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Aggregation functions on bounded partially ordered sets and their classification
Fuzzy Sets and Systems 175 (2011) 48 – 56 www.elsevier.com/locate/fss
Aggregation functions on bounded partially ordered sets and their classification Magda Komorníková, Radko Mesiar∗ Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 813 68 Bratislava, Slovakia Received 30 September 2010; received in revised form 19 January 2011; accepted 24 January 2011 Available online 4 February 2011
Abstract Several classes of aggregation functions on bounded partially ordered sets are introduced and discussed. Considering their conjunctive and/or disjunctive attitude, some classification approaches are proposed, including several types of averaging aggregation functions. Aggregation functions on special cases of lattices and chains are discussed, too. © 2011 Elsevier B.V. All rights reserved. Keywords: Averaging aggregation; Conjunctive aggregation; Disjunctive aggregation; Tolerant aggregation
1. Introduction Classification of objects considered in any domain is an important tool for the transparentness, better understanding of the considered domain, but also for construction and application of discussed objects. In the area of aggregation functions acting on real intervals, such a classification was proposed by Dubois and Prade at AGOP’2001 conference in Oviedo, see Dubois and Prade [12]. In Dubois–Prade approach, conjunctive, disjunctive, averaging and remaining aggregation functions were considered, defined by their relationship to Min and Max functions. The class C of all (n-ary) conjunctive functions (acting on a real interval [a, b]) is characterized by the inequality A ≤ Min, while the inequality A ≥ Max is characteristic for the disjunctive aggregation functions from the class D. Concerning the averaging aggregation functions, they should satisfy Min ≤ A ≤ Max. To exclude the trivial overlapping of conjunctive and averaging (disjunctive and averaging) aggregation functions, the class P of pure averaging aggregation functions consists of all averaging aggregation functions except Min and Max. Denoting A the class of all aggregation functions (n-ary, on real interval [a, b]), R = A\(C ∪ D ∪ P) consists of all remaining aggregation functions, which are neither conjunctive, nor disjunctive nor averaging. Thus this standard classification (C, D, P, R) forms a partition of the class A. As prototypical examples of members of these special classes, consider [0, 1] interval and the product (x1 , . . . , xn ) = x1 · x2 · . . . · xn (triangular norm, conjunctive aggregation function), bounded sum SL (x1 , . . . , xn ) = min{1, x1 + · · · + xn } (triangular conorm, disjunctive aggregation function), arithmetic mean M(x1 , . . . , xn ) = (1/n)(x1 +· · ·+xn ) (mean, pure averaging aggregation function) and the 3--operator E(x1 , . . . , xn ) = (x1 , . . . , xn )/((1 − x1 , . . . , 1 − xn ) + (x1 , . . . , xn )) (uninorm, remaining aggregation function). In several domains we need to classify the aggregation of more complex ∗ Corresponding author.
E-mail addresses: [email protected] (M. Komorníková), [email protected] (R. Mesiar). 0165-0114/$ - see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2011.01.015
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objects, which rarely form a chain, but they can be considered as elements of some (bounded) lattice or poset (we will use this abbreviation for a partially ordered set throughout this paper), see e.g., [5,6]. This is, for example, the case of aggregation of fuzzy sets (intersection, union), of distribution functions (convolution), etc. However, such a classification of aggregation functions on posets is missing in the literature so far. Obviously, we cannot repeat the approach of Dubois and Prade once Min and Max are not defined. The aim of this paper is to fill the above-mentioned gap. Note that some first steps towards the classification of aggregation functions on posets can be found in our paper Mesiar and Komorníková [17]. This paper is organized as follows. In the next chapter, aggregation functions on posets are recalled and exemplified. In Section 3, conjunctive and disjunctive classifications of aggregation functions on a bounded poset are proposed and discussed. In Section 4, some general classifications are introduced. Section 5 is devoted to the classification of aggregation functions on a bounded lattice (chain). In Section 6, aggregation functions on product posets/lattices are discussed. Finally, some concluding remarks are added. 2. Aggregation functions on bounded posets Aggregation functions on real intervals are special functions characterized by the monotonicity and the boundary conditions. Recall that an aggregation function A : [0, 1]n → [0, 1] (an extended aggregation function A : n∈N [0, 1]n → [0, 1]) is supposed to be increasing, i.e., A(x) ≤ A(y) whenever x ≤ y, and A(0, . . . , 0) = 0, A(1, . . . , 1) = 1. The framework of aggregation functions [0, 1] can be modified into any closed interval [a, b] ⊆ [−∞, ∞], or even to any interval I ⊆ [−∞, ∞] (with modified boundary conditions, see [13]). Obviously, aggregation functions can be introduced to act on any (partially) ordered structure with bounds. Definition 1. Let (P, ≤, 0, 1) be a bounded poset (partially ordered set). Let n ∈ N be fixed. A mapping A : P n → P is called an (n-ary) aggregation function on P whenever it is increasing, A(x) ≤ A(y) whenever x ≤ y (i.e., x1 ≤ y1 , . . . , xn ≤ yn )
(1)
and it satisfies boundary conditions A(0, . . . , 0) = 0, A(1, . . . , 1) = 1. (2) A mapping B : n∈N P n → P is called an extended aggregation function on P whenever B|P n is an n-ary aggregation function on P for any n ∈ N . For a more detailed discussion of aggregation on posets/lattices we recommend Demirci [6]. Special types of aggregation functions on posets, especially triangular norms and conorms, are studied, for example, in [2,4,7–11,14,15,18–20]. Note that if P = [0, 1] is equipped with the standard ordering of reals, Definition 1 turns into the classical definition of an aggregation function [1,3,13]. Example 1. Consider, for example, the diamond lattice D = {0, a, b, 1} visualized in Fig. 1. Then a mapping A : D → D is a unary aggregation function on D if and only if A(0) = 0 and A(1) = 1 (i.e., the values A(a) and A(b) can be chosen arbitrarily). Define mapping B : D 2 → D as follows (for x, y ∈ D): B(x, y) = 0 if 0 ∈ {x, y}, B(x, y) = 1 if 1 ∈ {x, y} and 0 ∈ / {x, y}, B(x, x) = x if x ∈ D, B(a, b) = B(b, a). Then B is well defined and it is a binary aggregation function on D.
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1
a
b
0 Fig. 1. Hasse diagram of the diamond lattice D.
Finally, we introduce a ternary aggregation function C : D 3 → D as follows: C(x, y, z) = u whenever the triple {x, y, z} contains at least two times u; C(x, y, z) = 0 whenever {x, y, z} = {0, a, b} and C(x, y, z) = 1 whenever {x, y, z} = {1, a, b}. Observe that all algebraic properties of aggregation functions acting on [0, 1] as introduced in Grabisch et al. [13], see also Beliakov et al. [1], Calvo [3], can be straightforwardly introduced also for aggregation functions acting on bounded posets. This is, for example, the case of: • • • • • • • • •
symmetry; associativity; annihilator (zero element); weak annihilator; neutral element (unit element); bisymmetry; idempotency; internality (i.e., A(x1 , . . . , xn ) ∈ {x1 , . . . , xn }); decomposability.
Recall that some of these properties concern the extended aggregation functions (decomposability, general bisymmetry, general associativity, etc.), while some of them are meaningful for aggregation functions with a fixed arity only. Note that the binary aggregation function B : D 2 → D acting on the diamond lattice D and introduced in Example 1 is associative, symmetric (and thus also bisymmetric), idempotent, 0 is its annihilator, and if B(a, b) ∈ {a, b} then B is also internal. On the other hand, the introduced ternary aggregation functions C : D 3 → D is symmetric, idempotent and internal. 3. Conjunctive and disjunctive classification Let n ∈ N and a poset (P, ≤, 0, 1) be fixed. For a given aggregation function A : P n → P and x = (x1 , . . . , xn ) ∈ P n we denote: g A (x) = card{i|xi ≥ A(x)} and s A (x) = card{i|xi ≤ A(x)}.
