Domination of Aggregation Operators and Preservation of Transitivity

Abstract

Aggregation processes are fundamental in any discipline where the fusion of information is of vital interest. For aggregating binary fuzzy relations such as equivalence relations or fuzzy orderings, the question arises which aggregation operators preserve specific properties of the underlying relations, e.g. T-transitivity. It will be shown that preservation of T-transitivity is closely related to the domination of the applied aggregation operator over the corresponding t-norm T. Furthermore, basic properties for dominating aggregation operators, not only in the case of dominating some t-norm T, but dominating some arbitrary aggregation operator, will be presented. Domination of isomorphic t-norms and ordinal sums of t-norms will be treated. Special attention is paid to the four basic t-norms (minimum t-norm, product t-norm, Łukasiewicz t-norm, and the drastic product).

Generalized convexities related to aggregation operators of fuzzy sets

“…Moreover, it also shows that the intersection of a finite collection of T -convex fuzzy sets based on the t-norm T is again T -convex, as the minimum t-norm dominates any other t-norm (see [14]). …”

Aggregation Operators and Commuting

“…Because of the preservation properties of dominance during isomorphic transformations (see also [34]) we immediately can state the following result:…”

“…This paper is devoted to a mathematical investigation of commuting aggregation operators which are used, e.g., in utility theory [15], but also in extension theorems for functional equations [33]. Very often, the commuting property is instrumental in the preservation of some property during an aggregation process, like transitivity when aggregating preference matrices or fuzzy relations (see, e.g., [13], [34]), or some form of additivity when aggregating set functions (see, e.g., [15]). In fact, early examples of commuting appear in probability theory for the merging of probability distributions.…”

General results on the decomposition of transitive fuzzy relations

“…[29,34]) and its usefulness has been demonstrated several times (see e.g. [12,35]). It can be straightforwardly generalized to conjunctors [33].…”