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Aggregation Operators on Partially Ordered Sets and Their Categorical Foundations

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Aggregation Operators on Partially Ordered Sets and Their Categorical Foundations KYBERNETIKA—VOLUME 4 2 (2006), NUMBER 3, PAGES 261–277 AGGREGATION OPERATORS ON PARTIALLY ORDERED SETS AND THEIR CATEGORICAL FOUNDATIONS Mustafa Demirci In spite of increasing studies and investigations in the field of aggregation operators, there are two fundamental problems remaining unsolved: aggregation of L-fuzzy set- theoretic notions and their justification. In order to solve these problems, we will formu- late aggregation operators and their special types on partially ordered sets with universal bounds, and introduce their categories. Furthermore, we will show that there exists a strong connection between the category of aggregation operators on partially ordered sets with universal bounds (Agop) and the category of partially ordered groupoids with uni- versal bounds (Pogpu). Moreover, the subcategories of Agop consisting of associative aggregation operators, symmetric and associative aggregation operators and associative aggregation operators with neutral elements are, respectively, isomorphic to the subcate- gories of Pogpu formed by partially ordered semigroups, commutative partially ordered semigroups and partially ordered monoids in the sense of Birkhoff. As a justification of the present notions and results, some relevant examples for aggregations operators on partially ordered sets are given. Particularly, aggregation process in probabilistic metric spaces is also considered. Keywords: category theory, aggregation operator, associative aggregation operator, par- tially ordered groupoid, partially ordered semigroup, partially ordered monoid AMS Subject Classification: 18A15, 03G10, 06F05, 08C05, 03E72 1. INTRODUCTION Aggregation operators are an essential mathematical tool for the combination of several data as a single outcome. Although they have been studied intensively in the last years [4, 11, 15], there are still some fundamental questions remaining unan- swered. Because of the fact that aggregation operators have been formulated based on a particular partially ordered set ([0, 1], ) (or more generally, ([a, b], ) for some ≤ ≤ a, b R)[4, 11, 15], the present literature does not provide an aggregation process for L∈-fuzzy set theoretic notions that are playing a central role in the development of various branches of fuzzy mathematics [10], where L is a partial ordered set with universal bounds (posetu, for short) in general. This is one of the fundamental prob- lems. In order to solve this problem, we will formulate aggregation operators based 262 M. DEMIRC˙ I˙ on a posetu L, and extend the algebraic properties of symmetry, associativity and the neutral element of aggregation operators to this setting. Furthermore, we design some relevant examples, and show that aggregation operators on posetus are suffi- ciently rich and abstract notions for unifying various kinds of aggregation operator under the same framework. In addition to the above problem, since aggregation operators are some kinds of generalization of binary operation on L with some further properties, it is natural to ask the connections between aggregation operators and such binary operations on L. This question can also be stated for the connection between the subclass of ag- gregation operators consisting of symmetric aggregation operators (resp. associative aggregation operators, aggregation operators with neutral elements) and binary op- erations on L with some suitable algebraic properties. These questions lead us to the necessity for considering the category of aggregation operators on posetus (Agop) and its subcategories of associative aggregation operators (Asagop), of symmetric and associative aggregation operators (Smasagop), and of associative aggregation operators with neutral elements (or simply monoidal aggregation operators) ( Mon- agop). If we take into consideration the category of partially ordered groupoids with universal bounds (Pogpu) and its subcategories of partially ordered semigroups (Posmgu), of commutative partially ordered semigroups (Cmposmgu), and of partially ordered monoids (Pomonu), then the second main problem can be ab- stracted to the categorical relation between Agop (resp. Asagop, Smasagop and Monagop) and Pogpu (resp. Posmgu, Cmposmgu and Pomonu). All of these categories are summarized in Table 1. As an answer to this problem, we will prove that Pogpu is isomorphic to an isomorphism-closed and full subcategory of Agop. Furthermore, Asagop, Smasagop and Monagop are, respectively, isomorphic to Posmgu, Cmposmgu and Pomonu. In addition to these categorical connections, we will also look at subobjects and fibres in Agop towards the end of this paper, and establish a meaningful