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Let A = {A : P n → P|A is an aggregation function}, and define two mappings , : A → {0, 1, . . . , n} by (A) = inf{g A (x)|x ∈ P n }, (A) = inf{s A (x)|x ∈ P n }. Observe that sup{g A (x)|x ∈ P n } = sup{s A (x)|x ∈ P n } = n due to the boundary conditions (2) forcing g A ((0, . . . , 0)) = g A ((1, . . . , 1)) = s A ((0, . . . , 0)) = s A ((1, . . . , 1)) = n, independently of A. Both functions and allow to introduce a classification of aggregation functions from A. Proposition 1. C = {C0 , C1 , . . . , Cn } and D = {D0 , D1 , . . . , Dn } given by Ci = −1 ({i}) and Di = −1 ({i}), i = 0, 1, . . . , n are partitions of A. Remark 1. (i) An aggregation function A : P n → P belongs to the class Cn (Dn ) if and only if for all x = (x1 , . . . , xn ) ∈ P n it holds xi ≥ A(x)(xi ≤ A(x)) for each i ∈ {1, . . . , n}. In the case of standard aggregation functions on [0, 1] (or any real interval) this means that A is conjunctive (disjunctive). Therefore, aggregation functions from the classCn will n−1 Ci be called strongly conjunctive, and we define Cs = Cn . Moreover, the aggregation functions from Cw = i=1 will be called weakly conjunctive. Finally, Ca = C0 is the class of aggregation functions admitting the existence of x ∈ P n such that for each i ∈ {1, . . . , n}, either xi < A(x) or xi ⊥ A(x) (xi is incomparable to A(x)); these aggregation functions will be called anticonjunctive. Similarly, the classes Ds , Dw and Da of strongly disjunctive, weakly disjunctive and antidisjunctive aggregation functions can be introduced dually. (ii) The concepts of a conjunctive classification and of a disjunctive classification are dual in the following sense: for a fixed poset (P, ≤, 0, 1) one can define a dual poset simply reversing the order, i.e., (Q, , 0 Q , 1 Q ) is given by Q = P, 0 Q = 1, 1 Q = 0, and for x, y ∈ Q, x y if and only if x ≥ y. Evidently, each aggregation function A : P n → P can be considered as an aggregation function on Q, too. However, then for all x ∈ P n = Q n Q Q it holds g AP (x) = s A (x), s AP (x) = g A (x), and thus A ∈ CiP (DiP ) in the case of poset (P, ≤, 0, 1) if and only if Q Q A ∈ Di (Ci ) in the case of poset (Q, , 0 Q , 1 Q ). (iii) Recently, Marichal [16] has introduced the concept of k-tolerant and k-intolerant aggregation functions on real intervals. Our approach is a modification and an extension of Marichal’s one to the case of posets. n Indeed, if Di . P = [a, b] is a real interval, and k ∈ {1, . . . , n}, then k-tolerant aggregation functions form the class i=n−k+1 In particular, 1-tolerant aggregation functions are exactly the strongly disjunctive aggregation functions. Similarly, n Ci , and 1-intolerant aggregation functions are just the k-intolerant aggregation functions form the class i=n−k+1 strongly conjunctive aggregation functions. (iv) Order norms on posets introduced by De Cooman and Kerre [5] are typical examples of strongly conjunctive and strongly disjunctive aggregation functions. For unary aggregation functions, strongly disjunctive and strongly conjunctive aggregation functions overlap, Ds ∩ Cs = {i P }, where i P is the identity on P. To avoid this undesirable effect, we will deal with n > 1 since now. Moreover, we will suppose also car d P ≥ n to avoid the necessity of the repetition of arguments for any x ∈ P n . In the next considerations we will not repeat these constrains, though we will deal with them. Proposition 2. Let A : P n → P be a fixed aggregation function. Then (A) + (A) ≤ n + 1, and if P is not a chain, then (A) = n implies (A) = 0, and (A) = n implies (A) = 0. Proof. Choose an x ∈ P n such that the set {x1 , . . . , xn } has cardinality n, i.e., all coordinate values x1 , . . . , xn are different. Denote e A (x) = card{i|xi = A(x)} and in A (x) = card{i|xi ⊥ A(x)}. Then e A (x) ∈ {0, 1} and g A (x) + s A (x) + in A (x) = n + e A (x). Consequently, (A) + (A) ≤ n + 1. If (A) = 0 or (A) = 0, then necessarily e A (x) = 0 for some x ∈ P n , and evidently (A) + (A) ≤ n.
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Suppose that P is not a chain, i.e., there are a, b ∈ P such that a ⊥ b, and that (A) = n. Then g A ((a, b, . . . , b)) = n, i.e., A((a, b, . . . , b)) ≤ a and A((a, b, . . . , b)) ≤ b. Consequently A((a, b, . . . , b)) = c cannot satisfy neither c ≥ a nor c ≥ b, i.e., s A ((a, b, . . . , b)) = 0 = (A). The rest of the proof runs similarly. The last result has an important consequence. Corollary 1. A strongly conjunctive (disjunctive) aggregation function A is necessarily antidisjunctive (anticonjunctive). Observe that the class A of all n-ary aggregation functions on a fixed poset P is itself a bounded partially ordered set with inherited ordering ≤, A1 ≤ A2 whenever A1 (x) ≤ A2 (x) for all x ∈ P n , and with the top element A∗ : P n → P, 0 if x = (0, . . . , 0), ∗ A (x) = 1 else, and the bottom element A∗ : P n → P, 1 if x = (1, . . . , 1), A∗ (x) = 0 else. Evidently, A∗ ∈ Ds is strongly disjunctive and A∗ ∈ Cs is strongly conjunctive. Moreover, the mappings , : A → {0, 1, . . . , n} are monotone. Indeed, is decreasing, A1 ≤ A2 implies (A1 ) ≥ (A2 ), while is increasing, A1 ≤ A2 implies (A1 ) ≤ (A2 ). Example 2. Recall aggregation functions B and C from Example 1. B is both weakly conjunctive and weakly disjunctive, (B) = (B) = 1. Similarly, (C) = (C) = 1. 4. General classifications Conjunctive and disjunctive classifications (and also their refinements) may lead to an unacceptably large number of possible classes. As observed in several papers, human beings have no problem with 3–7 classes to be distinguished. The already mentioned Dubois–Prade classification consists of four classes. Following their approach, and fixing the classes of strongly conjunctive and strongly disjunctive aggregation functions (due to Corollary 1, they are disjoint) we have to consider the counterpart of the class of (purely) averaging aggregation functions. We can either identify them with the weakly conjunctive aggregation functions, see Remark 1(i), and to introduce the coarsened conjunctive classification CC = {Cs , Cw , Ds , RC }, where the remaining aggregation functions form the class RC = C0 \Ds of all anticonjunctive aggregation functions which are not strongly disjunctive. Or we can introduce a coarsened disjunctive classification DC = {Cs , Dw , Ds , R D } with R D = D0 \Cs . Nevertheless, not to support neither the conjunctive nor the disjunctive approach, we propose to consider the two next concepts of averaging aggregation functions. Definition 2. Let an aggregation function A : P n → P be given. Then 1. A is called strongly averaging whenever it is both weakly conjunctive and weakly disjunctive, A ∈ As = Cw ∩ Dw ; 2. A is called weakly averaging whenever it is either weakly conjunctive or weakly disjunctive, A ∈ Aw = Cw ∪ Dw . Based on Definition 2, we introduce two general classifications of aggregation functions on posets visualized in Fig. 2. Definition 3. 1. The partition {Cs , Aw , Ds , C0 ∩ D0 } will be called a weak classification of the class A of all n-ary aggregation functions on a fixed poset (P, ≤, 0, 1). 2. The partition {Cs , As , Ds , Cw \Dw , Dw \Cw , C0 ∩ D0 } will be called a strong classification of the class A of all n-ary aggregation functions on a fixed poset (P, ≤, 0, 1).
