Shift invariant binary aggregation operators

Fuzzy Sets and Systems 142 (2004) 51 – 62 www.elsevier.com/locate/fss

Shift invariant binary aggregation operators J. L&azaroa;∗ , T. R*uckschlossov&ab , T. Calvoa a

Department of Computer Science, University of Alcala, E-28871 Alcala de Henares, Madrid, Spain b Department of Mathematics, STU Bratislava, Bratislava 813 68, Slovakia

Abstract When applying an information fusion method, usually the output results are required to be a scalar value. In multiple situations, the application of the fusion method to input values shifted by a constant should result in an output that is shifted likewise. In this paper, we present a study of the shift invariance property of aggregation operators. The relationship between shift invariant binary aggregation operators and 1-Lipschitz aggregation operators is shown, and a full description of shift invariant aggregation operators is given. c 2003 Elsevier B.V. All rights reserved.
Keywords: Aggregation operators; 1-Lipschitz property; Shift invariant property

1. Introduction Information aggregation is a crucial issue in the construction of many intelligent systems. The necessity of applying some aggregation process appears in neural networks, fuzzy logic controllers, vision systems, expert systems, etc. Aggregation problems are in general heterogeneous; here we consider aggregation problems, where from several input values one output value should be obtained. Among other considerations, it is possible to mention the aggregation process of in8nity real inputs (see e.g. [4,8,9,18]), of inputs from some ordinal scales [6,7], of complex inputs (probability distributions [17,20], fuzzy sets [21], etc.) but all these topics are out of our interest and therefore we will not deal with them. In most cases, the aggregation operators are de8ned on a pure axiomatic basis and are interpreted either as logical connectives (such as t-norms and t-conorms) or as averaging operators allowing a compensation e>ect (such as the arithmetic mean). However, it can be observed in the literature on this topic that the above classes of aggregation operators di>er from some others (see e.g. [5,19,22]). ∗

Corresponding author. E-mail addresses: [email protected] (J. L&azaro), [email protected] (T. R*uckschlossov&a), [email protected] (T. Calvo). c 2003 Elsevier B.V. All rights reserved. 0165-0114/$ - see front matter  doi:10.1016/j.fss.2003.10.031

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Whatever the case be, in many real situations shift of input data results in the same shift of the output value, see [2,12]. Since not every operator ful8ll this property, the study of the aggregation operators which allows this transformation (called shift invariant aggregation operators) is of great interest to solve its associated problems. The aim of this paper is to show the characterization of shift invariant aggregation operators. The paper is organized as follows. In Section 2, the characterization of shift invariant binary aggregation operators is given. In Section 3 we present some speci8c examples of this kind of operators. The relationship between shift invariance and 1-Lipschitz properties is studied in Section 4. After that, in Section 5, we establish a characterization theorem of shift invariant n-ary aggregation operators. Finally, we summarize the results in the conclusion section. 2. Characterization of shift invariant binary aggregation operators
A mapping A: n∈N [0; 1] n → [0; 1] is an aggregation operator (see [2,9,10,14]) if it holds the following properties: (Ag1) A(x1 ; : : : ; xn )6A(y1 ; : : : ; yn ) if (x1 ; : : : ; xn )6(y1 ; : : : ; yn ). (Ag2) A(x) = x for all x ∈ [0; 1]. (Ag3) A(0; : : : ; 0) = 0; A(1; : : : ; 1) = 1. Denition 1. An aggregation operator A: following equality holds:




n∈N [0; 1]

n

→ [0; 1] is said to be shift invariant when the

A(x1 + c; : : : ; xn + c) = c + A(x1 ; : : : ; xn ); for any n ∈ N, (x1 ; : : : ; xn ) ∈ [0; 1] n , c ∈ [0; 1] and (x1 + c; : : : ; xn + c) ∈ [0; 1] n . Observe that a shift invariant aggregation operator is idempotent. Denition 2. Given an aggregation operator B:
S B : n ∈ N [0; 1] n → [0; 1] as




n∈N [0; 1]

n

→ [0; 1] we can de8ne the operator

S B (x1 ; : : : ; xn ) = u + B(x1 − u; : : : ; xn − u); where u = min{x1 ; : : : ; xn }. Note that, for a given aggregation operator B, the operator S B need not be an aggregation operator, as we can see in the following example. Example 1. Taking the aggregation operator B de8ned as  0 if y¡ 12 ; B(x; y) = y if y¿ 12 :

