Handling of neutral information by aggregation operators

Fuzzy Sets and Systems 158 (2007) 861 – 880 www.elsevier.com/locate/fss

Handling of neutral information by aggregation operators G. Beliakova,∗ , T. Calvob , A. Praderac a School of Engineering and Information Technology, Deakin University, 221 Burwood Hwy, Burwood 3125, Australia b Departamento de Ciencias de la Computación, Universidad de Alcalá, 28871-Alcalá de Henares, Madrid, Spain c Departamento de Arq. y Tecnología de Computadores y CCIA, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain

Received 18 February 2006; received in revised form 4 August 2006; accepted 1 November 2006 Available online 29 November 2006

Abstract We generalize the notion of a neutral element of aggregation operators. Our construction involves tuples of values that are neutral with respect to the result of aggregation. Neutral tuples are useful to model situations in which information from different sources, or preferences of several decision makers, cancel each other. We examine many popular classes of aggregation operators in respect to their neutral sets, and also construct new aggregation operators with predefined neutral sets. © 2006 Elsevier B.V. All rights reserved. Keywords: Aggregation operators; Neutral element; Neutral set; Neutral information

1. Introduction Aggregation of pieces of information coming from different sources is an important task in expert and decision support systems, multicriteria decision making and group decision making. Aggregation operators are mathematical objects that perform precisely this type of information fusion. For a recent extensive overview of different classes of aggregation operators see [7]. Frequently these pieces of information contradict each other, and in some cases need to be canceled out. For example, in group decision making, two members of a five-member jury may be in favor of a decision, and two may be against it. In this case the decision is based solely on the vote of the remaining fifth member. If the members of the jury are allowed to express the strength of their opinion, or have different voting power, their votes can cancel out in more complicated ways. For instance if two members are in favor of a decision, one weakly, the other one strongly, and two others are both moderately against, we still have the fifth member deciding the outcome. In company shareholders meetings, votes that cancel each other also depend on the number of shares each shareholder possesses. In politics, the balance of power is the term referring to one, or few members of minor parties whose vote or opinion is crucial, when the votes of members of major parties cancel each other. Under some aggregation rules, the outcome will be given exactly by the preferences of the minority parties, whereas other aggregation rules will modify these preferences. Similarly, in expert systems there may be certain pieces of evidence in favor of a hypothesis, and certain pieces not supporting it, so that in total the hypothesis is neither supported nor rejected. In this case some additional evidence may be sought, which will be decisive. A classical example of an expert system with such behavior is MYCIN [6]. ∗ Corresponding author. Tel.: +61 3 95669854.

E-mail addresses: [email protected] (G. Beliakov), [email protected] (T. Calvo), [email protected] (A. Pradera). 0165-0114/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2006.11.002

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In the context of medical applications, there is specific knowledge which particular inputs may or may not cancel each other, and the aggregation operator may not be symmetric. In the above mentioned examples, the “pros’’ and “cons’’ cancel each other, and the outcome in some sense is neutral with respect to (w.r.t.) this information (evidence, opinions). We shall refer to it as neutral information. Within the framework of aggregation operators, the values that cancel each other will be referred to as neutral tuples, and the set made of all the neutral tuples of an aggregation operator will be called its neutral set. The illustrative examples provided above suggest the variety of practical situations in which we would like to model cancelative behavior of the aggregation procedure. It is interesting to examine known aggregation operators in respect to this behavior, and also to develop construction procedures if we want to obtain an aggregation operator with a predefined cancelative behavior. The purpose of this paper is to study aggregation operators from the point of view of handling neutral information. We generalize the standard notion of a neutral element and study some known families of aggregation operators and identify their neutral sets. We shall also present a method of construction of aggregation operators with the desired neutral set. The rest of the paper is organized as follows. The next two sections give the basic definitions of aggregation operators and neutral sets, and provide a number of general results. In Section 4 we analyze the most important families of aggregation operators and establish their neutral sets. Some other families are treated in the appendix. In Section 5 we solve the inverse problem: how to build an aggregation operator with a predefined neutral set. We conclude the article with a short summary. 2. Preliminaries Let I denote the basic domain [0, 1]. Definition 1 (Calvo et al. [7]). An aggregation operator is a function F :



n∈N I

n

→ I such that

(i) F (x1 , . . . , xn )F (y1 , . . . , yn ) whenever xi yi for all i ∈ {1, . . . , n}. (ii) F (t) = t for all t ∈ I. (iii) F (0, . . . , 0) = 0 and F (1, . . . , 1) = 1.       n-times n-times Each aggregation operator F can be represented by a family of n-ary operators F (n) : I n → I given by F (n) (x1 , . . . , xn ) = F (x1 , . . . , xn ). This representation allows to define most of the properties of aggregation operators: Definition 2. Let F be an aggregation operator and (F (n) )n∈N the corresponding family of n-ary operations. (i) F is called symmetric, idempotent, strictly monotone (on the whole domain) or continuous if, for each n 2, the n-ary operation F (n) is symmetric, idempotent, strictly monotone or continuous, respectively. (ii) An element e ∈ I is called a neutral element and an element a ∈ I is called an annihilator of F if for each n 2, for each i ∈ {1, . . . , n} and for all x1 , . . . , xn ∈ I, we have, respectively F (n) (x1 , . . . , xi−1 , xi , xi+1 , . . . , xn ) = F (n−1) (x1 , . . . , xi−1 , xi+1 , . . . , xn ) whenever xi = e, F (n) (x1 , . . . , xi−1 , xi , xi+1 , . . . , xn ) = a whenever xi = a. 3. Neutral information 3.1. Definitions According to Definition 2, a neutral element of an aggregation operator F is a value e ∈ I that can be omitted, without influencing the final output, from any position of any input vector. In the following, in order to cope with

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larger pieces of neutral information, we generalize the standard definition of neutral element to the case of tuples  = (1 , . . . , m ) ∈ I m , m ∈ N = {1, 2, . . .}. In general, F is not symmetric, and, unlike the neutral element, the values that cancel each other may depend on their positions. To denote the subsets of components of a vector x ∈ I n we shall employ the following notation: if I = {I1 , . . . , Im } ⊂ {1, . . . , n} is an index set with cardinality 0
Definition 3. Let F be an aggregation operator and let I ⊂ {1, . . . , n}, n > 1, be an index set such that 0 < |I| = m, with an associated permutation P . Then • A tuple  ∈ I m is neutral for F at level n w.r.t. I when F (n) (x) = F (n−m) (xI )

(1)

holds for all x ∈ I n such that xI = . • The set made of all the tuples  ∈ I m which are neutral for F at level n w.r.t. I, will be denoted by Em (F, n, I) and will be called the neutral set of F at level n w.r.t. I. Example 1. Let F be an aggregation operator, n = 3, I = {2, 3} with P = (2, 1) and  = (0, 1) ∈ I 2 . Then  is neutral for F at level 3 w.r.t. I, i.e.,  ∈ E2 (F, 3, I), if F (3) (t, 1, 0) = F (1) (t) = t holds for any t ∈ I . The above definition implies that when aggregating n values with F , the information contained in a given tuple , if appearing in the positions indicated by (I, P ), does not affect the final output. Of course, the same could happen—as it is the case of the standard neutral element—independently of the positions that the components of  occupy in the input vector x. The next definition accommodates this situation. Definition 4. Let F be an aggregation operator and let m, n ∈ N, m < n. Then • A tuple  ∈ I m is neutral for F at level n when, for any index set I ⊂ {1, . . . , n} such that |I| = m,  is neutral for F at level n w.r.t. I. • The set made of all the tuples  ∈ I m which are neutral for F at level n will be denoted by Em (F, n) and will be called the m-neutral set of F at level n. Example 2. Choosing, as in the previous example, n = 3 and  = (0, 1) ∈ I 2 , now  is neutral for F at level 3, i.e.,  ∈ E2 (F, 3), if F (3) (P (t, 0, 1)) = F (1) (t) = t, where P (x) is any permutation of the components of x, holds for any t ∈ I. Remark 1. If F is an aggregation operator and m, n ∈ N, m < n, then  Em (F, n, I). (i) Em (F, n) = I ⊂{1,...,n},|I |=m