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Fig. 2. Strong classification on the class A described by (a) class symbols, (b) values of (( A), ( A)), here r, s are arbitrary values from {1, . . . , n −1}.
Observe that Dw \As = Dw \Cw , Cw \As = Cw \Dw and {As , Cw \As , Dw \As } is a partition of the class Aw of all weakly averaging aggregation functions (i.e., the strong classification is a refinement of the weak classification), and thus Fig. 2 visualizing the strong classification contains implicitly the visualization of the weak classification, too. 5. Classification of aggregation functions on bounded chains and bounded lattices Suppose the case (P, ≤, 0, 1) where ≤ is a total (linear) order, i.e., P is a bounded chain. Then the inequality A(x) ≥ xi for some i ∈ {1, . . . , n} is equivalent to A(x) ≥ Min(x). Similarly, A(x) ≤ Max(x) and A(x) ≤ xi are equivalent. Thus the strong averaging aggregation function A is characterized by Min ≤ A ≤ Max inequalities, coinciding with Dubois–Prade approach to averaging aggregation functions. Moreover, (A) = 0 ((A) = 0) is equivalent to (A) = n((A) = n) in this case. Summarizing, we have the next result verifying that our approach extends the original Dubois–Prade classifications from [12]. Proposition 3. Let (P, ≤, 0, 1) be a bounded chain. Then the strong classification and the weak classification of (n-ary) aggregation functions on P coincide and they both correspond to Dubois–Prade classification (C, D, P, R) introduced in [12]. Considering a bounded lattice, the situation is completely different. First of all, we can introduce the class Al of all lattice-averaging aggregation functions paraphrasing the approach of Dubois and Prade, i.e., A ∈ Al if and only if Min ≤ A ≤ Max, and A ∈ / {Min, Max}. Proposition 4. Let A : L n → L be an aggregation function on a bounded lattice (L , ≤, 0, 1). Then: (i) if A is strongly averaging then A is also lattice-averaging (i.e., As ⊆ Al ); (ii) the fact that A is weakly averaging brings no information about the possible lattice-averagingness of A, and vice versa (i.e., the classes Aw and Al are in a general position, in general). Proof. (i) Also in the lattice case, if xi ≤ A(x) for some i ∈ {1, . . . , n}, then A(x) ≥ Min(x) (xi ≥ A(x) implies A(x) ≤ Max(x)). Thus each strongly averaging aggregation function A is necessarily also lattice-averaging. (ii) Consider the lattice (L = [0, 1]3 , ≤, (0, 0, 0), (1, 1, 1)) equipped with the standard partial order ≤. Define two aggregation functions A, B : L 3 → L by x1 + y1 + z 1 x2 + y2 + z 2 x3 + y3 + z 3 A((x1 , x2 , x3 ), (y1 , y2 , y3 ), (z 1 , z 2 , z 3 )) = , , 3 3 3 and B((x1 , x2 , x3 ), (y1 , y2 , y3 ), (z 1 , z 2 , z 3 )) =
x1 + y1 + z 1 , x2 y2 z 2 , x3 y3 z 3 . 3
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Then A((0, 21 , 1), ( 21 , 1, 0), (1, 0, 21 )) = ( 21 , 21 , 21 ) and thus (A) = (A) = 0, A is not weakly averaging, A ∈ / Aw . On the other hand, it is evident that A is lattice-averaging. Concerning B, B(( 41 , 24 , 43 ), ( 24 , 43 , 41 ), ( 43 , 41 , 24 )) = 3 3 , 32 ), while Min(( 41 , 24 , 43 ), ( 24 , 43 , 41 ), ( 43 , 41 , 24 )) = ( 41 , 41 , 41 ). Hence B ∈ / Al and (B) ≤ 2. Moreover, at ( 21 , 32 least one element x1 , y1 or z 1 is greater or equal to (x1 + y1 + z 1 )/3, while all elements x2 , y2 , z 2 (x3 , y3 , z 3 ) are greater than or equal to x2 y2 z 2 (x3 y3 z 3 ), proving that (B) > 0. Consequently, B ∈ Aw . Due to the last proposition, in the case of bounded lattices which are not chains we have three different general classifications of aggregation functions: • weak classification {Cs , Ds , Aw , C0 ∩ D0 }; • strong classification {Cs , Ds , As , Cw \As , Dw \As , C0 ∩ D0 }; • lattice classification {Cs , Ds , Al , R}, where R = A\(Cs ∪ Ds ∪ Al ). Note that each strongly conjunctive aggregation function A satisfies A(x) ≤ xi (for each i and each x) and thus A ≤ Min, and thus Cs ∩ Al = ∅. Similarly, Ds ∩ Al = ∅. 6. Aggregation functions on product posets/lattices For any non-empty index set I, consider a system ((Pi , ≤i , 0i , 1i ))i∈I of bounded posets (lattices). The product poset (lattice) (P, ≤, 0, 1) is then defined as P = ×i∈I Pi (Cartesian product), 0 = (0i )i∈I , 1 = (1i )i∈I , and for x = (xi )i∈I , y = (yi )i∈I ∈ P we have x ≤ y whenever xi ≤i yi for all i ∈ I , see e.g., Birkhoff [2]. Aggregation functions on product lattices were already discussed in [4,14]. A special class of decomposable aggregation functions on product posets/lattices can be defined by means of aggregation functions ( Ai )i∈I acting on component posets/lattices Pi , i ∈ I . Indeed, then A : P n → P is given by (1)
(n)
(1)
(n)
A((xi )i∈I , . . . , (xi )i∈I ) = (Ai (xi , . . . , xi ))i∈I . Several properties of single aggregation functions Ai are inherited by the global aggregation function A. For example, if each Ai is a triangular norm on Pi , i ∈ I , then A is a triangular norm on P, see De Baets and Mesiar [4]. However, not all triangular norms on P are decomposable [14]. The next result is straightforward. Proposition 5. Let A : P n → P be a decomposable aggregation function, A = (Ai )i∈I , acting on a product poset P = ×i∈I Pi . Then A is strongly conjunctive (strongly disjunctive) if and only if each Ai : Pin → Pi , i ∈ I is strongly conjunctive (strongly disjunctive). Proposition 5 cannot be extended for strongly (weakly) averaging aggregation functions. However, if P is a product lattice, then the l-averaging property is preserved by decomposable aggregation functions. Example 3. Consider the lattice L = [0, 1]3 as in the proof of Proposition 4, and define a decomposable aggregation function C : L 3 → L by C((x1 , x2 , x3 ), (y1 , y2 , y3 ), (z 1 , z 2 , z 3 ))=(Med(x1 , y1 , z 1 ), Med(x2 , y2 , z 2 ), Med(x3 , y3 , z 3 )). Evidently, median function Med : [0, 1]3 → [0, 1] is strongly averaging. On the other side, consider x ∈ L 3 , x = ((0, 0.3, 0.6), (0.3, 0.6, 0), (0.6, 0, 0.3)). Then C(x) = (0.3, 0.3, 0.3), and thus gC (x) = sC (x) = 0. Therefore, (C) = (C) = 0 and hence C is not strongly averaging, neither weakly averaging. However, C is l-averaging. Example 3, as well as the aggregation function A introduced in the proof of Proposition 4, shows also that the weak conjunctivity (the weak disjunctivity) of decomposable aggregation functions on product posets/lattices is not guaranteed by the weak conjunctivity (the weak disjunctivity) of all component aggregation functions. The strongest result in this spirit is the next one. Proposition 6. Let A : P n → P be a decomposable aggregation function, A = (Ai )i∈I , acting on a product poset P = ×i∈I Pi . If there is j ∈ I such that A j : P jn → P j is weakly conjunctive (weakly disjunctive), and for each i ∈ I, i j, Ai : Pin → Pi is strongly conjunctive (strongly disjunctive), then A is weakly conjunctive (weakly disjunctive).