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it is easy to see that its corresponding operator S B is not an aggregation operator because of S B is not non-decreasing. For this, it is enough to do the following computation: S B (0:2; 0:8) = 0:2 + B(0; 0:6) = 0:8; S B (0:4; 0:8) = 0:4 + B(0; 0:4) = 0:4: Moreover, the operator S B may have output values exceeding 1, as in the next example. Example 2. Suppose that B is the strongest binary aggregation operator, i.e., B(x; y) = 1 for all (x; y) di>erent from (0; 0). Then S B (x; x) = x and, if x = y, S B (x; y) = 1 + min{x; y}. Obviously, its range is RanS B = [0; 2[ and S B is not monotone. Due to Marichal [12], see also [2], we can characterize all shift invariant aggregation operators as follows:
Proposition 1. An aggregation operator A: n∈N [0; 1] n → [0; 1] is shift invariant if and only if there is some aggregation operator B such that A = S B , i.e., for all n ∈ N, (x1 ; : : : ; xn ) ∈ [0; 1] n A(x1 ; : : : ; xn ) = u + B(x1 − u; : : : ; xn − u);

(1)

where u = min{x1 ; : : : ; xn }. Evidently, A is shift invariant if and only if A = S A . In [2], the following open problem was given: characterize all aggregation operators B such that S B is an aggregation operator. Observe that if we solve this problem, we will also immediately know all shift invariant aggregation operators. Note also that (1) implies that S B is idempotent and S B (0; : : : ; 0) = 0, S B (1; : : : ; 1) = 1, S B (x) = x, for all x ∈ [0; 1], for all aggregation operators B. Hence only the monotonicity of S B should be checked (and then obviously RanS B = [0; 1], since if there are elements x1 ; : : : ; xn ∈ [0; 1] such that S B (x1 ; : : : ; xn )¿1 then S B cannot be monotone because of the fact that S B (1; : : : ; 1) = 1). In this section, we show the solution of the above stated open problem for binary aggregation operators A: [0; 1] 2 → [0; 1], i.e., we 8x in the rest of the section n = 2. Let B: [0; 1] 2 → [0; 1] be a binary aggregation operator, i.e., B(x; y)6B(x ; y ) whenever x6x , y6y and B(0; 0) = 0, B(1; 1) = 1. To check the monotonicity of the operator A = S B , we have to show that for all x; y ∈ [0; 1],  ∈ [0; 1 − x],  ∈ [0; 1 − y], it holds A(x; y)6A(x + ; y + ):

(2)

Suppose 8rst that x6y, i.e., following (1), A(x; y) = x + B(0; y − x). • If y + ¿x + , then A(x + ; y + ) = x +  + B(0; y +  − (x + )) and then (2) turns to x + B(0; y − x)6x +  + B(0; y +  − x − ):

(3)

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However, under all above requirements, (3) is true for all x6y,  ∈ [0; 1 − x],  ∈ [0; 1 − y] such that x + 6y +  if and only if B(0; v) − B(0; u)6v − u

for all u; v; v − u ∈ [0; 1]:

(4)

Observe that, since S B (0; y) = 0 + B(0; y), property (4) is equivalent to S B (0; v) − S B (0; u)6v − u

for all u; v; v − u ∈ [0; 1]:

• Further, if y + ¡x + , then A(x + ; y + ) = y +  + B(x +  − (y + ); 0), and then (2) turns to x + B(0; y − x)6y +  + B(x +  − y − ; 0), i.e., B(0; y − x)6(y − x) + ( + B(x +  − y − ; 0)); which is obviously ful8lled because of (4). Analogously, assuming now x¿y, we can show the equivalence of the monotonicity of A = S B with the condition B(v; 0) − B(u; 0)6v − u

for all u; v; v − u ∈ [0; 1];