(ii) If  = (1 , . . . , m ) ∈ Em (F, n), then  = ((1) , . . . , (m) ) ∈ Em (F, n) for any permutation  = ((1), . . . , (m)) of (1, . . . , m). (iii) If F is symmetric, then it obviously suffices to have  ∈ Em (F, n, I) for some I in order to automatically have  ∈ Em (F, n). Coming back to Definition 3, note now that it refers to just one specific dimension, n, of the aggregation operator F . Similarly to the way in which the standard neutral element is defined, we could think of tuples  ∈ I m remaining neutral for any dimension (as long as such dimension contains the positions given by (I, P )):

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Definition 5. Let F be an aggregation operator and let I ⊂ {1, 2, . . .} be an index set such that |I| = m, with the associated permutation P . Then • A tuple  ∈ I m is neutral for F w.r.t. I when, for any n  max(|I| + 1, max(I)),  is neutral for F at level n w.r.t. I. • The set made of all the tuples  ∈ I m which are neutral for F w.r.t. I will be denoted by Em (F, I) and will be called the neutral set of F w.r.t. I. Example 3. Choosing, as in Example 1, I = {2, 3} with P = (2, 1), then  = (0, 1) ∈ I 2 is neutral for F w.r.t. I, i.e.,  ∈ E2 (F, I), if F (3) (t1 , 1, 0) = F (1) (t1 ) = t1 holds for any t1 ∈ I , F (4) (t1 , 1, 0, t2 ) = F (2) (t1 , t2 ) holds for any t1 , t2 ∈ I , F (5) (t1 , 1, 0, t2 , t3 ) = F (3) (t1 , t2 , t3 ) holds for any t1 , t2 , t3 ∈ I , etc. Remark 2. If F is an aggregation operator, I ⊂ {1, 2, . . .} is an index set such that |I| = m, then  Em (F, I) = Em (F, n, I). n  max(|I |+1,max(I ))

Definitions 4 and 5 have been obtained from Definition 3 after independently introducing a stronger demand on two different aspects: the position of the neutral information within the input vector and the dimension of the latter, respectively. If these two aspects are taken into account simultaneously, the result can be stated as follows: Definition 6. Let F be an aggregation operator and let m ∈ N. Then • A tuple  ∈ I m is neutral for F when, for any n > m and for any index set I ⊂ {1, . . . , n} such that |I| = m,  is neutral for F at level n w.r.t. I. • The set made of all the tuples  ∈ I m which are neutral for F will be denoted by Em (F ) and will be called the m-neutral set of F . Example 4. The tuple  = (0, 1) ∈ I 2 is neutral for F if: ∀n > 2, ∀(x1 , . . . , xn ) ∈ I n , F (n) (x1 , . . . , xn ) = F (n−2) (x1 , . . . , xi−1 , xi+1 , . . . , xj −1 , xj +1 , . . . , xn ) whenever there exist i, j ∈ {1, . . . , n} such that xi = 0, xj = 1. Remark 3. If F is an aggregation operator and m ∈ N, then  (i) Em (F ) = Em (F, n, I). I ⊂{1,...,n},|I |=m

(ii) When choosing m = 1, Definition 6 recovers the standard definition of the neutral element, i.e.:  {e} if F has neutral element e ∈ I, E1 (F ) = ∅ otherwise. Remark 4. The concept of neutral tuple for F could have been alternatively defined using either Definitions 4 or 5, that is, the two following statements hold: (i)  ∈ I m is neutral for F if and only if  is neutral for F at level n for any n > m, that is  Em (F, n). Em (F ) = n>m

(ii)  ∈ I m is neutral for F if and only if  is neutral for F w.r.t. I for any index set I ⊂ {1, 2, . . .} such that |I| = m, that is  Em (F, I). Em (F ) = I ⊂{1,2,...},|I |=m

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When referring to the set made of all neutral tuples for a given aggregation operator F , regardless of their dimension, we will use the following: Definition 7. The neutral set of an aggregation operator F , denoted by E(F ), is the set made of all the tuples  ∈ I m , m ∈ N, which are neutral for F , i.e., Em (F ). E(F ) = m∈N

3.2. Basic properties Let us now discuss some general properties of neutral tuples. We can first of all notice that the concatenation of two neutral tuples provides a new neutral tuple. Indeed, if given  = (1 , . . . , p ) ∈ I p and  = (1 , . . . , q ) ∈ I q , p, q ∈ N,  is used to denote the tuple of dimension p + q built as (1 , . . . , p , 1 , . . . , q ), then we have Proposition 1. Let F be an aggregation operator. If ,  ∈ E(F ), then  ∈ E(F ). Proof. Suppose that  ∈ I p and  ∈ I q for some p, q ∈ N. Then we have to prove that for any n > p + q and for any I ⊂ {1, . . . , n} such that |I| = p + q it is  ∈ Ep+q (F, n, I), i.e., F (n) (x) = F (n−(p+q)) (xI ) for any x ∈ I n such that xI = . But if xI = , then there exist J , K such that xJ =  and (xJ )K = . Then F (n) (x) = F (n−p) (xJ ) (because  ∈ E(F ) and xJ = ) = F ((n−p)−q) ((xJ )K ) =F

(n−(p+q))

(xI )

(because  ∈ E(F ) and (xJ )K = )

(because (xJ )K = xI ).



Proposition 2. Let F be an aggregation operator with a non-empty neutral set Em (F, n, I) at level n. Then for any  ∈ Em (F, n, I) and any x ∈ I n such that xI =  and xI = (0, 0, . . . , 0): F (n) (x) = 0 = min(x). Also for any  ∈ Em (F, n, I) and any x ∈ I n such that xI =  and xI = (1, 1, . . . , 1): F (n) (x) = 1 = max (x). Proof. In the first case F (n) (x) = F (n−m) (0, 0, . . . , 0) = 0, and in the second F (n) (x) = F (n−m) (1, 1, . . . , 1) = 1.



p ∧ q Corollary 1. Let M∨ p (F ) = {x ∈ I |F (x) = max(x)}, and Mq (F ) = {x ∈ I |F (x) = min(x)}. Then

{x ∈ I p |xI ∈ Em (F, p, I), xI = (1, . . . , 1)} ⊆ M∨ p (F ) and {x ∈ I q |xI ∈ Em (F, q, I), xI = (0, . . . , 0)} ⊆ M∧ q (F ) for all p, q > m. Proposition 3. Let F be an aggregation operator with m-neutral tuples  and , such that   componentwise. Then all  ∈ I m such that   are also m-neutral tuples of F . Proof. Follows from the monotonicity of aggregation operators.  When considering concrete examples, we will show that there are aggregation operators—like, for example, arithmetic mean—that have empty neutral sets. However, Proposition 1 allows us to establish that there are at least some important

