutral element eA 2 [0, 1]. Let B be an aggregation function and a 2 [0, 1]. If A is a–B-migrative, then exactly one of the following three cases holds: (i) B(0, a) < eA < B(a, 1); (ii) eA 6 B(0, a) and B(a, 1) = 1; (iii) eA P B(a, 1) and B(0, a) = 0.
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As well as exactly one of the following three cases: (iv) B(a, 0) < eA < B(1, a); (v) eA 6 B(a, 0) and B(1, a) = 1; (vi) eA P B(1, a) and B(a, 0) = 0. Proof. Suppose that eA 6 B(0, a), then it follows from Proposition 7(i) that A is constant on {eA} [B(a, 1), 1]. Since eA is the neutral element of A, this is only possible if B(a,1) = 1. The case eA P B(a, 1) can be treated similarly. The second group of cases follows in the same way from Proposition 7(ii). h Proposition 12. Let A be an aggregation function with neutral element eA 2 [0, 1] and let B be an aggregation function. If A is 0–B-migrative and 1–B-migrative, then both B(0, 1) and B(1, 0) belong to {0, 1}. Proof. From Corollary 8 and Remark 1, it follows that A is constant on [0, B(0, 1)] [B(0, 1), 1]. From Proposition 11, it follows that if B(0, 1) < eA, then B(0, 1) = 0, whereas if B(0, 1) > eA, then B(0, 1) = 1. Finally, if B(0, 1) = eA, then either eA = 0 or eA = 1. Summarizing, we find that B(0, 1) 2 {0, 1}. In a similar way, it follows that B(1, 0) 2 {0, 1}. h Finally, if both A and B have a neutral element, we have the following result. Proposition 13. Let A and B be two aggregation functions with neutral element eA 2 [0, 1] and eB 2 [0, 1], respectively, and let a 2 [0, 1]. If A is a–B-migrative, then (i) if eA 6 eB, then B(x, a) 6 A(x, a) and B(a, x) 6 A(a, x) for any x 2 [0, 1]; (ii) if eA P eB, then B(x, a) P A(x, a) and B(a, x) P A(a, x) for any x 2 [0, 1]. Proof. Let us show (i). Using the a–B-migrativity of A, it follows that
Bðx; aÞ ¼ AðBðx; aÞ; eA Þ 6 AðBðx; aÞ; eB Þ ¼ Aðx; Bða; eB ÞÞ ¼ Aðx; aÞ and
Bða; xÞ ¼ AðeA ; Bða; xÞÞ 6 AðeB ; Bða; xÞÞ ¼ AðBðeB ; aÞ; xÞ ¼ Bða; xÞ; and thus (i). h Corollary 14. Let A and B be two aggregation functions with the same neutral element e 2 [0, 1]. If A is B-migrative, then A = B and A is associative. 3.4. Influence of absorbing elements With respect to the existence of an absorbing element, we have the following proposition, which enlarges the number of examples of a–B-migrative aggregation functions. Proposition 15. Let A be an aggregation function such that A(a, 0) = A(0, a) = A(a, 1) = A(1, a) for some a 2 [0, 1]. For any aggregation function B with absorbing element a, it holds that A is a–B-migrative. Proof. Note that A takes a constant value, say c, on [0, a] [a, 1] [ [a, 1] [0, a]. It then easily follows that
c ¼ Aða; yÞ ¼ AðBðx; aÞ; yÞ and
c ¼ Aðx; aÞ ¼ Aðx; Bða; yÞÞ: Hence, A is a–B-migrative. h 3.5. Transformation of migrative aggregation functions Isomorphic transformations have a preserving effect on a–B-migrativity, as stated in the following theorem: Theorem 16. Consider an aggregation function B, a 2 [0, 1] and a monotone bijection u: [0, 1] ? [0, 1]. An aggregation function A is a–B-migrative if and only if Au is u1(a)–Bu-migrative.