(5)

which can be again presented as S B (v; 0) − S B (u; 0)6v − u

for all u; v; v − u ∈ [0; 1]:

We have just solved the given open problem for binary aggregation operators, since the nondecreasing property of S B can be obtained studying its value in the left and down edges of the [0; 1] 2 domain. Theorem 1. A binary aggregation operator A: [0; 1] 2 → [0; 1] is a shift invariant operator if and only if there is a binary aggregation operator B: [0; 1] 2 → [0; 1] such that for all u; v ∈ [0; 1], max{|B(0; u) − B(0; v)|; |B(u; 0) − B(v; 0)|}6|u − v|

(6)

and A = S B . Proof. SuKciency was already shown above. To see the necessity, suppose that (6) is violated. Without any loss of generality, we may suppose that B(0; u) − B(0; v)¿u − v for some u; v ∈ [0; 1]. But in this case we have S B (0; u) = B(0; u)¿u − v + B(0; v) = S B (u − v; u), i.e., the operator S B is not monotone. Remark 1. Note that because of construction (1), only the values of B on lower boundaries of the unit square are important in the case of binary operators, and then only (6) is responsible for the monotonicity of S B . On the reminder of the unit square, i.e., on ]0; 1] 2 , the values of the aggregation operator B are irrelevant.

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3. Specic examples For a given aggregation operator B, we now present some speci8c examples of shift invariant operators. (1) If B has 0 as annihilator, as t-norms, the aggregation operator A coincides with the minimum t-norm. This is obtained from the de8nition of A = S B and observing that B(x − min{x; y}; y − min{x; y}) has the value 0 in at least one of its arguments. (2) In a similar way, if B has 0 as neutral element, as t-conorms, we obtain the maximum t-conorm. In this case, B(x − min{x; y}; y − min{x; y}) = max{x; y} − min{x; y}, which means A(x; y) = min{x; y} + max{x; y} − min{x; y} = max{x; y}. (3) If B is a weighted mean, i.e., B(x; y) = ax + (1 − a)y, where a ∈ [0; 1], it is easy to show that A(x; y) = S B (x; y) = B(x; y). The reason for this result is that B was a shift invariant aggregation operator itself. (4) Finally, consider B = med a , i.e., B(x; y) = med(a; x; y). As the domain of B is DomB = [0; 1] 2 , if a60 we get S B (x; y) = B(x; y) = min{x; y}, and in the case a¿1 the operator S B (x; y) = B(x; y) = max{x; y} is obtained. For a ∈ ]0; 1[, if x¿y we have S B (x; y) = y + B(x − y; 0)  y + x − y if x − y¡a; = y+a if x − y¿a;  x if x − y¡a; = y + a if x − y¿a and when x¡y we get S B (x; y) = x + B(0; y − x)  x + y − x if y − x¡a; = x+a if y − x¿a;  y if y − x¡a; = x + a if y − x¿a: We can join these results in B

S (x; y) =



max{x; y} min{x; y} + a

if |x − y|¡a; if |x − y|¿a;

i.e., S B (x; y) = min{max{x; y}; min{x; y} + a}. We can also order the shift invariant binary aggregation operators with a “greater than or equal” relation observing that this partial order only depends on the usual partial order of unary functions over the same domain, i.e., f6g if and only if f(x)6g(x) for all x ∈ Dom, applied to the unary functions B(x; 0) and B(0; y). To prove this, suppose that there exist two shift invariant binary