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families of aggregation operators with non-empty neutral sets (moreover, with non-empty m-neutral sets for any dimension m ∈ N): Proposition 4. Let F be an aggregation operator with neutral element e ∈ I . Then for any m ∈ N, (e, . . . , e) ∈ Em (F ).    m-times Further, the neutral set Em (F, n, I) is not empty for every I ⊂ {1, . . . , n}, m = |I|. Proof. e ∈ E1 (F ) by definition (see Remark 3). Then, applying Proposition 1 to  =  = e we get (e, e) ∈ E2 (F ). The same result, applied to  = (e, e) and  = e, shows that (e, e, e) ∈ E3 (F ), and, similarly, we get that, in general, (e, . . . , e) ∈ Em (F ) for any m 1. Since    m-times  Em (F ) = Em (F, n, I), I ⊂{1,...,n},|I |=m

all Em (F, n, I) = ∅.  However, one may have some Em (F, n, I)  = ∅ holding for m 1, but no neutral element (e.g., projection operators in Section 4.4). Thus, we have established that aggregation operators with a standard neutral element e ∈ I , such as triangular norms, triangular conorms or uninorms, will have non-empty neutral sets, including, at least, the tuples in the form (e, . . . , e); later we shall see that neutral sets may have elements with a more complicated structure. Observe now that neutral tuples have the property that their aggregation, by means of F , always provides the same output: Proposition 5. Let F be an aggregation operator. Then for any ,  ∈ E(F ) it is F () = F (). Proof. Let us suppose that it is  = (1 , . . . , p ) ∈ I p and  = (1 , . . . , q ) ∈ I q for some p, q ∈ N. Then: (i)  ∈ E(F ) implies, in particular, F (p+q) (1 , . . . , p , 1 , . . . , q ) = F (q) (). (ii) Similarly,  ∈ E(F ) implies F (p+q) (1 , . . . , p , 1 , . . . , q ) = F (p) (). From (i) and (ii) we get F (p) () = F (q) (), i.e., F () = F ().



By choosing p = q = 1, the above proposition recovers a well-known result, which establishes the uniqueness of the standard neutral element. Remark 5. Proposition 5 applies to aggregation operators with standard neutral element: if F has a neutral element e ∈ I , then for any  ∈ E(F ): F () = e. However, the converse is not true, i.e., F () = e, does not imply that  is a neutral tuple. For example, the 3 −  operator (Example 5) with the convention 00 = 21 . F (0, 1) = 21 but (0, 1) is not a neutral tuple as F (x, 0, 1) = 21  = x if x = 21 .  Recall also (see for example [7]) that given an aggregation operator F : n∈N [c, d]n → [c, d] and a mono tone bijection  : [a, b] → [c, d], the operator F : n∈N [a, b]n → [a, b], defined as (F (n) ) (x1 , . . . , xn ) = −1 (F (n) ((x1 ), . . . , (xn ))), is in turn an aggregation operator, usually known as the -transform of F . We may, therefore, wonder about the relationship between the neutral information associated to F and the one related to its -transform F . The next proposition describes this relationship:  Proposition 6. Let F : n∈N [c, d]n → [c, d] be an aggregation operator and let  : [a, b] → [c, d] be a monotone bijection. If I ⊂ {1, . . . , n}, n > 1, is an index set such that 0 < |I| = m, then for any  ∈ [c, d]m :  ∈ Em (F, n, I) if and only if

−1 () ∈ Em (F , n, I),

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where, if  ∈ [c, d]m represents the vector (1 , . . . , m ), then −1 () ∈ [a, b]m denotes the vector (−1 (1 ), . . . , −1 (m )). Proof. It suffices to note that (x)I = (xI ) for any bijection .  Take [a, b] = [c, d] = [0, 1] and  : [0, 1] → [0, 1] as the duality transformation (t) = d (t) = 1 − t for any t ∈ [0, 1]. In this case, the d -transform of a given aggregation operator F , Fd , is known as the dual of F , and the consequence of Proposition 6 is that the neutral tuples of a given operator may be directly obtained from the ones of its dual operator.

4. Neutral sets of aggregation operators 4.1. Conjunctive and disjunctive operators Conjunctive aggregation operators, i.e., those verifying F  min, constitute an important class of operators that includes already mentioned triangular norms (subsequently abbreviated as t-norms) and copulas (see [12,15]). With regards to their neutral sets, the next result proves that these sets are either empty or they are limited to just one specific tuple: Proposition 7. (i) Let F be a conjunctive aggregation operator. Then for any n>1 and any ordered index set I ⊂ {1, . . . , n} such that |I| = m, it is Em (F, n, I) ⊆ {(1, . . . , 1)}.    m-times (ii) If F has neutral element e = 1, then Em (F, n, I) = Em (F ) = {(1, . . . , 1)}.    m-times Proof. (i) Let us suppose that  = (1 , . . . , m ) ∈ Em (F, n, I) and let us choose x ∈ I n such that xI =  and xI = (1, . . . , 1). Then Proposition 2 establishes that F (n) (x) = 1. Now, since F is conjunctive, this implies 1 = F (n) (x) min(x) = min(1 , . . . , m ), that is,  = (1, . . . , 1). (ii) If F has neutral element 1 then it is necessarily conjunctive. To see this, let min(x) = xi . Because of the monotonicity property and the fact that e = 1 acts as a neutral element, it is F (n) (x) F (n) (1, . . . , 1, xi , 1, . . . , 1) = xi = min(x) for any x ∈ I n , i.e., F  min. Hence Em (F, n, I) ⊆ {(1, . . . , 1)} for any m ∈ N. In order to obtain the    m-times equality, we apply Proposition 4 (and also Remark 3).  Disjunctive aggregation operators, that is, those verifying F  max, are the dual operators of the conjunctive ones, so similar results may be obtained by duality, using Proposition 6. Proposition 8. (i) Let F be a disjunctive aggregation operator. Then for any n>1 and any ordered index set I ⊂ {1, . . . , n} such that |I| = m, it is Em (F, n, I) ⊆ {(0, . . . , 0)}.    m-times (ii) If F has neutral element 0, then Em (F, n, I) = Em (F ) = {(0, . . . , 0)}.    m-times Thus, it appears that both conjunctive and disjunctive aggregation operators, in particular t-norms and t-conorms, are not very interesting as far as neutral information is concerned, as they only have trivial neutral sets.

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4.2. Uninorms Let us now consider uninorms, and some other types of aggregation operators. We start with representable uninorms which are defined with the help of an additive generator [11,9], a strictly monotone bijection g : I → [−∞, ∞], F (n) (x1 , . . . , xn ) = g −1 (g(x1 ) + · · · + g(xn )). We use the convention ∞ + (−∞) = −∞. We recall that uninorms are associative commutative aggregation operators with the neutral element e ∈ [0, 1]. Uninorms are necessarily discontinuous on I n , except when e = 0 or 1, in which case they coincide with t-norms or t-conorms. Representable binary uninorms are continuous on I 2 \{(0, 1), (1, 0)}, (for n variables discontinuity happens at all faces of I n whose coordinates contain at least one 0 and one 1). The neutral element of representable uninorms is the zero of g, g(e) = 0. Uninorms are frequently used in fuzzy systems modeling [19]. A notable example is MYCIN’s aggregation operator, which turn out to be a representable uninorm [18,8]. Proposition 9. Let F be a representable uninorm, with the generator g. Then