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Proof. We will prove the conjunctive case only, as the disjunctive runs similarly. For any x = (x1 , . . . , xn ) ∈ P n , xk = (k) (1) (n) (1) (n) (xi )i∈I , k = 1, . . . , n, it holds g A (x) ≥ n − i∈I (n − g Ai ((xi , . . . , xi ))) = g A j ((x j , . . . , x j )), and hence (A) ≥ (A j ). On the other side, it is obvious that (A) ≤ inf{(Ai )|i ∈ I } = (A j ), showing the equality (A) = (A j ). As far as A j is weakly conjunctive, so is A. To illustrate Proposition 6, consider the aggregation function B introduced in the proof of Proposition 4. Decomposable aggregation function B is linked to aggregation functions B1 , B2 , B3 : [0, 1]3 → [0, 1], B1 = M, B2 = B3 = , i.e., B2 and B3 are strongly conjunctive and B1 is weakly conjunctive. Therefore, B is weakly conjunctive (hence also weakly averaging). 7. Concluding remarks We have introduced and discussed some possible classifications of aggregation functions on bounded posets (lattices), extending and generalizing the ideas of Dubois and Prade [12] and Marichal [16] proposed for aggregation functions acting on real intervals. Observe that the distributive lattices admit an introduction of order statistics. However, Marichal’s approach extended from the real intervals (chains) to these lattices does not cover our approach (compare Example 3, for example). Note that there are some sufficient conditions ensuring the belongingness of considered aggregation functions into a relevant class. So, for example, let A : P n → P be an aggregation function with neutral element 1 (0). Then necessarily A is strongly conjunctive (strongly disjunctive). Internality of A (i.e., A(x) ∈ {x1 , . . . , xn } for all x ∈ P n ) forces (A) ≥ 1 and (A) ≥ 1, thus if P is not a chain, A is necessarily strongly averaging. Observe also that any kind of averaging we have introduced ensures the idempotency of A. It is well known that if P is a chain then also the reverse claim is valid (note that then all introduced concepts of averaging coincide). In general, if P is a lattice, the idempotency of an aggregation function different from Min and Max is equivalent to the lattice-averaging concept 1
c
d
a
b
0 Fig. 3. Hasse diagram of a poset P.
Table 1 The aggregation function A. A
0
a
b
c
d
1
0 a b c d 1
0 0 0 0 0 0
0 a 0 a a a
0 0 b b b b
0 a b c a c
0 a b a d d
0 a b c d 1
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Table 2 The aggregation function B. B
0
a
b
c
d
1
0 a b c d 1
0 0 0 0 0 0
0 a 0 a a a
0 0 b b b b
0 a b c b c
0 a b b d d
0 a b c d 1
(but neither to the strong averaging nor to the weak averaging). Moreover, if P is a lattice, then the only idempotent strongly conjunctive (strongly disjunctive) aggregation function is Min (Max). This is not true on a general poset, where we can have several strongly conjunctive (strongly disjunctive) idempotent aggregation functions. As an example, consider a poset P visualized in Fig. 3, and aggregation functions A, B : P 2 → P described in Tables 1 and 2. Acknowledgments The research summarized in this paper was supported by the Grants APVV-0012-07, VEGA 1/0080/10 and LPP-0111-09. References [1] G. Beliakov, A. Pradera, T. Calvo, Aggregation Functions: A Guide for Practitioners, Springer-Verlag, Berlin, Heidelberg, 2007. [2] G. Birkhoff, Lattice Theory, American Mathematical Society Colloquium Publications, third ed., vol. XXV, American Mathematical Society, Providence, RI, 1967. [3] 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: Properties, Classes and Construction Methods, Aggregation Operators: New Trends and Applications, Physica-Verlag, Heidelberg, New York, 2002, pp. 1–104. [4] B. De Baets, R. Mesiar, Triangular norms on product lattices, Fuzzy Sets and Systems 104 (1999) 61–76. [5] G. De Cooman, E.E. Kerre, Order norms on bounded partially ordered sets, The Journal of Fuzzy Mathematics 2 (1994) 281–310. [6] M. Demirci, Aggregation operators on partially ordered sets and their categorical foundations, Kybernetika 42 (2006) 261–277. [7] G. Deschrijver, Arithmetic operators in interval-valued fuzzy set theory, Information Sciences 177 (14) (2007) 2906–2924. [8] G. Deschrijver, Characterizations of (weakly) Archimedean t-norms in interval-valued fuzzy set theory, Fuzzy Sets and Systems 160 (6) (2009) 778–801. [9] G. Deschrijver, Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory, Fuzzy Sets and Systems 160 (21) (2009) 3080–3102. [10] G. Deschrijver, E.E. Kerre, Uninorms in L ∗ -fuzzy set theory, Fuzzy Sets and Systems 148 (2004) 243–262. [11] G. Deschrijver, E. Kerre, Aggregation operators in interval-valued fuzzy and Atanassov’s intuitionistic fuzzy set theory, Studies in Fuzziness and Soft Computing 220 (2008) 183–203. [12] D. Dubois, H. Prade, On the use of aggregation operations in information fusion processes, Fuzzy Sets and Systems 142 (2004) 143–161. [13] M. Grabisch, J.-L. Marichal, R. Mesiar, E. Pap, Aggregation Functions, Cambridge University Press, Cambridge, 2009. [14] S. Jenei, B. De Baets, On the direct decomposability of t-norms on product lattices, Fuzzy Sets and Systems 139 (3) (2003) 699–707. [15] F. Karacal, D. Khadjiev, ∨-distributive and infinitely ∨-distributive t-norms on complete lattices, Fuzzy Sets and Systems 151 (2) (2005) 341–352. [16] J.-L. Marichal, k-intolerant capacities and Choquet integrals, European Journal of Operational Research 177 (3) (2007) 1453–1468. [17] R. Mesiar, M. Komorníková, Classification of aggregation functions on bounded partially ordered sets, in: Proceedings of the SISY’ 2010, Subotica, September 10–11, 2010, pp. 13–16. [18] S. Saminger-Platz, E.P. Klement, R. Mesiar, On extensions of triangular norms on bounded lattices, Indagationes Mathematicae 19 (1) (2008) 135–150. [19] I.B. Turksen, Interval-valued fuzzy sets and ‘compensatory AND’, Fuzzy Sets and Systems 51 (3) (1992) 295–307. [20] D. Zhang, Triangular norms on partially ordered sets, Fuzzy Sets and Systems 153 (2) (2005) 195–209.