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aggregation operators B0 and B1 such that B0 (x; 0)6B1 (x; 0) and B0 (0; y)6B1 (0; y). Then, it is clear that B0 (x; y) = u + B0 (x − u; y − u)6u + B1 (x − u; y − u) = B1 (x; y), where u = min{x; y}, for all x; y ∈ [0; 1], or equivalently, B0 6B1 . With this partial order, we can show that the smallest shift invariant binary aggregation operator is the minimum t-norm (as B(x; 0) = B(0; y) = 0 is the smallest unary function in [0; 1]) and that the greatest is the maximum t-conorm (i.e., the greater function ful8lling (5) in the 8rst case and that satisfying (4) in the second case) is the identity function. To show this, consider a function B1 ful8lling (5) such that id¡B1 , i.e., there exists at least one point x0 such that B1 (x0 ; 0)¿x0 . But in this case, we have the following contradiction: x0 ¡B1 (x0 ; 0) − B1 (0; 0)6x0 − 0 = x0 . 4. Shift invariance and 1-Lipschitz property 1-Lipschitz property of aggregation operators was recently discussed and studied in [2,3,11]. Recall that (6) means that the functions f; g: [0; 1] → [0; 1], f(u) = B(0; u), g(v) = B(v; 0) are 1-Lipschitz. Further, an operator C: [0; 1] 2 → [0; 1] is 1-Lipschitz if and only if for all u; v; u ; v ∈ [0; 1], |C(u; v) − C(u ; v )|6|u − u | + |v − v |: Evidently, each 1-Lipschitz aggregation operator B: [0; 1] 2 → [0; 1] ful8lls (6), i.e., A = S B is a shift invariant aggregation operator. On the other hand, let B be any aggregation operator ful8lling (6). We can de8ne a new operator ˜ [0; 1] 2 → [0; 1] by the next formula: B: ˜ y) = med(xy; f(y); f(1) + x(1 − f(1)); g(x); g(1) + y(1 − g(1))); B(x;

(7)

where med is the median operator and f; g were already de8ned above as f(y) = B(0; y), g(x) = B(x; 0). Then: ˜ 0) = med(0; 0; f(1)·(1−x)+x; g(x); g(1)) = g(x) = B(x; 0) as far as f(1)·(1−x)+x¿x¿g(x) (1) B(x; ˜ y) = f(y) = B(0; y). because of (6), and g(1)¿g(x) due to the monotonicity of B. Similarly, B(0; B˜ B Consequently, S = S . (2) B˜ is a 1-Lipschitz aggregation operator. Indeed, following [11], it is enough to check the partial derivatives of B˜ wherever they exists. 1-Lipschitz property is equivalent to the boundedness of these partial derivatives by 1. However, the result of median operator in (7) is necessarily one the arguments, and hence ˜ if they exist, always ful8ll: partial derivatives of B, ˜ y) @B(x; ∈ {y; 1 − f(1); g (x); 0}; @x ˜ y) @B(x; ∈ {x; f (y); 1 − g(1); 0}: @y The 1-Lipschitz property of f and g means that g (x)61 and f (y)61 in all points where these derivatives exist. Now, the 1-Lipschitz property of B˜ is evident.

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All these arguments can be summarized as follows: Theorem 2. A binary aggregation operator A: [0; 1] 2 → [0; 1] is a shift invariant aggregation operator if and only if there is a 1-Lipschitz aggregation operator B: [0; 1] 2 → [0; 1] such that A = S B . Moreover, each shift invariant binary aggregation operator A is necessarily 1-Lipschitz. Proof. We only have to prove the 1-Lipschitz property of A = S B when B is 1-Lipschitz. Again, this can be seen by means of partial derivatives. Supposing the existence of partial derivatives of A, we have A(x; y) = x +B(0; y −x) for x6y, and hence @A(x; y)=@x = 1 − (@B(0; y − x)=@x) ∈ [0; 1]. Similarly @A(x; y)=@y = (@B(0; y − x)=@y) ∈ [0; 1].

Corollary 1. Given an aggregation operator A, S A is a shift invariant aggregation operator if and only if S A is 1-Lipschitz. Note that not all 1-Lipschitz aggregation operators are also shift invariant aggregation operators, as can be easily shown considering B(x; y) = xy since it is not idempotent. Now, we give examples of some shift invariant operators as well as operators B˜ obtained by means of (7). ˜ y) = xy, and hence S B˜ (x; y) = Example 3. Taking f(x) = g(x) = 0 for all x ∈ [0; 1], we obtain B(x; min{x; y}. Example 4. When f(x) = g(x) = x for all x ∈ [0; 1], the corresponding operator B˜ results to be ˜ y) = S B˜ (x; y) = max{x; y}. B(x; Examples 3 and 4 present other ways to obtain the smallest and greatest shift invariant aggregation operators, showing that di>erent aggregation operators may lead to the same shift invariant aggregation operator. ˜ y) = y = S B˜ (x; y). Example 5. For f(x) = x, g(y) = 0 for all x; y ∈ [0; 1], we get B(x; Example 6. Considering f(x) = g(x) = x 2 =2, x ∈ [0; 1], it follows that:  2 x   if 2y¡x;     2 B˜ = y2  if 2x¡y;    2   xy otherwise ˜