⎫ ⎧

⎬ ⎨ 

Em (F ) = x ∈ (0, 1)m

g(xi ) = 0 . ⎭ ⎩

i=1,...,m

Proof. From the definition of neutral tuples we have  n  n     −1 −1 g g(xi ) = g g(xi ) . i∈ /I

i=1

Then for xi , i ∈ I distinct from 0 and 1 we should have n 

g(xi ) =

i=1



g(xi ) +

i∈I



g(xi ) =

i∈ /I



g(xi ),

i∈ /I

 which implies the necessity of i∈I g(xi ) = 0. Sufficiency is straightforward. The tuples involving 0 and 1 are excluded since F (0, 1, t) = t for example, will violate associativity. The reasoning is valid for any m, n > m, and any I.  Example 5. 3 −  operator [7, p. 19], given by  xi  F (n) (x1 , x2 , . . . , xn ) =  , xi + (1 − xi ) with the convention 00 = 0 is a representable uninorm with an additive generator g(x) = log(x/(1 − x)) and neutral element e = 21 . The neutral set Em (F ) containing the tuples  ∈ (0, 1)m is identified from   i = (1 − i ). i=1,...,m

i=1,...,m

In particular, when m = 2, we have an explicit formula E2 (F ) = { ∈ (0, 1)2 |1 + 2 = 1}. Thus, we also have, by concatenating neutral tuples as in Proposition 1 { ∈ (0, 1)m |i + j = 1, k = 21 , k  = i, j } ⊆ Em (F ),

m > 2, and also

{ ∈ (0, 1)m |i + j = 1, k + l = 1, all i, j, k, l distinct} ⊆ Em (F ), m = 4, etc.

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Proposition 10. Let F be a representable uninorm, with the neutral element e = g(t) = −g(1 − t). Then

869 1 2

and a generator g, satisfying

E2 (F ) = { ∈ (0, 1)2 |1 + 2 = 1} and the tuples { ∈ (0, 1)m |i + j = 1, k = 21 , k  = i, j } ⊆ Em (F ),

m > 2,

{ ∈ (0, 1)m |i + j = 1, k + l = 1, all i, j, k, l distinct} ⊆ Em (F ), m = 4, etc. Proof. Clearly, g(1 ) + g(2 ) = g(1 ) + g(1 − 1 ) = 0. By applying Proposition 1 we obtain the other tuples.



Note that 3 −  operator in Example 5 satisfies the condition of Proposition 10. Remark 6. When dealing with aggregation of pieces of information that may be in favor or against, it is customary to employ aggregation operators on a bipolar scale [−1, 1]. Negative values of the arguments are often referred to as negative information, whereas positive values are referred to as positive information. There is an isomorphism between unipolar and bipolar scales, and one can easily construct a bipolar n-ary aggregation operator f from a unipolar operator f˜ by taking, e.g.,   xn + 1 x1 + 1 ,..., − 1. f (x1 , . . . , xn ) = 2f˜ 2 2 Of course, one can also use the scale [0, 1] and treat the values below c, a fixed point of a strong negation N (c) = c, as negative information, and values above c as positive information. The question which scale to use is mainly related to interpretability of the input/output values. We mention bipolar scales because they are used in expert systems literature (e.g., [6]). Proposition 6 allows one to translate characterizations of neutral sets from one scale to another. On the bipolar scale [−1, 1], the conditions of Proposition 10 are e = 0 and g(t) = −g(−t), and the characterization equation changes to 1 + 2 = 0. Example 6. The MYCIN’s aggregation operator is a representable uninorm on a bipolar scale with the generator [6,8]  ln(1 + t) if t < 0, g(t) = − ln(1 − t) otherwise. On I = [0, 1] it is given as  ln(2t) g(t) = − ln(2(1 − t))

1 , 2 otherwise. if t <

On bipolar scale the neutral sets are characterized by

⎧ ⎫

 ⎨ ⎬ 

(1 + i ) = (1 − i ) . Em (F ) =  ∈ (−1, 1)m

⎩ ⎭

i <0 i  0 Clearly, the neutral element e = 0 and g(t) = −g(−t), hence some neutral tuples are given explicitly as in Proposition 10 (see Remark 6), E2 (F ) = { ∈ (−1, 1)2 |1 + 2 = 0} and { ∈ (−1, 1)m |i + j = 0, k = 0, k = i, j } ⊆ Em (F ),

m > 2,

{ ∈ (−1, 1)m |i + j = 0, k + l = 0, all i, j, k, l distinct} ⊆ Em (F ), m = 4, etc.

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Example 7. PROSPECTOR’s aggregation operator is a representable uninorm on a bipolar scale with the generator [8]   1+t g(t) = ln . 1−t The neutral sets are

⎧

 ⎨

Em (F ) =  ∈ (−1, 1)m

(1 + i ) = ⎩

i=1,...,m

 i=1,...,m

(1 − i )

⎫ ⎬ ⎭

.

The neutral element e = 0 and g(t) = −g(−t), conditions of Proposition 10 and Remark 6 are satisfied, and we obtain explicitly some of the same neutral tuples as in MYCIN’s operator. However, in general, except for m = 2, the neutral sets are different. Remark 7. We analyzed two other related classes of aggregation operators, namely nullnorms and quasi-linear T–S operators, and concluded that they have empty neutral sets. 4.3. Other generated aggregation operators Consider now another class of unipolar aggregation operators similar to representable uninorms [13,14]. Let g : I → [a, b], −∞ < a < b < ∞ be a monotone increasing function with zero e ∈ I . Define F (n) (x1 , . . . , xn ) = g (−1) (g(x1 ) + · · · + g(xn )),

(2)

where g (−1) denotes the pseudoinverse. The function (2) is continuous on I n , but it is not associative. Further, on [e, 1]n it coincides with a (scaled) nilpotent t-conorm and on [0, e]n it coincides with a (scaled) nilpotent t-norm. As with uninorms, e is its neutral element, and when e = 1 or e = 0 we obtain t-norms and t-conorms as limiting cases. We have an analogue of Proposition 9. Proposition 11. Let F be a function given by (2), with an additive generator g. Then

⎧ ⎫

⎨ ⎬ 

Em (F ) = x ∈ I m

g(xi ) = 0 . ⎩ ⎭

i=1,...,m We also have an analogue of Proposition 10 for this type of operators, which helps characterize some of the neutral tuples explicitly. Example 8 (Calvo et al. [7], Mesiar [13,14]). Let g(t) = t − 21 . Then F is an ordinal sum of Lukasiewicz t-norm and t-conorm, given by    n  1  1 F (x) = max 0, min 1, + xi − . 2 2 i=1  The neutral set Em (F ) = { ∈ I m | m i=1 i = m/2}. It is interesting to consider the same operator on bipolar scale, i.e.,   n   F (x) = max −1, min 1, . xi i=1

The neutral set is given as 

m 

 Em (F ) =  ∈ [−1, 1] i = 0 .

m

i=1

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4.4. Averaging operators An aggregation operator is called an averaging if it is bounded by minimum and maximum. Averaging aggregation operators are idempotent. A distinguished class of averaging operators is the class of weighted quasi-arithmetic means ([1,7]): Definition 8. An aggregation operator is a weighted quasi-arithmetic mean if, for each n ∈ N, it can be written as  F

(n)

(x) = g

−1

n 

 win g(xi ) ,

i=1 n where n g : [0, 1] → [−∞, +∞] is a continuous strictly monotone function and wn = (w1n , . . . , wnn ) ∈ I verifies i=1 win = 1.