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Proof. By definition, it holds that
Au ðBu ðx; u1 ðaÞÞ; yÞ ¼ Au ðu1 ðBðuðxÞ; aÞÞ; yÞ ¼ u1 ðAðBðuðxÞ; aÞ; uðyÞÞÞ: Now, if A is a-B-migrative, then it follows that
u1 ðAðBðuðxÞ; aÞ; uðyÞÞÞ ¼ u1 ðAðuðxÞ; Bða; uðyÞÞÞÞ ¼ Au ðx; u1 Bða; uðyÞÞÞ ¼ Au ðx; Bu ðu1 ðaÞ; yÞÞ: Hence, Au is u1(a)–Bu-migrative. The implication from right to left follows by applying the implication from left to right to A⁄ :¼ Au, u⁄ :¼ u1 and a⁄ :¼ u(a). h Corollary 17. Consider an aggregation function B and a 2 [0, 1]. An aggregation function A is a–B-migrative if and only if its dual Ad is (1 a)–Bd-migrative. 4. Migrativity and additive generators Several interesting results for a-migrative t-norms (and associative copulas) can be found in [4,14,15]. The following result is of interest for the present discussion [14]. Theorem 18. Let t be an additive generator of a strict t-norm T and a 2 ]0, 1[. Then T is a-migrative if and only if there exists a continuous, strictly decreasing function t0: [a, 1] ? [0, 1], with t0(a) < +1 and t0(1) = 0, such that
tðxÞ ¼ kt0 ðaÞ þ t0
x ; k
a
for any k 2 N and any x 2 ]ak+1, ak]. This theorem expresses that the additive generator t is fully determined by its restriction to the interval [a, 1]. Both strict t-norms and strict t-conorms are isomorphic to the algebraic product TP. Hence, due to Theorem 16, all results for a-migrativity can be translated into corresponding results for a–T-migrativity, with T a strict t-norm, and a–S-migrativity, with S is a strict t-conorm. This leads to the following corollaries: Corollary 19. Let t be an additive generator of a strict t-norm T and a 2 ]0, 1[. A continuous t-norm T1 is a–T-migrative if and only if it is generated by an additive generator t1: [0, 1] ? [0, 1] such that
t1 ðxÞ ¼ kc þ hðt1 ðtðxÞ ktðaÞÞÞ; for any k 2 N and any x 2 ]t1((k + 1)t(a), t1(kt(a))], where h is a decreasing bijection h: [a, 1] ? [0, c], with c = t(a) a positive constant. The above corollary expresses that the additive generator t1 is built up from its restriction to the interval [a, 1], which can be chosen arbitrarily. Corollary 20. Let s be an additive generator of a strict t-conorm S and a 2 ]0, 1[. A continuous t-conorm S1 is a–S-migrative if and only if it is generated by an additive generator s1: [0, 1] ? [0, 1] such that
s1 ðxÞ ¼ kc þ hðs1 ðsðxÞ ksðaÞÞÞ for any k 2 N and any x 2 [s1(ks(a)), s1((k + 1)s(a))[, where h is an increasing bijection h: [0, a] ? [0, c], with c = s(a) a positive constant. As expressed in Corollary 14, T being a t-norm, the only T-migrative t-norm is T itself; similarly, S being a t-conorm, the only S-migrative t-conorm is S itself. As Corollaries 19 and 20 concern strict t-norms and strict t-conorms, it is natural to consider representable uninorms next. Since any uninorm comes in a conjunctive and disjunctive variant, it is sufficient to consider a–U-migrativity for conjunctive representable uninorms only. Theorem 21. Let u be an additive generator of a conjunctive representable uninorm U, with neutral element e = u1(0), and a 2 ]0, 1[. A conjunctive representable uninorm U1 with the same neutral element e is a–U-migrative if and only if it is generated by an additive generator u1: [0, 1] ? [1, 1] such that: (i) u1 is arbitrary if a = e; (ii) if a > e, then
u1 ðxÞ ¼ nc þ hðn1 ðnðxÞ nuðaÞÞÞ; for any n 2 Z and any x 2 [u1(nu(a)), u1((n + 1)u(a))], where h is an increasing bijection h: [e, a] ? [0, c], with c = u(a) a positive constant. Note that h is the restriction of u1 to [e, a].