and S B (x; y) = min{x; y} + (x − y) 2 =2. We now summarize all obtained results in the next corollary:

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Fig. 1. Representation of a shift invariant binary aggregation operator.

Corollary 2. A binary aggregation operator A: [0; 1] 2 → [0; 1] is a shift invariant operator if and only if there are two 1-Lipschitz functions f; g: [0; 1] → [0; 1], f(0) = g(0) = 0, so that  x + f(y − x) if x6y; A(x; y) = y + g(x − y) otherwise: Further, A is a symmetric shift invariant aggregation operator if and only if A(x; y) = min{x; y} + f(|x − y|); for some 1-Lipschitz function f: [0; 1] → [0; 1], f(0) = 0. Fig. 1 shows a shift invariant binary aggregation operator and its associated 1-Lipschitz functions. 5. Characterization of shift invariant n-ary aggregation operators As seen in Theorem 1, the binary operator A = S B is an aggregation operator if and only if max{|B(0; u) − B(0; v)|; |B(u; 0) − B(v; 0)|}6|u − v| for all u; v ∈ [0; 1], i.e., the di>erence of the outputs is bounded by the di>erence of the inputs when B is applied on its lower boundaries. Extending this idea to the n-dimensional case, given an aggregation operator B: [0; 1] n → [0; 1] we de8ne the operators Bi : [0; 1] n−1 → [0; 1] as Bi (x1 ; : : : ; xn−1 ) = B(x1 ; : : : ; 0i ; : : : ; xn−1 ), i.e., Bi is the operator B applied on its ith lower boundary. Obviously, Bi is an aggregation operator. For each i, we bound the di>erence of the outputs of Bi by the maximum of the di>erence of the inputs → → |B (− u ) − B (− v )|6 max {|u − v |} i

i

16r 6n−1

r

r

and combine all these inequalities in one → → u ) − Bi (− v )|6 max {|ur − vr |}: max |Bi (− 16i6n

16r 6n−1

We will show that this result is the condition needed to ensure that A = S B is a shift invariant aggregation operator.

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Theorem 3. An aggregation operator A: [0; 1] n → [0; 1] is a shift invariant operator if and only if → → there is an aggregation operator B: [0; 1] n → [0; 1] such that for all − u ;− v ∈ [0; 1] n−1 , → → u ) − Bi (− v )|6 max max |Bi (−

16i6n

16r 6n−1

{|ur − vr |}

(8)

and A = S B . Proof. We are going to prove that, given an aggregation operator B, the condition S B is non→ → → → decreasing if and only if (8) holds. This is equivalent to show that, given − x ;− y with − x 6− y, → → tx + Bi (− x − tx )6ty + Bj (− y − ty ) ⇔ (8)

(9)

→ → v ∈ [0; 1] n , Bi de8ned as above, and Bi (− v − tv ) = B(v1 − tv ; : : : ; vn − tv ), being tv = minr {vr } for all − i.e., we suppose that tv = vi (we use Bi in order to simplify the notation to show the argument equal to zero in B). Using this simpli8cation and without loss of generality, we can suppose tx = xi ; ty = yj . → To prove the suKciency of (9), we de8ne a new vector − z where zr = min{(xr − tx ); (yr − ty )}. With this, we have that → → → → z 6− y − ty ⇒ B(− z )6B(− y − ty ), (1) zr 6(yr − ty )∀r ⇒ − → − → (2) both − z and x − tx have zero in the same component, i.e., zi = xi − tx = 0, so they can be compared in the same boundary of B.