The above definition includes two important classes of commonly used operators: weighted arithmetic means (obtained when choosing g(t) = t) and quasi-arithmetic means (obtained when taking the weights win = 1/n for all n ∈ N, i ∈ {1, . . . , n}). The neutral sets of weighted quasi-arithmetic means are, in general, empty, except in some specific cases described in the following proposition: Proposition 12. Let F be a weighted quasi-arithmetic mean. Then for any n ∈ N and any I ⊂ {1, . . . , n} such that |I| = m it is Em (F, n, I) = ∅ 

if and only if

(i) win = 0 (ii) wI j n = wjp

∀i ∈ I, ∀j ∈ {1, . . . , p},

where p = n − m. In addition, under the above conditions Em (F, n, I) = I m . The proof is given in the Appendix. The projections to the first and to the last coordinates, given, respectively, by PF (x1 , . . . , xn ) = x1 and PL (x1 , . . . , xn ) = xn , are important examples of averaging operators [7] that appear to have non-empty neutral sets. Indeed, both can be seen as weighted arithmetic means with weighting vectors of the form (1, 0, . . . , 0) and (0, . . . , 0, 1), / I and respectively. Then, according to Proposition 12, PF will have non-empty neutral sets Em (PF , n, I) = I m , if 1 ∈ 0 < m n − 1, and Em (PF , n, I) = ∅ if 1 ∈ I. It is similar for PL , and thus projection operators deliver the largest possible neutral sets with any index set I : |I| = n − 1. Of course, projection operators discard all information given by the components xI , and thus may seem to be of little practical interest. However, they help to prove that the absence of a neutral element does not imply that the neutral sets are empty. On the other hand, Proposition 12 shows that weighted quasi-arithmetic means with strictly positive weighting vectors have always empty neutral sets (this is the case, in particular, of any quasi-arithmetic mean). We can generalize this result for other averaging aggregation operators: Proposition 13. Let F be an averaging aggregation operator bounded by min(x)
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We will now deal with another important class of averaging operators: Definition 9 (Calvo et al. [7], Yager [20]). An aggregation operator is a generalized ordered weighted averaging (OWA) if, for each n ∈ N, it can be written as  n   (n) −1 win g(x(i) ) , F (x) = g i=1 n where n g : [0, 1] → [−∞, +∞] is a continuous strictly monotone function, wn = (w1n , . . . , wnn ) ∈ I verifies i=1 win = 1 and (x(1) , . . . , x(n) ) is a vector obtained from x by arranging its components in a non-decreasing order.

Proposition 13 may be applied, in particular, to generalized OWA operators with strictly positive weighting vectors and generating function satisfying Ran(g) ⊂ R, thus proving the existence of generalized OWA operators with empty neutral sets. However, the next result shows that this is not always the case (note that, since generalized OWA are symmetric, we can directly deal with sets Em (F, n) instead of sets Em (F, n, I)): Proposition 14. Let F be a generalized OWA operator. Then for any m, n ∈ N such that p = n − m > 0 it is  ∈ Em (F, n) if and only if there exists r, s 0 such that m = r + s and: (i) If r  = 0, then win = 0 ∀i ∈ {1, . . . , r}. If s  = 0, then win = 0 ∀i ∈ {n − s + 1, . . . , n}. (ii)  ∈ {0, 1}m and contains exactly r zeros and s ones. (iii) (w(r+1)n , . . . , w(n−s)n ) = (w1p , . . . , wpp ). The proof is given in the Appendix. Obviously, Proposition 14 applies to OWA operators, obtained by choosing g(t) = t, and then also to min and max, since both of them are special cases of OWA operators with weighting vectors (1, 0, . . . , 0) and (0, . . . , 0, 1), respectively. In these cases Proposition 14 recovers the fact (see Section 4.1) that (1, . . . , 1) and (0, . . . , 0) are neutral, respectively, for min and max. Definition 10 (Grabisch [10]). Choquet integral based aggregation operator w.r.t. a fuzzy measure v is given by Cv (x) =

n 

x(i) [v({j |xj x(i) }) − v({j |xj x(i+1) })],

(3)

i=1

where (x(1) , x(2) , . . . , x(n) ) is a non-decreasing permutation of the input x, and x(n+1) = ∞ by convention. Choquet integral based operators may have non-trivial neutral sets, as can be seen from the following example. However, full characterization of its neutral sets is an open problem. Example 9. Consider fuzzy measure given by v({1}) = a, v({1, 3}) = c,

v({2}) = 0,

v({3}) = b,

v({2, 3}) = d,

v({1, 2}) = 1,

v({1, 2, 3}) = 1,

where a, b, c, d ∈ I , a, b c, b d. The neutral set E2 (F, 3, {2, 3}) = {(1, 0)}, i.e., Cv (t, 1, 0) = t. Note that values a, b, c, d can be chosen fairly arbitrarily. 4.5. Self-dual operators Definition 11 (Calvo et al. [7]). An aggregation operator is called self-dual if F (x) = 1 − F (1 − x) for all x ∈ I n .

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Self-dual operators have been characterized by Silvert [16] as those that are symmetric sums F (x) =

A(x) , A(x) + A(1 − x)

with the convention 00 = 21 , where A is an aggregation operator. In some cases self-dual aggregation operators will have a neutral element (necessarily e = operator  xi (n)  F (x1 , x2 , . . . , xn ) =  , xi + (1 − xi )

(4)

1 2 ).

For example, the

with the convention 00 = 21 . This aggregation operator coincides with the 3 −  uninorm in the interior of the unit cube, but it is not associative [7]. There are many other self-dual aggregation operators with the neutral element e = 21 . To construct them, take as A any non self-dual aggregation operator with e = 21 (if A is self-dual, we get F = A). Clearly, if A has the neutral element e = 21 , so does F . Proposition 15. Let F be a self-dual aggregation operator given by (4). Then (i) If F has the neutral element e = 21 then it has non-empty neutral sets Em (F ). (ii) A necessary condition for a tuple  to be a neutral tuple is A() = A(1 − ). (iii) If  is a neutral tuple, then  = 1 −  is also a neutral tuple. Proof. (i) Follows from Proposition 4. (ii) From Remark 5 F () = 21 . Then A() = 21 (A() + A(1 − )), and the result follows. (iii) Evident.  5. Construction methods In the previous sections we have seen that different classes of aggregation operators have quite distinct neutral sets, ranging from empty sets, to finite and infinite neutral sets, to the whole cube I m , m < n. However, these neutral sets may not be sufficient for applications, in which one may have specific knowledge of which inputs should cancel each other. In addition, even if there are existing aggregation operators with the desired neutral set, they may not be suitable for unrelated reasons. If we have a given aggregation operator F , in many cases we can characterize its neutral sets explicitly. The goal of this section is to solve the opposite problem: given a desired neutral set, how to design an aggregation operator with this neutral set. We concentrate on building an n-ary Lipschitz continuous aggregation operator F (n) (x) of a fixed dimension n. In the following we will assume that the aggregation operator is Lipschitz-continuous, with a Lipschitz constant M in some norm · p . Such aggregation operators are of significant practical interest, as they provide stable output w.r.t. inaccuracies in the values of arguments. p-stable, 1-Lipschitz, kernel aggregation operators, copulas and quasi-copulas are special classes of Lipschitz aggregation operators with Lipschitz constant M = 1. In this section we construct the largest and the smallest Lipschitz aggregation operators with the desired neutral set, and will also identify the optimal one. 5.1. General construction Let the desired neutral set of an aggregation operator be given as Em (F, n, I) = {x ∈ I m |h(x) = 0}. We have seen in the previous section that frequently h is given as the opposite diagonal, i.e., m i=1 xi = 1 in the unipolar case, or

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m

i=1 xi = 0 in the bipolar scale. Such a neutral set seems to be quite intuitive, it includes tuples that should cancel each other, like (1, −1), (1, 1, −1, −1), (p, −p, q, −q), and so on (we will restrict ourselves to bipolar scale from now on, it will simplify the equations, and transformation to the unipolar scale is easily done with Proposition 6, see Remark 6). A general method of construction of the largest and the smallest monotone Lipschitz functions interpolating a given k n k k set of data was given in [2–5]. Suppose we want to interpolate the data set D = {(xk , y k )}K k=1 , x ∈ I , y ∈ I, y = n k F (x ) with a Lipschitz function with the Lipschitz constant M. The upper and lower bounds on the values of such a function are given by

u (x) = min{y k + M (x − xk )+ p }, k

l (x) = max{y k − M (xk − x)+ p }, k

(5)

where z+ denotes the positive part of vector z: z+ = (¯z1 , . . . , z¯ n ), with z¯ i = max{zi , 0}, and the optimal interpolant, the one that minimizes the error in the worst case scenario is given by the central scheme [17] as g(x) = 21 (l (x) + u (x)).