Aggregation functions on bounded partially ordered sets and their classification Magda Komorníková, Radko Mesiar∗ Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 813 68 Bratislava, Slovakia Received 30 September 2010; received in revised form 19 January 2011; accepted 24 January 2011 Available online 4 February 2011
Abstract Several classes of aggregation functions on bounded partially ordered sets are introduced and discussed. Considering their conjunctive and/or disjunctive attitude, some classification approaches are proposed, including several types of averaging aggregation functions. Aggregation functions on special cases of lattices and chains are discussed, too. © 2011 Elsevier B.V. All rights reserved. Keywords: Averaging aggregation; Conjunctive aggregation; Disjunctive aggregation; Tolerant aggregation
1. Introduction Classification of objects considered in any domain is an important tool for the transparentness, better understanding of the considered domain, but also for construction and application of discussed objects. In the area of aggregation functions acting on real intervals, such a classification was proposed by Dubois and Prade at AGOP’2001 conference in Oviedo, see Dubois and Prade [12]. In Dubois–Prade approach, conjunctive, disjunctive, averaging and remaining aggregation functions were considered, defined by their relationship to Min and Max functions. The class C of all (n-ary) conjunctive functions (acting on a real interval [a, b]) is characterized by the inequality A ≤ Min, while the inequality A ≥ Max is characteristic for the disjunctive aggregation functions from the class D. Concerning the averaging aggregation functions, they should satisfy Min ≤ A ≤ Max. To exclude the trivial overlapping of conjunctive and averaging (disjunctive and averaging) aggregation functions, the class P of pure averaging aggregation functions consists of all averaging aggregation functions except Min and Max. Denoting A the class of all aggregation functions (n-ary, on real interval [a, b]), R = A\(C ∪ D ∪ P) consists of all remaining aggregation functions, which are neither conjunctive, nor disjunctive nor averaging. Thus this standard classification (C, D, P, R) forms a partition of the class A. As prototypical examples of members of these special classes, consider [0, 1] interval and the product (x1 , . . . , xn ) = x1 · x2 · . . . · xn (triangular norm, conjunctive aggregation function), bounded sum SL (x1 , . . . , xn ) = min{1, x1 + · · · + xn } (triangular conorm, disjunctive aggregation function), arithmetic mean M(x1 , . . . , xn ) = (1/n)(x1 +· · ·+xn ) (mean, pure averaging aggregation function) and the 3--operator E(x1 , . . . , xn ) = (x1 , . . . , xn )/((1 − x1 , . . . , 1 − xn ) + (x1 , . . . , xn )) (uninorm, remaining aggregation function). In several domains we need to classify the aggregation of more complex ∗ Corresponding author.
E-mail addresses: [email protected] (M. Komorníková), [email protected] (R. Mesiar). 0165-0114/$ - see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2011.01.015
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objects, which rarely form a chain, but they can be considered as elements of some (bounded) lattice or poset (we will use this abbreviation for a partially ordered set throughout this paper), see e.g., [5,6]. This is, for example, the case of aggregation of fuzzy sets (intersection, union), of distribution functions (convolution), etc. However, such a classification of aggregation functions on posets is missing in the literature so far. Obviously, we cannot repeat the approach of Dubois and Prade once Min and Max are not defined. The aim of this paper is to fill the above-mentioned gap. Note that some first steps towards the classification of aggregation functions on posets can be found in our paper Mesiar and Komorníková [17]. This paper is organized as follows. In the next chapter, aggregation functions on posets are recalled and exemplified. In Section 3, conjunctive and disjunctive classifications of aggregation functions on a bounded poset are proposed and discussed. In Section 4, some general classifications are introduced. Section 5 is devoted to the classification of aggregation functions on a bounded lattice (chain). In Section 6, aggregation functions on product posets/lattices are discussed. Finally, some concluding remarks are added. 2. Aggregation functions on bounded posets Aggregation functions on real intervals are special functions characterized by the monotonicity and the boundary conditions. Recall that an aggregation function A : [0, 1]n → [0, 1] (an extended aggregation function A : n∈N [0, 1]n → [0, 1]) is supposed to be increasing, i.e., A(x) ≤ A(y) whenever x ≤ y, and A(0, . . . , 0) = 0, A(1, . . . , 1) = 1. The framework of aggregation functions [0, 1] can be modified into any closed interval [a, b] ⊆ [−∞, ∞], or even to any interval I ⊆ [−∞, ∞] (with modified boundary conditions, see [13]). Obviously, aggregation functions can be introduced to act on any (partially) ordered structure with bounds. Definition 1. Let (P, ≤, 0, 1) be a bounded poset (partially ordered set). Let n ∈ N be fixed. A mapping A : P n → P is called an (n-ary) aggregation function on P whenever it is increasing, A(x) ≤ A(y) whenever x ≤ y (i.e., x1 ≤ y1 , . . . , xn ≤ yn )
(1)
and it satisfies boundary conditions A(0, . . . , 0) = 0, A(1, . . . , 1) = 1. (2) A mapping B : n∈N P n → P is called an extended aggregation function on P whenever B|P n is an n-ary aggregation function on P for any n ∈ N . For a more detailed discussion of aggregation on posets/lattices we recommend Demirci [6]. Special types of aggregation functions on posets, especially triangular norms and conorms, are studied, for example, in [2,4,7–11,14,15,18–20]. Note that if P = [0, 1] is equipped with the standard ordering of reals, Definition 1 turns into the classical definition of an aggregation function [1,3,13]. Example 1. Consider, for example, the diamond lattice D = {0, a, b, 1} visualized in Fig. 1. Then a mapping A : D → D is a unary aggregation function on D if and only if A(0) = 0 and A(1) = 1 (i.e., the values A(a) and A(b) can be chosen arbitrarily). Define mapping B : D 2 → D as follows (for x, y ∈ D): B(x, y) = 0 if 0 ∈ {x, y}, B(x, y) = 1 if 1 ∈ {x, y} and 0 ∈ / {x, y}, B(x, x) = x if x ∈ D, B(a, b) = B(b, a). Then B is well defined and it is a binary aggregation function on D.
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1
a
b
0 Fig. 1. Hasse diagram of the diamond lattice D.
Finally, we introduce a ternary aggregation function C : D 3 → D as follows: C(x, y, z) = u whenever the triple {x, y, z} contains at least two times u; C(x, y, z) = 0 whenever {x, y, z} = {0, a, b} and C(x, y, z) = 1 whenever {x, y, z} = {1, a, b}. Observe that all algebraic properties of aggregation functions acting on [0, 1] as introduced in Grabisch et al. [13], see also Beliakov et al. [1], Calvo [3], can be straightforwardly introduced also for aggregation functions acting on bounded posets. This is, for example, the case of: • • • • • • • • •
symmetry; associativity; annihilator (zero element); weak annihilator; neutral element (unit element); bisymmetry; idempotency; internality (i.e., A(x1 , . . . , xn ) ∈ {x1 , . . . , xn }); decomposability.
Recall that some of these properties concern the extended aggregation functions (decomposability, general bisymmetry, general associativity, etc.), while some of them are meaningful for aggregation functions with a fixed arity only. Note that the binary aggregation function B : D 2 → D acting on the diamond lattice D and introduced in Example 1 is associative, symmetric (and thus also bisymmetric), idempotent, 0 is its annihilator, and if B(a, b) ∈ {a, b} then B is also internal. On the other hand, the introduced ternary aggregation functions C : D 3 → D is symmetric, idempotent and internal. 3. Conjunctive and disjunctive classification Let n ∈ N and a poset (P, ≤, 0, 1) be fixed. For a given aggregation function A : P n → P and x = (x1 , . . . , xn ) ∈ P n we denote: g A (x) = card{i|xi ≥ A(x)} and s A (x) = card{i|xi ≤ A(x)}.