definition of subaggregation operator as a special subobject in Agop. 2. AGGREGATION OPERATORS ON PARTIALLY ORDERED SETS Let L = (L, , , ) denote a posetu, i. e. (L, ) is a partially ordered set (poset, for brevity)≤ with⊥ the> least element and the≤ greatest element . The ordering ⊥ > structure on L can be coordinatwisely extended to Ln (n N+), i. e. for the relation on Ln, defined by ∈ ≤ [(α , . , α ) (β , . , β )] [( i = 1, . , n)(α β )], 1 n ≤ 1 n ⇔ ∀ i ≤ i and for the elements (n) = ( ,..., ), (n) = ( ,..., ) of Ln, the four-tuple L = (Ln, , (n), (n⊥)) forms⊥ a poset⊥ with> universal> bounds> (n) and (n). n ≤ ⊥ > ⊥ > n Definition 1. A mapping A : + L L is called an aggregation operator on n N L = (L, , , ) if the following conditions∈ → are fulfilled: ≤ ⊥ > ∪ (AG.1) A preserves the order on Ln for all n N+, i. e. ∈ [(α , . , α ) (β , . , β )] [A(α , . , α ) A(β , . , β )] 1 n ≤ 1 n ⇒ 1 n ≤ 1 n AGOPS on Posets and Their Categorical Foundations 263 (AG.2) A is the identity mapping id on L, i. e. A(α) = α for all α L, L ∈ (AG.3) A preserves universal bounds, i. e. (n) (n) + A( ) = and A( ) = for all n N . ⊥ ⊥ > > ∈ For n 2, a mapping B : Ln L is called an n -ary aggregation operator on L = (≥L, , , ) iff the conditions→ (AG.1) and (AG.3) are satisfied. A 1-ary aggregation operator≤ ⊥ > B : L L is the identity mapping id on L. → L An aggregation operator A on L can be identified by a family of n-ary aggregation + operators An n N . This means that for a given aggregation operator A { | ∈ } + on L, we may associate a family of n-ary aggregation operators An n N to A, which is defined by A (α , . , α ) = A(α , . , α ). Conversely,{ | if∈ A } n 1 n 1 n { n | n N+ is a family of n-ary aggregation operators on L, then we can define an ∈ } aggregation operator A on L by A(α1, . , αn) = An(α1, . , αn). It is obvious that the connection between the aggregation operators and the families of n -ary aggregation operators are bijective. Thus an aggregation operator and its associated + family of n-ary aggregation operators An n N can be conceived as the same things. { | ∈ } Aggregation operators on posetus in the sense of Definition 1 provide an abstract and a useful mathematical tool for unifying various kinds of aggregation operator under the same framework, and they also enable us to apply the notion of aggregation operator in several branch of mathematics. In order to clarify this, we design the following example: Example 2. (a) For the particular posetu I = ([0, 1], , 0, 1), an (n-ary) aggrega- tion operator on I is called an (n-ary) aggregation operator≤ in [4, 11, 15]. Thus an (n -ary) aggregation operator in the sense of Definition 1 is a straightforward extension of (n-ary) aggregation operator in the sense of [4, 11, 15] to a general posetu. (b) Let [0, 1] denote the set of all closed subintervals of [0, 1], i. e. [0, 1] = I I [a, b] a, b [0, 1], a b . If we consider the partial ordering w on [0, 1] (called the{ weak| interval-ordering∈ ≤ } [12]) defined by ¹ I [a, b] [c, d] ((a c) and (b d)) , ¹w ⇔ ≤ ≤ then [0, 0] and [1, 1] are obviously the least and greatest elements of the poset ( [0, 1], ). An interval-valued aggregation operator (or an extended aggregation I ¹w operator) in [12] is nothing else but an aggregation operator on ( [0, 1], w, [0, 0], [1, 1]). I ¹ 2 (c) Consider the set L∗ = (x , x ) [0, 1] x + x 1 . The relation on { 1 2 ∈ | 1 2 ≤ } ≤L∗ L∗, defined as (x , x ) (y , y ) ((x y ) and (y x )) , 1 2 ≤L∗ 1 2 ⇔ 1 ≤ 1 2 ≤ 2 264 M. DEMIRC˙ I˙ is a partial ordering [6, 7]. 0L∗ = (0, 1) and 1L∗ = (1, 0) are the least and greatest elements of the poset (L∗, ). An aggregation operator on (L∗, , 0 , 1 ) is ≤L∗ ≤L∗ L∗ L∗ called an aggregation operator on L∗ in [6]. (d) Let [0, 1] stand for the set of all normal convex fuzzy subsets of [0, 1] (i. e. a normal convexNC fuzzy subset µ of [0, 1] is a mapping µ : [0, 1] [0, 1] satisfying the conditions that µ(x) = 1 and µ(x ) µ(x ) µ(x ) for→ all x , x , x [0, 1] x [0,1] 1 ∧ 3 ≤ 2 1 2 3 ∈ with x x x ∈[13]). Define the relation on [0, 1] by 1 ≤ 2 ≤ ∨3 v NC µ ν µ ν = µ, v ⇔ u where : [0, 1] [0, 1] [0, 1] is the extended minimum operation min on [0u, 1]NC [8, 13],× i. e. N C → N C NC g [ (µ, ν)] (x)= min(µ, ν) (x)= µ(x ) ν(x ) x , x [0, 1], x=min x , x . u { 1 ∧ 2 | 1 2 ∈ { 1 2}} [ ] ∨ Then ( [0, 1],g) forms a poset [13] . Furthermore the fuzzy subsets 01, 11 of [0, 1] , definedNC by v 1, if x = 0 1, if x = 1 0 (x) = and 1 (x) = 1 0, otherwise 1 0, otherwise, { } { } are the least and greatest elements of ( [0, 1], ). We will show in Remark 3 that NC v aggregation operators on ( [0, 1], , 01, 11) enable us to aggregate type 2 fuzzy sets with normal convex fuzzyNC grades.v (e) A non-decreasing function F :[ , ] [0, 1], satisfying the properties that F is left-continuous on (0, ), F (0) =−∞ 0 and∞ F→( ) = 1, is called a distance distri- bution function on [ , ∞][11, 16].
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