(6)

In our case, we will calculate the bounds that result from the conditions F (n) (x) = F (1) (xj ) = xj for xI ∈ Em (F, n, I), j ∈ / I and m = n − 1, which will be applied together with the bounds resulting from other data, such as conditions F (1) = 1, F (0) = 0, etc. A detailed treatment of this method is given in [4,5]. Let us formally state the problem. Take an index set I of cardinality n − 1, j ∈ / I. Let us use the notation z = (z1 , . . . , zj −1 , t, zj +1 , . . . , zn ), such that zI ∈ Em (F, n, I) and zj = t. Given the desired neutral set Em (F, n, I) = {x ∈ I m |h(x) = 0}, |I| = m = n − 1, which implies the condition F (z) = t, and also the Lipschitz constant M in some norm (necessarily M 1), compute the upper and lower bounds on F given in (5), and the optimal aggregation operator (6). The bounds (5) translate into u (x) =

min

z|h(zI )=0,t∈I

{t + M (x − z)+ p },

l (x) =

max

z|h(zI )=0,t∈I

{t − M (z − x)+ p }.

(7)

5.2. Construction for a specific neutral set While Eq. (7) provide generic formulae to calculate both bounds, they are not well suited for practical calculations, as they involve numerical solution of a constrained optimization problem. In the remainder of this section we compute these bounds explicitly for a specific case. Consider the upper bound in (7). Using the method of Lagrange multipliers, convert it to min

 0,z∈[−1,1]n

(t + M (x − z)+ + h(zI )).

(8)

 We remind that zj = t. Let us now consider a special case h(z) = i∈I zi , mentioned earlier, i.e., the opposite diagonal of the aggregation operator. We shall use a standard · p -norm. For p > 1, after differentiating w.r.t. and components of zI , we have the following system of the necessary conditions,  zi = 0, i∈I p−1

KM(x1 − z1 )+ + = 0, .. . p−1 KM(xj −1 − zj −1 )+ + = 0, p−1

KM(xj +1 − zj +1 )+ + = 0, .. . p−1 KM(xn − zn )+ + = 0,

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 p p with K = −( i∈I (xi − zi )+ + (xj − t)+ )(p−1)/p from which p−1

p−1

(x1 − z1 )+

p−1

p−1

= · · · = (xj −1 − zj −1 )+ = (xj +1 − zj +1 )+ = · · · = (xn − zn )+ .  Note that if h(x) = i∈I xi 0, then there exists z|h(z) = 0, ∀i ∈ I : xi zi , meaning (xi − zi )+ = 0, hence the minimum w.r.t. z is achieved at such a point, and we obtain u (x) = min {t + M(xj − t)+ }, t∈[−1,1]

which yields the solution u (x)= xj . n Thus, we consider the case i∈I xi > 0. This effectively restricts the domain to {z ∈ [−1, 1] |∀i ∈ I zi xi and z = 0}, and such a subset is non-empty. Together with the necessary conditions of a minimum, this translates i∈I i into the system of equations x1 − z1 = · · · = xj −1 − zj −1 = xj +1 − zj +1 = · · · = xn − zn .  In conjunction with i∈I zi = 0 solving for z1 we get z1 =

x1 (n − 2) − x2 − · · · − xn . n−1

By resolving the rest of the equations w.r.t. zi we obtain a generic formula, which identifies all but the j th component of the optimal z. zi = xi −

1  xi , n−1

i ∈ I.

(9)

i∈I

  For p = 1 it is not difficult to check that the expression i∈I (xi − zi )+ , when  restricted to i∈I zi = 0 and zi xi , is a constant function, hence the set of its minimizers is {z|∀i ∈ I zi xi , i∈I zi = 0}. Evidently, z given in (9) belongs to this set, and hence we can use (9) for p 1. Substituting the values of zi in (7) we obtain p

u (x) = min (t + M((xj − t)+ + )1/p ), t∈[−1,1]

  p where = i∈I (xi − zi )+ = (n − 1)1−p ( i∈I xi )p . To identify the minimum w.r.t. t we use the following: Proposition 16. Let 0, M 1, p 1, a ∈ [−1, 1] and p

fa (t) = t + M((a − t)+ + )1/p . The minimum of fa (t) is achieved at • t ∗ = −1, if M = 1; • t ∗ = a, if p = 1 and M > 1;

• t ∗ = med{−1, a − ( M p/(p−1) )1/p , a} otherwise, −1 and its value is ⎧ −1 + M( + (a + 1)p )1/p ⎪ ⎪ ⎨ min fa (t) = a + (M p/(p−1) − 1)(p−1)/p 1/p ⎪ ⎪ ⎩ a + M 1/p The proof is given in the Appendix.

if t ∗ = −1, if t ∗ = a − if t ∗ = a.



M p/(p−1) − 1

1/p ,

(10)

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  p Thus, u (x) is given in (10), with a = xj and = (n − 1)1−p (  i∈I xi ) , when i∈I xi 0, or by u (x) = xj otherwise. Similar development for l (x) yields l (x) = xj when i∈I xi 0, otherwise the maximum in (7) is achieved at • t ∗ = 1, if M = 1; • t ∗ = xj , if p = 1 and M > 1, or

)1/p , 1} otherwise, • t ∗ = med{xj , xj + ( M p/(p−1) −1 and the value is ⎧ xj − M 1/p ⎪ ⎪ ⎨ l (x) = xj − (M p/(p−1) − 1)(p−1)/p 1/p ⎪ ⎪ ⎩ 1 − M( + (1 − xj )p )1/p

if t ∗ = xj , if

t∗

= xj +

if t ∗ = 1.