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Let A = {A : P n → P|A is an aggregation function}, and define two mappings , : A → {0, 1, . . . , n} by (A) = inf{g A (x)|x ∈ P n }, (A) = inf{s A (x)|x ∈ P n }. Observe that sup{g A (x)|x ∈ P n } = sup{s A (x)|x ∈ P n } = n due to the boundary conditions (2) forcing g A ((0, . . . , 0)) = g A ((1, . . . , 1)) = s A ((0, . . . , 0)) = s A ((1, . . . , 1)) = n, independently of A. Both functions and allow to introduce a classification of aggregation functions from A. Proposition 1. C = {C0 , C1 , . . . , Cn } and D = {D0 , D1 , . . . , Dn } given by Ci = −1 ({i}) and Di = −1 ({i}), i = 0, 1, . . . , n are partitions of A. Remark 1. (i) An aggregation function A : P n → P belongs to the class Cn (Dn ) if and only if for all x = (x1 , . . . , xn ) ∈ P n it holds xi ≥ A(x)(xi ≤ A(x)) for each i ∈ {1, . . . , n}. In the case of standard aggregation functions on [0, 1] (or any real interval) this means that A is conjunctive (disjunctive). Therefore, aggregation functions from the classCn will n−1 Ci be called strongly conjunctive, and we define Cs = Cn . Moreover, the aggregation functions from Cw = i=1 will be called weakly conjunctive. Finally, Ca = C0 is the class of aggregation functions admitting the existence of x ∈ P n such that for each i ∈ {1, . . . , n}, either xi < A(x) or xi ⊥ A(x) (xi is incomparable to A(x)); these aggregation functions will be called anticonjunctive. Similarly, the classes Ds , Dw and Da of strongly disjunctive, weakly disjunctive and antidisjunctive aggregation functions can be introduced dually. (ii) The concepts of a conjunctive classification and of a disjunctive classification are dual in the following sense: for a fixed poset (P, ≤, 0, 1) one can define a dual poset simply reversing the order, i.e., (Q, , 0 Q , 1 Q ) is given by Q = P, 0 Q = 1, 1 Q = 0, and for x, y ∈ Q, x y if and only if x ≥ y. Evidently, each aggregation function A : P n → P can be considered as an aggregation function on Q, too. However, then for all x ∈ P n = Q n Q Q it holds g AP (x) = s A (x), s AP (x) = g A (x), and thus A ∈ CiP (DiP ) in the case of poset (P, ≤, 0, 1) if and only if Q Q A ∈ Di (Ci ) in the case of poset (Q, , 0 Q , 1 Q ). (iii) Recently, Marichal [16] has introduced the concept of k-tolerant and k-intolerant aggregation functions on real intervals. Our approach is a modification and an extension of Marichal’s one to the case of posets. n Indeed, if Di . P = [a, b] is a real interval, and k ∈ {1, . . . , n}, then k-tolerant aggregation functions form the class i=n−k+1 In particular, 1-tolerant aggregation functions are exactly the strongly disjunctive aggregation functions. Similarly, n Ci , and 1-intolerant aggregation functions are just the k-intolerant aggregation functions form the class i=n−k+1 strongly conjunctive aggregation functions. (iv) Order norms on posets introduced by De Cooman and Kerre [5] are typical examples of strongly conjunctive and strongly disjunctive aggregation functions. For unary aggregation functions, strongly disjunctive and strongly conjunctive aggregation functions overlap, Ds ∩ Cs = {i P }, where i P is the identity on P. To avoid this undesirable effect, we will deal with n > 1 since now. Moreover, we will suppose also car d P ≥ n to avoid the necessity of the repetition of arguments for any x ∈ P n . In the next considerations we will not repeat these constrains, though we will deal with them. Proposition 2. Let A : P n → P be a fixed aggregation function. Then (A) + (A) ≤ n + 1, and if P is not a chain, then (A) = n implies (A) = 0, and (A) = n implies (A) = 0. Proof. Choose an x ∈ P n such that the set {x1 , . . . , xn } has cardinality n, i.e., all coordinate values x1 , . . . , xn are different. Denote e A (x) = card{i|xi = A(x)} and in A (x) = card{i|xi ⊥ A(x)}. Then e A (x) ∈ {0, 1} and g A (x) + s A (x) + in A (x) = n + e A (x). Consequently, (A) + (A) ≤ n + 1. If (A) = 0 or (A) = 0, then necessarily e A (x) = 0 for some x ∈ P n , and evidently (A) + (A) ≤ n.
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Suppose that P is not a chain, i.e., there are a, b ∈ P such that a ⊥ b, and that (A) = n. Then g A ((a, b, . . . , b)) = n, i.e., A((a, b, . . . , b)) ≤ a and A((a, b, . . . , b)) ≤ b. Consequently A((a, b, . . . , b)) = c cannot satisfy neither c ≥ a nor c ≥ b, i.e., s A ((a, b, . . . , b)) = 0 = (A). The rest of the proof runs similarly. The last result has an important consequence. Corollary 1. A strongly conjunctive (disjunctive) aggregation function A is necessarily antidisjunctive (anticonjunctive). Observe that the class A of all n-ary aggregation functions on a fixed poset P is itself a bounded partially ordered set with inherited ordering ≤, A1 ≤ A2 whenever A1 (x) ≤ A2 (x) for all x ∈ P n , and with the top element A∗ : P n → P, 0 if x = (0, . . . , 0), ∗ A (x) = 1 else, and the bottom element A∗ : P n → P, 1 if x = (1, . . . , 1), A∗ (x) = 0 else. Evidently, A∗ ∈ Ds is strongly disjunctive and A∗ ∈ Cs is strongly conjunctive. Moreover, the mappings , : A → {0, 1, . . . , n} are monotone. Indeed, is decreasing, A1 ≤ A2 implies (A1 ) ≥ (A2 ), while is increasing, A1 ≤ A2 implies (A1 ) ≤ (A2 ). Example 2. Recall aggregation functions B and C from Example 1. B is both weakly conjunctive and weakly disjunctive, (B) = (B) = 1. Similarly, (C) = (C) = 1. 4. General classifications Conjunctive and disjunctive classifications (and also their refinements) may lead to an unacceptably large number of possible classes. As observed in several papers, human beings have no problem with 3–7 classes to be distinguished. The already mentioned Dubois–Prade classification consists of four classes. Following their approach, and fixing the classes of strongly conjunctive and strongly disjunctive aggregation functions (due to Corollary 1, they are disjoint) we have to consider the counterpart of the class of (purely) averaging aggregation functions. We can either identify them with the weakly conjunctive aggregation functions, see Remark 1(i), and to introduce the coarsened conjunctive classification CC = {Cs , Cw , Ds , RC }, where the remaining aggregation functions form the class RC = C0 \Ds of all anticonjunctive aggregation functions which are not strongly disjunctive. Or we can introduce a coarsened disjunctive classification DC = {Cs , Dw , Ds , R D } with R D = D0 \Cs . Nevertheless, not to support neither the conjunctive nor the disjunctive approach, we propose to consider the two next concepts of averaging aggregation functions. Definition 2. Let an aggregation function A : P n → P be given. Then 1. A is called strongly averaging whenever it is both weakly conjunctive and weakly disjunctive, A ∈ As = Cw ∩ Dw ; 2. A is called weakly averaging whenever it is either weakly conjunctive or weakly disjunctive, A ∈ Aw = Cw ∪ Dw . Based on Definition 2, we introduce two general classifications of aggregation functions on posets visualized in Fig. 2. Definition 3. 1. The partition {Cs , Aw , Ds , C0 ∩ D0 } will be called a weak classification of the class A of all n-ary aggregation functions on a fixed poset (P, ≤, 0, 1). 2. The partition {Cs , As , Ds , Cw \Dw , Dw \Cw , C0 ∩ D0 } will be called a strong classification of the class A of all n-ary aggregation functions on a fixed poset (P, ≤, 0, 1).