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(iii) if a < e, then
u1 ðxÞ ¼ nd þ hðn1 ðnðxÞ nuðaÞÞÞ; for any n 2 Z and any x 2 [u1((n + 1)u(a)), u1(nu(a))], where h is an increasing bijection h: [a, e] ? [d, 0], with d = d(a) a negative constant. Note that h is the restriction if u1 to [a, e]. Proof. Sufficiency is immediate. To check necessity, let U1 be an a-U-migrative conjunctive representable uninorm with neutral element e = u1(0). Evidently, if a = e, any aggregation function is e–U-migrative. Suppose that a 2 ]e, 1[ (the case a 2 ]0, e[ is similar) and let u1: [0, 1] ? [1, 1] be an additive generator of U1. Since U1 also has neutral element e, it holds that u1(e) = 0, and the a–U-migrativity of U1 implies that for any x, y 2 [0, 1] it holds that
u1 ðu1 ðuðxÞ þ uðaÞÞÞ þ u1 ðyÞ ¼ u1 ðxÞ þ u1 ðu1 ðuðaÞ þ uðyÞÞÞ: 1
Let us consider the increasing bijection g = u1 u then (10) can be rewritten as
ð10Þ
: [1, 1] ? [1, 1], with g(0) = 0. If we denote p = u(x) and r = u(y),
gðp þ uðaÞÞ þ gðrÞ ¼ gðpÞ þ gðr þ uðaÞÞ
ð11Þ
for any p, r 2 [1, 1]. Following [1], this type of Cauchy equation has the following solutions. For any increasing bijection k: [0, u(a)] ? [0, c], with c a positive constant, it holds that the function g: [1, 1] ? [1, 1] defined by
gðpÞ ¼ nc þ kðp nuðaÞÞ; for any n 2 Z and any p 2 [nu(a), (n + 1)u(a)] is a solution. Now, it suffices to take into account that u1 = g u, h = k uj[e, a], and the result follows. h Remark 3. It is also worthwhile to point out the following observations: (i) The functional Eq. (11) can be seen as u(a)- + -migrativity on R of a function G : R2 ! R defined by G(x, y) = g(x) + g(y). (ii) Under the same assumptions of Theorem 21, it holds that
ðnÞ ðnÞ U 1 x; aU ¼ U x; aU ; for any x 2 [0, 1] and any n 2 Z, where: ðnÞ (a) if n > 0, then aU ¼ Uða; . . . ; aÞ (the argument a being repeated n times); ð0Þ (b) aU ¼ e; ðnÞ ðnÞ ðnÞ (c) if n < 0, then aU is the unique solution of the equation Uðx; aU Þ ¼ e, i.e., aU ¼ u1 ðnuðaÞÞ. (iii) For a representable uninorm U with neutral element e, the functions T, S: [0, 1]2 ? [0, 1] defined by
Tðx; yÞ ¼
Uðex; eyÞ e
Sðx; yÞ ¼
Uðe þ ð1 eÞx; e þ ð1 eÞyÞ e 1e
and
are a strict t-norm and a strict t-conorm, respectively (see [17–19]). If we denote the underlying strict t-norm and strict tconorm of another representable uninorm U1 with neutral element e by T1 and S1, respectively, then the above results imply that e (a) if U1 is a–U-migrative with a > e, then S1 is b–S-migrative with b ¼ a1e ; (b) if U1 is a–U-migrative with a < e, then T1 is l–T-migrative with l ¼ ae .
Example 4. Let us consider the best known representable uninorm, the so-called the 3-p operator U3p, introduced by [28] and defined by
U 3p ðx; yÞ ¼
xy ; xy þ ð1 xÞð1 yÞ
1 with the convention 00 ¼ 0. The 3-p operator is a conjunctive representable uninorm with neutral element e¼ 2 and additive x generator u: [0, 1] ? [1, 1] defined by uðxÞ ¼ log 1x . Let a ¼ 34 and consider the function h : 12 ; 34 ! 0; 14 defined by hðxÞ ¼ x 12. Following Theorem 21(ii), we construct a function u1: [0, 1] ? [1, 1] such that u1 ð0Þ ¼ 1; u1 ð1Þ ¼ 1; u1 ðxÞ ¼ hðxÞ ¼ x 12 if x 2 12 ; 34 , and
H. Bustince et al. / Information Sciences 191 (2012) 76–85
u1 ðxÞ ¼
85
n2 x þ n ; 4 3 ð1 xÞ þ x
for any n 2 Z and any x 2
h
3n 3nþ1
i 3nþ1 . Then u1 generates a representable conjunctive uninorm U1 that is 3/4–U-migrative. ; 3nþ1 þ1
Moreover, the underlying strict t-conorm S of U is the so-called Einstein sum given by
Sðx; yÞ ¼
xþy : 1 þ xy
As for the uninorm U1, the underlying strict t-conorm S1 has an additive generator s1: [0, 1] ? [0, 1] given by s1 ðxÞ ¼ u1 ðe þ ð1 eÞxÞ ¼ u1 xþ1 and is 1/2–S-migrative. 