M p/(p−1) − 1

1/p ,

(11)

The actual bounds on the values of a Lipschitz aggregation operator F are computed as in [4,5] Bu (x) = min{u (x), M x , 1}, Bl (x) = max{l (x), −M x , −1}, where u , l are given in (7) (explicitly in (10) and (11)), and the other expressions arise from the interpolation conditions F (−1, . . . , −1) = −1, F (1, . . . , 1) = 1. In the presence of other requirements (conjunctive or disjunctive behavior, annihilator, etc.) the bounds are further tightened as detailed in [5]. In the case when there exist neutral tuples w.r.t. several index sets I, for example by taking I j = {1, . . . , n} \ {j } j j and running j = 1, . . . , n, we obtain the bounds as in (7) for each fixed j , call them u , l , and then take the pointwise minimum and maximum u = min{1u , . . . , nu },

l = max{1l , . . . , nl },

and then compute the bounds Bu , Bl as earlier. The optimal aggregation operator is then computed as the half-sum of Bu and Bl . Example 10. Consider 1-Lipschitz aggregation operator with the neutral set E2 (F, 3, {1, 2}) = {z ∈ [−1, 1]3 |z1 +z2 = 0}. Applying (10) we have  if x1 + x2 < 0, x3 u (x) = x1 + x2 + x3 otherwise. Therefore, the largest aggregation operator with this neutral set is  if x1 + x2 < 0, x3 Bu (x) = min(x1 + x2 + x3 , 1) otherwise. If we desire to have a symmetric aggregation operator, then we take E2 (F, 3) = {z ∈ [−1, 1]3 |zi + zj = 0, i  = j }, cf. Example 8. It is sufficient to build the function B(x(1) , x(2) , x(3) ), where x(i) is the ith largest component of x. By taking the minimum over all possible u (x) we get ⎧ if x(1) + x(2) < 0, ⎨ min(x) Bu (x) = min(x(2) , x1 + x2 + x3 ) if x(1) + x(2) 0, but x(1) + x(3) < 0, ⎩ otherwise. min(x1 + x2 + x3 , 1) That is, we have Lukasiewicz t-conorm in the positive octant and minimum in the negative octant. The lower bound is found analogously as ⎧ if x(2) + x(3) > 0, ⎨ max(x) Bl (x) = max(x(2) , x1 + x2 + x3 ) if x(2) + x(3) 0, but x(1) + x(2) > 0, ⎩ otherwise. max(x1 + x2 + x3 , −1)

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Table 1 The neutral sets of various families of aggregation operators Aggregation operator

Neutral set Em (F, n, I )

Triangular norms Triangular conorms Uninorms Nullnorms Other generated operators Quasi-linear T–S operators Means

{(1, 1, . . . , 1)} {(0, 0, . . . , 0)} Non-empty non-trivial neutral sets Empty Non-empty non-trivial neutral sets Empty Empty, if all weights are strictly positive largest neutral set otherwise Empty, if all weights are strictly positive otherwise  ∈ {0, 1}m Empty, if all weights are strictly positive otherwise  ∈ {0, 1}m Non-empty if there is neutral element Non-empty in some cases Non-empty, largest neutral set

OWA Generalized OWA Symmetrical sums Choquet integrals Projection operators

To transform to the [0, 1] scale we use Proposition 6, and we obtain E2 (F, 3) = {z ∈ [0, 1]3 |zi + zj = 1, i  = j } and a combination of max and a scaled Lukasiewitz t-conorm or their duals as the result. 6. Conclusion We extended the notion of the neutral element of aggregation operators to neutral tuples. Such tuples are useful when modeling cancelative behavior of aggregation procedures. Neutral tuples may have non-trivial structure, like the opposite diagonal for uninorms, and may be present in asymmetric aggregation operators. We examined the most common classes of aggregation operators and established their neutral sets, summarized in Table 1. These results will be useful when selecting suitable aggregation operators for specific applications. In some applications the neutral tuples may be specified a priori, and then the task is to build a suitable aggregation operator with such neutral tuples. We developed a general method of construction of Lipschitz aggregation operators (in particular p-stable, 1-Lipschitz and kernel aggregation operators) with the desired neutral set. We also particularized this method for the special case of the opposite diagonal being the neutral set, and obtained an explicit solution. Our future research will involve a parallel study of absorbing sets, which generalize the notion of the annihilator.

Acknowledgements This work was supported by the projects MTM2004-3175, BFM2003-05308, TIC2002-11942-E and TIC200309001-C02-02 from Ministerio de Educación y Ciencia, Spain, and PRIB-2004-9250, Govern de les Illes Balears.

Appendix A. Proof of Proposition 12 (Neutral sets of quasi-arithmetic means) Proof. If the conditions (i) and (ii) hold, it is clearly Em (F, n, I) = I m . Conversely, let us suppose that Em (F, n, I)  = ∅, i.e., there exists  ∈ I m such that  ∈ Em (F, n, I). By definition, this means  i∈I

win g(i ) +

p  j =1

wI j n g(xI j ) =

p  j =1

wjp g(xI j )

(A.1)

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for any x ∈ I n such that xI = . Given a, b ∈]0, 1[, a  = b, this must be true, in particular, for xI = (a, . . . , a) and xI = (b, . . . , b). Then we have ⎤ ⎡ p   win g(i ) = g(a) ⎣1 − wI j n ⎦ i∈I

j =1

⎡ = g(b) ⎣1 −

p  j =1

⎤ wI j n ⎦ .

p Since g is strictly monotone, this implies j =1 wI j n = 1, and, therefore, win = 0 for any i ∈ I, i.e., condition (i) must be satisfied. But then Eq. (A.1) becomes p  j =1

wI j n g(xI j ) =

p  j =1

wjp g(xI j )

for any xI ∈ I p . In particular, for each j ∈ {1, . . . , p} we can choose xI such that g(xI i )  = 0 if i = j and g(xI i ) = 0 otherwise (if g is such that 0 ∈ / Ran(g), recall—see e.g., [7]—that any linear transformation of g generates the same weighted quasi-arithmetic mean). We then obtain wI j n = wjp for any j ∈ {1, . . . , p}, which is condition (ii).  Appendix B. Proof of Proposition 14 (Neutral sets of OWA) Proof. If the three conditions (i)–(iii) are satisfied then  is clearly a neutral tuple. Conversely, let us suppose that  ∈ Em (F, n). By definition, this means n  i=1

win g(x(i) ) =

p  j =1

(B.1)

wjp g(xI (j ) )

for any x ∈ I n such that xI = , where I, |I| = m, is an arbitrary index set. Let us prove that  contains only zeros and ones. Let us assume that  has j zeroes, s ones, m = j + s + 1, and one of the components 0 < k < 1. We note that the generating function g(t) is defined up to an arbitrary linear transformation (see e.g., [7]), therefore, with no loss of generality we assume that g(k ) < 0, and g increasing. Let us show first that at least one of the weights win , j + 1 < i < n − s + 1 is strictly positive. Take xI = (c, c, . . . , c), k < c < 1, i.e., x() = (0, . . . , 0, k , c . . . , c, 1, . . . , 1). Then from (B.1) j 

win g(0) + wj +1,n g(k ) +

i=1

n−s  i=j +2

win g(c) +

n  i=n−s+1

win g(1) =

p 

wip g(c) = g(c).

i=1

But if win = 0, j + 1 < i < n − s + 1, the expression on the left is constant, and that on the right depends on c, hence contradiction. Thus, let wqn be the first non-zero weight, i.e., win = 0, i = j + 1, . . . , q − 1, q < n − s + 1 and wqn > 0. Eq. (B.1) must hold for any xI , let us take the following vectors x1I = (a, a, a, . . . , a), x2I = (b, a, a, . . . , a),

x3I = (b, b, a, . . . , a), ... xn−m+1 = (b, b, b, . . . , b), I

with 0 < b < k < a < 1, and g(a) = 0.