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Fig. 2. Strong classification on the class A described by (a) class symbols, (b) values of (( A), ( A)), here r, s are arbitrary values from {1, . . . , n −1}.
Observe that Dw \As = Dw \Cw , Cw \As = Cw \Dw and {As , Cw \As , Dw \As } is a partition of the class Aw of all weakly averaging aggregation functions (i.e., the strong classification is a refinement of the weak classification), and thus Fig. 2 visualizing the strong classification contains implicitly the visualization of the weak classification, too. 5. Classification of aggregation functions on bounded chains and bounded lattices Suppose the case (P, ≤, 0, 1) where ≤ is a total (linear) order, i.e., P is a bounded chain. Then the inequality A(x) ≥ xi for some i ∈ {1, . . . , n} is equivalent to A(x) ≥ Min(x). Similarly, A(x) ≤ Max(x) and A(x) ≤ xi are equivalent. Thus the strong averaging aggregation function A is characterized by Min ≤ A ≤ Max inequalities, coinciding with Dubois–Prade approach to averaging aggregation functions. Moreover, (A) = 0 ((A) = 0) is equivalent to (A) = n((A) = n) in this case. Summarizing, we have the next result verifying that our approach extends the original Dubois–Prade classifications from [12]. Proposition 3. Let (P, ≤, 0, 1) be a bounded chain. Then the strong classification and the weak classification of (n-ary) aggregation functions on P coincide and they both correspond to Dubois–Prade classification (C, D, P, R) introduced in [12]. Considering a bounded lattice, the situation is completely different. First of all, we can introduce the class Al of all lattice-averaging aggregation functions paraphrasing the approach of Dubois and Prade, i.e., A ∈ Al if and only if Min ≤ A ≤ Max, and A ∈ / {Min, Max}. Proposition 4. Let A : L n → L be an aggregation function on a bounded lattice (L , ≤, 0, 1). Then: (i) if A is strongly averaging then A is also lattice-averaging (i.e., As ⊆ Al ); (ii) the fact that A is weakly averaging brings no information about the possible lattice-averagingness of A, and vice versa (i.e., the classes Aw and Al are in a general position, in general). Proof. (i) Also in the lattice case, if xi ≤ A(x) for some i ∈ {1, . . . , n}, then A(x) ≥ Min(x) (xi ≥ A(x) implies A(x) ≤ Max(x)). Thus each strongly averaging aggregation function A is necessarily also lattice-averaging. (ii) Consider the lattice (L = [0, 1]3 , ≤, (0, 0, 0), (1, 1, 1)) equipped with the standard partial order ≤. Define two aggregation functions A, B : L 3 → L by x1 + y1 + z 1 x2 + y2 + z 2 x3 + y3 + z 3 A((x1 , x2 , x3 ), (y1 , y2 , y3 ), (z 1 , z 2 , z 3 )) = , , 3 3 3 and B((x1 , x2 , x3 ), (y1 , y2 , y3 ), (z 1 , z 2 , z 3 )) =
x1 + y1 + z 1 , x2 y2 z 2 , x3 y3 z 3 . 3
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Then A((0, 21 , 1), ( 21 , 1, 0), (1, 0, 21 )) = ( 21 , 21 , 21 ) and thus (A) = (A) = 0, A is not weakly averaging, A ∈ / Aw . On the other hand, it is evident that A is lattice-averaging. Concerning B, B(( 41 , 24 , 43 ), ( 24 , 43 , 41 ), ( 43 , 41 , 24 )) = 3 3 , 32 ), while Min(( 41 , 24 , 43 ), ( 24 , 43 , 41 ), ( 43 , 41 , 24 )) = ( 41 , 41 , 41 ). Hence B ∈ / Al and (B) ≤ 2. Moreover, at ( 21 , 32 least one element x1 , y1 or z 1 is greater or equal to (x1 + y1 + z 1 )/3, while all elements x2 , y2 , z 2 (x3 , y3 , z 3 ) are greater than or equal to x2 y2 z 2 (x3 y3 z 3 ), proving that (B) > 0. Consequently, B ∈ Aw . Due to the last proposition, in the case of bounded lattices which are not chains we have three different general classifications of aggregation functions: • weak classification {Cs , Ds , Aw , C0 ∩ D0 }; • strong classification {Cs , Ds , As , Cw \As , Dw \As , C0 ∩ D0 }; • lattice classification {Cs , Ds , Al , R}, where R = A\(Cs ∪ Ds ∪ Al ). Note that each strongly conjunctive aggregation function A satisfies A(x) ≤ xi (for each i and each x) and thus A ≤ Min, and thus Cs ∩ Al = ∅. Similarly, Ds ∩ Al = ∅. 6. Aggregation functions on product posets/lattices For any non-empty index set I, consider a system ((Pi , ≤i , 0i , 1i ))i∈I of bounded posets (lattices). The product poset (lattice) (P, ≤, 0, 1) is then defined as P = ×i∈I Pi (Cartesian product), 0 = (0i )i∈I , 1 = (1i )i∈I , and for x = (xi )i∈I , y = (yi )i∈I ∈ P we have x ≤ y whenever xi ≤i yi for all i ∈ I , see e.g., Birkhoff [2]. Aggregation functions on product lattices were already discussed in [4,14]. A special class of decomposable aggregation functions on product posets/lattices can be defined by means of aggregation functions ( Ai )i∈I acting on component posets/lattices Pi , i ∈ I . Indeed, then A : P n → P is given by (1)
(n)
(1)
(n)
A((xi )i∈I , . . . , (xi )i∈I ) = (Ai (xi , . . . , xi ))i∈I . Several properties of single aggregation functions Ai are inherited by the global aggregation function A. For example, if each Ai is a triangular norm on Pi , i ∈ I , then A is a triangular norm on P, see De Baets and Mesiar [4]. However, not all triangular norms on P are decomposable [14]. The next result is straightforward. Proposition 5. Let A : P n → P be a decomposable aggregation function, A = (Ai )i∈I , acting on a product poset P = ×i∈I Pi . Then A is strongly conjunctive (strongly disjunctive) if and only if each Ai : Pin → Pi , i ∈ I is strongly conjunctive (strongly disjunctive). Proposition 5 cannot be extended for strongly (weakly) averaging aggregation functions. However, if P is a product lattice, then the l-averaging property is preserved by decomposable aggregation functions. Example 3. Consider the lattice L = [0, 1]3 as in the proof of Proposition 4, and define a decomposable aggregation function C : L 3 → L by C((x1 , x2 , x3 ), (y1 , y2 , y3 ), (z 1 , z 2 , z 3 ))=(Med(x1 , y1 , z 1 ), Med(x2 , y2 , z 2 ), Med(x3 , y3 , z 3 )). Evidently, median function Med : [0, 1]3 → [0, 1] is strongly averaging. On the other side, consider x ∈ L 3 , x = ((0, 0.3, 0.6), (0.3, 0.6, 0), (0.6, 0, 0.3)). Then C(x) = (0.3, 0.3, 0.3), and thus gC (x) = sC (x) = 0. Therefore, (C) = (C) = 0 and hence C is not strongly averaging, neither weakly averaging. However, C is l-averaging. Example 3, as well as the aggregation function A introduced in the proof of Proposition 4, shows also that the weak conjunctivity (the weak disjunctivity) of decomposable aggregation functions on product posets/lattices is not guaranteed by the weak conjunctivity (the weak disjunctivity) of all component aggregation functions. The strongest result in this spirit is the next one. Proposition 6. Let A : P n → P be a decomposable aggregation function, A = (Ai )i∈I , acting on a product poset P = ×i∈I Pi . If there is j ∈ I such that A j : P jn → P j is weakly conjunctive (weakly disjunctive), and for each i ∈ I, i j, Ai : Pin → Pi is strongly conjunctive (strongly disjunctive), then A is weakly conjunctive (weakly disjunctive).