→ → x − tx )¿Bj (− y − ty ) since the other situation is As x6y, we only have to check the case Bi (− trivial. Then, we have → → → → Bi (− x − tx ) − Bj (− y − ty ) 6 Bi (− x − tx ) − Bi (− z ) 6 max |(xr − tx ) − zr | r

= |xs − tx − zs | − for a certain 16s6n. Attending at the de8nition of → z we have two possibilities: (1) If zs = xs − tx , then we have |xs − tx − zs | = 06ty − tx , (2) If zs = ys − ty , then |(xs − tx ) − (ys − ty )| = (xs − tx ) − (ys − ty ), since zs is the minimum of those values. This result is equivalent to (ty − tx ) − (ys − xs ), which is smaller than or equal to ty − tx and we have ended. → → x − tx )6ty + Bj (− y − The necessity is proven in the following way: as our hypothesis is tx + Bi (− → − → − ty ) for all x 6 y , we may suppose without loss of generality the situation tx = xi , ty = yi , i.e., → → → → → → Bi (− x − tx ) − Bi (− y − ty )6ty − tx , and also − u ¿− v (when − u ¡− v the proof is analogous). With this, → − → − → − → − |Bi ( u ) − Bi ( v )| = Bi ( u ) − Bi ( v ). → → → → → → To use the hypothesis we need − x and − y such that − x − x =− u and − y − y =− v . Denote i

i

$ = maxr {ur − vr }, and taking xr = ur ; yr = min{1; vr + $} we obtain xr 6yr ; ∀r ∈ {1 : : : n}; in this situation, taking into account that tx = 0, ty = $, we can follow the proof through the next chain of inequalities: → → → → Bi (− u ) − Bi (− v ) 6 Bi (− x ) − Bi (− y − $) 6 $ − 0 = max{ur − vr } = max{|ur − vr |}: r

r

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→ → As this idea is true for all i, we obtain maxi {|Bi (− u ) − Bi (− v )|}6 maxr {|ur − vr |} which is what we needed to prove. Remark 2. Note that (1) The binary case is simply a special case, since (8) in the 2-dimensional case is just the 1Lipschitz property of the boundary functions f and g. (2) For n-ary operators, the 1-Lipschitz property of B on its lower boundaries is not more suKcient (but it is still necessary) to ensure the monotonicity of the corresponding shift invariant operator S B . Take, e.g., the bounded sum B(x; y; z) = min{1; x + y + z} which is evidently a 1-Lipschitz aggregation operator; its corresponding ternary shift invariant operator S B : [0; 1]3 → R is given by S B (x; y; z) = min{1 + min{x; y; z}; x + y + z − 2 min{x; y; z}} which is clearly neither monotone nor bounded from above by the maximum operator. Its range is [0; 1:5] and S B (0:5; 1; 1) = 1:5. Remark 3. Observe that property (8) of an aggregation operator B is closely related to the kernel property recently introduced in [3,11]. Indeed, an n-ary aggregation operator B possesses the kernel → → → → property whenever for arbitrary − x ;− y from [0; 1] n , the inequality |B(− x ) − B(− y )|6 max16i6n {|xi − yi |} holds. Property (8) deals with one coordinate to be 8xed in zero, and therefore it can be called zero-kernel property. Evidently, each kernel aggregation operator has also zero-kernel property but not vice versa. → Example 7. For n¿2, let B be the standard product. Evidently, if at least one of inputs in vector − x n → − from [0; 1] is zero, then also B( x ) = 0, and thus B has zero-kernel property (and the corresponding shift invariant aggregation operator S B = min). However, B is not a kernel aggregation operator. Remark 4. As observed in [15], an n-ary aggregation operator B has the kernel property if and only if @B=@x1 + @B=@x2 + · · · + @B=@xn 61 in all points where all partial derivatives exist. Similarly, we can characterize the zero-kernel property, i.e., property (8). An n-ary aggregation operator B ful8lls the property (8) if and only if for all i ∈ {1; : : : ; n}, @Bi =@x1 + @Bi =@x2 + · · · + @Bi =@xn 61 in all points where all partial derivatives taken into account exist. Note that under requirement of di>erentiability of all Bi this fact was already shown in [1, Proposition 2]. However, our results are more general as no di>erentiability is supposed for B. Example 8. For n = 3, let B: [0; 1]3 → [0; 1] be given by B(x; y; z) = (xp + yp + z p )=3 for some positive p. Evidently, B is aggregation operator for all positive p; however, it is a kernel aggregation operator only for p = 1 (i.e., if B is an arithmetic mean), and it is a zero-kernel aggregation operator, i.e., B ful8lls (8), if and only if p ∈ [1; 32 ]. Because of the symmetry of B, also the corresponding shift invariant aggregation operator A = S B , p ∈ [1; 32 ], is symmetric and for 06x6y6z61 it is given by A(x; y; z) = x + ((y − x)p + (z − x)p )=3.