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879

By choosing an appropriate vector xl , l = 1, . . . , n − m + 1 from the collection above, we can always obtain such I a vector x, that its qth smallest component is k , i.e., x(q) = k . But then the expression n 

win g(x(i) ) =

j 

win g(0) +

n−s 

0g(b) + wqn g(k ) +

i=j +1

i=1

i=1

q−1 

0win +

i=q+1

n 

win g(1)

i=n−s+1

involves a non-zero term wqn g(k ) and is constant. On the right-hand side of (B.1) we have q−1 

wip g(b) +

p 

0wip = g(b)

i=q

i=1

q−1 

wip ,

i=1

i.e., it is either a non-constant function of b, or it is null (if all the weights wip = 0, i = 1, . . . , q − 1). In the former case we immediately have a contradiction, so consider the latter case, which implies j 

n 

win g(0) + wqn g(k ) +

i=1

win g(1) =

q−1 

wip = 0.

i=1

i=n−s+1

Now take another vector xl from the collection above, such that x(q−1) = k . We have I

j 

win g(0) + wq−1,n g(k ) +

i=1

n 

win g(1) = g(b)

i=n−s+1

q−2 

wip = 0.

i=1

But this leads to the conclusion that wq−1,n g(k ) = wqn g(k ), which is impossible since wq−1,n = 0 and wqn > 0. Hence we also have a contradiction. Assumption that more than one component of  is distinct from zero or one leads to a similar contradiction. Hence the necessity of item (ii) is proven. Thus, we assume in the rest of the proof that  = (0, . . . , 0, 1, . . . , 1 ) (or any permutation of this tuple).       r -times m−r times Now take the vector xI = (a, . . . , a), 0 < a < 1, such that g(a) = 0. We have F (x) =

r 

win g(0) +

i=1

n−m+r 

win g(a) +

i=r+1

n 

win g(1) =

i=n−m+r+1

p 

wjp g(a) = g(a) = 0.

j =1

Hence r 

n 

win g(0) +

i=1

win g(1) = 0.

i=n−m+r+1

Take another vector xI = (b, . . . , b), 0 < b < 1, b  = a. r n−m+r n    F (x) = win g(0) + win g(b) + i=1

=0+

i=r+1 n−m+r  i=r+1

win g(b) =

win g(1)

i=n−m+r+1 p 

wjp g(b) = g(b).

j =1

n  This implies that n−m+r i=1 win = 1, which means win = 0 for i = 1, . . . , r and i=r+1 win = 1, but win 0 and i = n − m + r + 1, . . . , n. Thus, the necessity of (i) is proven. Since (i) and (ii) are the necessary conditions, Eq. (B.1) becomes w(r+1)n g(x(1) ) + · · · + w(n−s)n g(x(p) ) = w1p g(x(1) ) + · · · + wpp g(x(p) ) for any x ∈ I p . Then choosing x such that g(x(1) )  = 0 and g(x(i) ) = 0 for any i  = 1 we get w(r+1)n = w1p (recall again that if 0 ∈ / Ran(g), g can always be replaced by a linear transformation). Since this can be done for any j ∈ {1, . . . , p}, condition (iii) is obtained. 

880

G. Beliakov et al. / Fuzzy Sets and Systems 158 (2007) 861 – 880

Appendix C. Proof of Proposition 16 Proof. If p = 1, then fa (t) = t + M( + (a − t)+ ) = t + M + M(a − t)+ . It is clear that when t a the function fx (t) = t + M is strictly increasing w.r.t. t and, therefore, its minimum is obtained at t ∗ = a. On the other hand, if t a then fa (t) = t (1 − M) + M + Ma and since M > 1, fa (t) would be decreasing and the minimum would be located at t ∗ = a; both cases provide min fa (t) = fa (a) = a + M . For the case p>1, note that for all t a, fa (t) is again strictly increasing in t (and it will have its minimum at t ∗ = a) so we only have to find the minimum of fa (t) on [−1, a]. The possible minimizers are the endpoints of this interval and the points fulfilling dfa (t)/dt = 0. The derivative is (p−1)/p  dfa (t) (a − t)p . =1−M dt

+ (a − t)p In the special case M = 1, if = 0 then fa (t) = t + (a − t) = a and mint fa (t) = fa (−1). If >0, fa (t) is increasing, and the minimum is achieved also at t = −1. For M>1, the critical points are t = −1, t = a − ( /(M p/(p−1) − 1))1/p , and t = a. Now, since fa (t) is a convex function it is clear that its minimum in [−1, a] is achieved at med(−1, a − ( /(M p/(p−1) − 1))1/p , a). The value of the minimum is easily obtained by substituting t in fa (t) by these values.  References [1] J. Aczel, On mean values, Bull. Amer. Math. Soc. 54 (1948). [2] G. Beliakov, Monotonicity preserving approximation of multivariate scattered data, BIT 45 (2005) 653–677. [3] G. Beliakov, Identification of general aggregation operators by Lipschitz approximation, in: M. Hamza (Ed.), The IASTED International Conference on Artificial Intelligence and Applications, ACTA Press, Innsbruck, Austria, 2005, pp. 230–233. [4] G. Beliakov, T. Calvo, Identification of general and double aggregation operators using monotone smoothing, in: E. Montseny, P. Sobrerillo (Eds.), EUSFLAT 2005, European Society for Fuzzy Logic and Technology, Barcelona, Spain, 2005, pp. 937–942. [5] G. Beliakov, T. Calvo, J. Lazaro, Pointwise construction of Lipschitz aggregation operators with specific properties, Internat. J. Uncertain., Fuzziness Knowledge-Based Systems, to appear. [6] B. Buchanan, E. Shortliffe, Rule-based expert systems, in: The MYCIN Experiments of the Stanford Heuristic Programming Project, AddisonWesley, Reading, MA, 1984. [7] T. Calvo, A. Kolesarova, M. Komornikova, 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, New York, 2002, pp. 3–104. [8] 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. [9] J. Fodor, R. Yager, A. Rybalov, Structure of uninorms, Internat. J. Uncertain., Fuzziness Knowledge-Based Systems 5 (1997) 411–427. [10] M. Grabisch, The interaction and Mobius representation of fuzzy measures on finite spaces, k-additive measures: a survey, in: M. Grabisch, T. Murofushi, M. Sugeno (Eds.), Fuzzy Measures and Integrals, Theory and Applications, Physica-Verlag, Heidelberg, 2000, pp. 70–93. [11] E. Klement, R. Mesiar, E. Pap, On the relationship of associative compensatory operators to triangular norms and conorms, Internat. J. Uncertain., Fuzziness Knowledge-Based Systems 4 (1996) 129–144. [12] E. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer, Dordrecht, 2000. [13] R. Mesiar, Compensatory operators based on triangular norms, in: H.-J. Zimmermann (Ed.), EUFIT, Aachen, 1995, pp. 131–135. [14] R. Mesiar, M. Komornikova, Triangular norm-based aggregation of evidence under fuzziness, in: B. Bouchon-Meunier (Ed.), Aggregation and Fusion of Imperfect Information, Physica-Verlag, Heidelberg, 1998, pp. 11–35. [15] R. Nelsen, An Introduction to Copulas, Springer, Berlin, 1998. [16] W. Silvert, Symmetric summation: a class of operations on fuzzy sets, IEEE Trans. Systems, Man and Cybernetics 9 (1979) 659–667. [17] J. Traub, H. Wozniakowski, A General Theory of Optimal Algorithms, Academic Press, New York, 1980. [18] A. Tsadiras, K. Margaritis, The MYCIN certainty factor handling function as uninorm operator and its use as a threshold function in artificial neurons, Fuzzy Sets and Systems 93 (1999) 263–274. [19] R. Yager, Uninorms in fuzzy systems modeling, Fuzzy Sets and Systems 122 (2001) 167–175. [20] R. Yager, Generalized OWA aggregation operators, Fuzzy Optimization and Decision Making 3 (2004) 93–107.