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Proof. We will prove the conjunctive case only, as the disjunctive runs similarly. For any x = (x1 , . . . , xn ) ∈ P n , xk = (k) (1) (n) (1) (n) (xi )i∈I , k = 1, . . . , n, it holds g A (x) ≥ n − i∈I (n − g Ai ((xi , . . . , xi ))) = g A j ((x j , . . . , x j )), and hence (A) ≥ (A j ). On the other side, it is obvious that (A) ≤ inf{(Ai )|i ∈ I } = (A j ), showing the equality (A) = (A j ). As far as A j is weakly conjunctive, so is A. To illustrate Proposition 6, consider the aggregation function B introduced in the proof of Proposition 4. Decomposable aggregation function B is linked to aggregation functions B1 , B2 , B3 : [0, 1]3 → [0, 1], B1 = M, B2 = B3 = , i.e., B2 and B3 are strongly conjunctive and B1 is weakly conjunctive. Therefore, B is weakly conjunctive (hence also weakly averaging). 7. Concluding remarks We have introduced and discussed some possible classifications of aggregation functions on bounded posets (lattices), extending and generalizing the ideas of Dubois and Prade [12] and Marichal [16] proposed for aggregation functions acting on real intervals. Observe that the distributive lattices admit an introduction of order statistics. However, Marichal’s approach extended from the real intervals (chains) to these lattices does not cover our approach (compare Example 3, for example). Note that there are some sufficient conditions ensuring the belongingness of considered aggregation functions into a relevant class. So, for example, let A : P n → P be an aggregation function with neutral element 1 (0). Then necessarily A is strongly conjunctive (strongly disjunctive). Internality of A (i.e., A(x) ∈ {x1 , . . . , xn } for all x ∈ P n ) forces (A) ≥ 1 and (A) ≥ 1, thus if P is not a chain, A is necessarily strongly averaging. Observe also that any kind of averaging we have introduced ensures the idempotency of A. It is well known that if P is a chain then also the reverse claim is valid (note that then all introduced concepts of averaging coincide). In general, if P is a lattice, the idempotency of an aggregation function different from Min and Max is equivalent to the lattice-averaging concept 1
c
d
a
b
0 Fig. 3. Hasse diagram of a poset P.
Table 1 The aggregation function A. A
0
a
b
c
d
1
0 a b c d 1
0 0 0 0 0 0
0 a 0 a a a
0 0 b b b b
0 a b c a c
0 a b a d d
0 a b c d 1
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M. Komorníková, R. Mesiar / Fuzzy Sets and Systems 175 (2011) 48 – 56
Table 2 The aggregation function B. B
0
a
b
c
d
1
0 a b c d 1
0 0 0 0 0 0
0 a 0 a a a
0 0 b b b b
0 a b c b c
0 a b b d d
0 a b c d 1
(but neither to the strong averaging nor to the weak averaging). Moreover, if P is a lattice, then the only idempotent strongly conjunctive (strongly disjunctive) aggregation function is Min (Max). This is not true on a general poset, where we can have several strongly conjunctive (strongly disjunctive) idempotent aggregation functions. As an example, consider a poset P visualized in Fig. 3, and aggregation functions A, B : P 2 → P described in Tables 1 and 2. Acknowledgments The research summarized in this paper was supported by the Grants APVV-0012-07, VEGA 1/0080/10 and LPP-0111-09. References [1] G. Beliakov, A. Pradera, T. Calvo, Aggregation Functions: A Guide for Practitioners, Springer-Verlag, Berlin, Heidelberg, 2007. [2] G. Birkhoff, Lattice Theory, American Mathematical Society Colloquium Publications, third ed., vol. XXV, American Mathematical Society, Providence, RI, 1967. [3] 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: Properties, Classes and Construction Methods, Aggregation Operators: New Trends and Applications, Physica-Verlag, Heidelberg, New York, 2002, pp. 1–104. [4] B. De Baets, R. Mesiar, Triangular norms on product lattices, Fuzzy Sets and Systems 104 (1999) 61–76. [5] G. De Cooman, E.E. Kerre, Order norms on bounded partially ordered sets, The Journal of Fuzzy Mathematics 2 (1994) 281–310. [6] M. Demirci, Aggregation operators on partially ordered sets and their categorical foundations, Kybernetika 42 (2006) 261–277. [7] G. Deschrijver, Arithmetic operators in interval-valued fuzzy set theory, Information Sciences 177 (14) (2007) 2906–2924. [8] G. Deschrijver, Characterizations of (weakly) Archimedean t-norms in interval-valued fuzzy set theory, Fuzzy Sets and Systems 160 (6) (2009) 778–801. [9] G. Deschrijver, Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory, Fuzzy Sets and Systems 160 (21) (2009) 3080–3102. [10] G. Deschrijver, E.E. Kerre, Uninorms in L ∗ -fuzzy set theory, Fuzzy Sets and Systems 148 (2004) 243–262. [11] G. Deschrijver, E. Kerre, Aggregation operators in interval-valued fuzzy and Atanassov’s intuitionistic fuzzy set theory, Studies in Fuzziness and Soft Computing 220 (2008) 183–203. [12] D. Dubois, H. Prade, On the use of aggregation operations in information fusion processes, Fuzzy Sets and Systems 142 (2004) 143–161. [13] M. Grabisch, J.-L. Marichal, R. Mesiar, E. Pap, Aggregation Functions, Cambridge University Press, Cambridge, 2009. [14] S. Jenei, B. De Baets, On the direct decomposability of t-norms on product lattices, Fuzzy Sets and Systems 139 (3) (2003) 699–707. [15] F. Karacal, D. Khadjiev, ∨-distributive and infinitely ∨-distributive t-norms on complete lattices, Fuzzy Sets and Systems 151 (2) (2005) 341–352. [16] J.-L. Marichal, k-intolerant capacities and Choquet integrals, European Journal of Operational Research 177 (3) (2007) 1453–1468. [17] R. Mesiar, M. Komorníková, Classification of aggregation functions on bounded partially ordered sets, in: Proceedings of the SISY’ 2010, Subotica, September 10–11, 2010, pp. 13–16. [18] S. Saminger-Platz, E.P. Klement, R. Mesiar, On extensions of triangular norms on bounded lattices, Indagationes Mathematicae 19 (1) (2008) 135–150. [19] I.B. Turksen, Interval-valued fuzzy sets and ‘compensatory AND’, Fuzzy Sets and Systems 51 (3) (1992) 295–307. [20] D. Zhang, Triangular norms on partially ordered sets, Fuzzy Sets and Systems 153 (2) (2005) 195–209.