2 5. Conclusion In this paper, we have introduced a generalization of the concepts of a-migrativity and migrativity. Generalized migrativity provides a link between Aczél’s general associativity equation [1] and the recursive equation studied in [3,9]. We have focused here on some specific properties only, but it should be clear that this approach deserves further theoretical investigations. Acknowledgements Authors were partially supported by Grants TIN2010-15055 and TIN2009-07901 from the Government of Spain, and by Grants APVV-0073-10 and MSM VZ 6198898701. References [1] J. Aczél, Lectures on Functional Equations and their Applications, Academic Press Inc., New York, 1966. [2] J. Aczél, V.D. Belousov, M. Hosszú, Generalized associativity and bisymmetry on quasigroups, Acta Mathematica Academiae Scientiarum Hungaricae 11 (1960) 127–136. [3] A. Amo, J. Montero, E. Molina, Representation of consistent recursive rules, European Journal of Operational Research 130 (2001) 29–53. [4] G. Beliakov, T. Calvo, On migrative means and copulas, in: Proceedings of Fifth International Summer School on Aggregation Operators, Palma de Mallorca, 2009, pp. 107–110. [5] M. Budincˇevic´, M. Kurilic´, A family of strict and discontinuous triangular norms, Fuzzy Sets and Systems 95 (1998) 381–384. [6] H. Bustince, J. Fernández, R. Mesiar, J. Montero, R. Orduna, Overlap functions, Nonlinear Analysis 72 (2010) 1488–1499. [7] H. Bustince, J. Montero, R. Mesiar, Migrativity of aggregation functions, Fuzzy Sets and Systems 160 (2009) 766–777. [8] T. Calvo, A. Kolesárová, M. Komorníková, R. Mesiar, Aggregation operators, properties, classes and construction methods, in: T. Calvo, G. Mayor, R. Mesiar (Eds.), Aggregation Operators New Trends and Applications, Physica-Verlag, Heidelberg, 2002, pp. 3–104. [9] V. Cutello, J. Montero, Recursive connective rules, International Journal of Intelligent Systems 14 (1999) 3–20. [10] B. De Baets, J. Fodor, van Melle’s combining function in MYCIN is a representable uninorm: an alternative proof, Fuzzy Sets and Systems 104 (1999) 133–136. [11] F. Durante, R. Ghiselli Ricci, Supermigrative semi-copulas and triangular norms, Information Sciences 179 (2009) 2689–2694. [12] F. Durante, P. Sarkoci, A note on the convex combinations of triangular norms, Fuzzy Sets and Systems 159 (2008) 77–80. [13] J. Fodor, B. De Baets, A single-point characterization of representable uninorms, Fuzzy Sets and Systems, submitted for publication, doi:10.1016/ j.fss.2011.12.001. [14] J. Fodor, I.J. Rudas, On continuous triangular norms that are migrative, Fuzzy Sets and Systems 158 (2007) 1692–1697. [15] J. Fodor, I.J. Rudas, On some classes of aggregation functions that are migrative, in: Proceedings of IFSA-EUSFLAT 2009, Lisbon, Portugal, 2009, pp. 653– 656. [16] J. Fodor, I.J. Rudas, Migrative t-norms with respect continuous ordinal sums, Information Sciences 181 (2011) 4860–4866. [17] J. Fodor, R. Yager, A. Rybalov, Structure of uninorms, International Journal of Uncertainty, Fuzziness and Knowledge-based Systems 5 (1997) 411–427. [18] E.P. Klement, R. Mesiar, E. Pap, On the relationship of associative compensatory operators to triangular norms and conorms, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 4 (1996) 129–144. [19] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000. [20] K.C. Maes, B. De Baets, Negation and affirmation: the role of involutive negators, Soft Computing 11 (2007) 647–654. [21] K.C. Maes, B. De Baets, Commutativity and self-duality: two tales of one equation, International Journal of Approximate Reasoning 50 (2009) 189–199. [22] K. Mak, Coherent continuous systems and the generalized functional equation of associativity, Mathematics of Operations Research 12 (1987) 597– 625. [23] Gy. Maksa, Quasisums and generalized associativity, Aequationes Mathematicae 69 (2005) 6–27. [24] R. Mesiar, H. Bustince, J. Fernandez, On the a-migrativity of semicopulas, quasi-copulas, and copulas, Information Sciences 180 (2010) 1967–1976. [25] J. Montero, V. López, D. Gómez, The role of fuzziness in decision making, in: D. Ruan et al. (Eds.), Fuzzy Logic: A Spectrum of Applied and Theoretical Issues, Springer, 2007, pp. 337–349. [26] J. Montero, D. Gómez, S. Muñoz, Fuzzy information representation for decision aiding, in: Proceedings of the IPMU Conference, Málaga, Spain, 2008, pp. 1425–1430. [27] B. Roy, Decision sciences or decision aid sciences, European Journal of Operational Research 66 (1993) 184–203. [28] R. Yager, A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems 80 (1996) 111–120.