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6. Conclusions We have characterized all aggregation operators leading to shift invariant binary aggregation operators and we have shown the relationship between shift invariant aggregation operators and 1-Lipschitz aggregation operators. Turning our attention to the case of n-ary shift invariant aggregation operators with n¿2, evidently for associative aggregation operators it is enough to check their binary form. However, because of Theorem 2, such associative aggregation operators are continuous, and then, taking into account results from [13], the only associative shift invariant aggregation operators are minimum, maximum and projection operators PF (projection to the 8rst coordinate) and PL (projection to the last coordinate). Evidently, for an arbitrary binary shift invariant aggregation operator A, also its n-ary extensions A∗ (x1 ; : : : ; xn ) = A(A(: : : A(A(x1 ; x2 ); x3 ); : : :); xn ) and A∗ (x1 ; : : : ; xn ) = A(x1 ; A(x2 ; A(: : : ; A(xn−1 ; xn )) : : :)) are shift invariant. However, not all n-ary shift invariant aggregation operators can be obtained in one of the above forms; the standard arithmetic mean is an example of this situation. Theorem 3 present a characterization for all shift invariant aggregation operators. Observe that while some classes of aggregation operators are shift invariant (and thus determined by values on boundaries), e.g., Choquet integral based aggregation operators and their subclasses (weighted means and OWA operators), in other distinguished classes of aggregation operators the shift invariance is a rather restrictive requirement. So, for example, in the class of t-norms (t-conorms), the only shift invariant operator is the minimum (maximum) aggregation operator, while in the class of uninorms (with neutral element e from ]0; 1[) there is no shift invariant operator. Concerning the quasi-arithmetic means (similar result holds for weighted quasi-arithmetic means), only the quasi-arithmetic mean generated by means of f(x) = e$x and the usual arithmetic mean are shift invariant, as can be seen in [16]. Acknowledgements This work has been partially supported by the Grant VEGA 1/8331/01, and the projects TIC20001368-C03-01 and BFN2000-1114. References [1] J. Aczel, D. Gronau, J. Schwaiger, Increasing solutions of the homogeneity equation and similar equations, J. Math. Anal. Appl. 182 (1994) 436–464. [2] T. Calvo, A. Koles&arov&a, M. Komorn&Rkov&a, R. Mesiar, Aggregation operators: basic concepts, issues and properties, in: T. Calvo, G. Mayor, R. Mesiar (Eds.), Aggregation Operators. New Trends and Applications, Physica-Verlag, Heidelberg, 2002, pp. 3–105. [3] T. Calvo, R. Mesiar, Stability of aggregation operators, Proc. EusSat’2001, Leicester, 2001, pp. 475 – 478. [4] D. Denneberg, Non-additive Measure and Integral, Kluwer Academic Publishers, Dordrecht, 1994. [5] J. Fodor, T. Calvo, Aggregation functions de8ned by t-norms and t-conorms, in: B. Bouchon-Meunier (Ed.), Aggregation and Fusion of Imperfect Information, Physica-Verlag, Heidelberg, 1998, pp. 36–48. [6] L. Godo, V. Torra, Extending Choquet integrals for aggregation of ordinal values, Proc. IPMU’2000, Madrid, 2000, pp. 410 – 417.

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