The fusion of transitive fuzzy relations preserving the transitivity is linked to the domination of the involved aggregation operator. The aim of this contribution is to investigate the domination of OWA operators over t-norms whereas the main emphasis is on the domination over the Łukasiewicz t-norm. The domination of OWA operators and related operators over continuous Archimedean t-norms will also be discussed.Keywords Domination, OWA operators
MotivationIn several applications of fuzzy logics and fuzzy systems the processing of data based on the strongest t-norm, i.e., the minimum T M suffers from the increase of uncertainty.Recall only the addition of fuzzy numbers where the final sum spreads equal to the sum of all incoming spreads. In order to avoid this undesirable effect, alternative approaches have to be taken into account. One of the most promising t-norms for reducing the enormous growth of uncertainty is the Łukasiewicz t-norm T L . The addition of triangular (trapezoidal) fuzzy numbers based on T L leads to output spreads equal to the maximal (left and right) incoming spreads -a property which is often required in models dealing with uncertainty. Moreover, the Łukas-iewicz t-norm T L is often applied when fuzzy rule based systems are designed with the purpose to reduce redundancy, and especially in clustering algorithms (likeness relations of Bezdek and Harris [2]). Recall also the approximate solutions of fuzzy relational equations [6] or several relations possessing T L -transitivity as a genuine counterpart of the classical triangle inequality of the Euclidean metric on R. Furthermore, when solving complex problems we sometimes need either to refine several fuzzy relations or to introduce their Cartesian product. However, we expect that the new fuzzy relation will be again T L -transitive if the original fuzzy relations have also been T L -transitive.As it has been shown in [11], the preservation of T-transitivity of fuzzy relations during an aggregation process is guaranteed if the involved aggregation operator dominates the corresponding t-norm T. Several special operators dominating the Łukasiewicz t-norm are already known, e.g. T L itself, the minimum and the arithmetic mean, see [11]. The later two are special cases of so called OWA operators, one of the most important family of aggregation operators applied in many domains (see [14,16]), which are used for summarizing singular data into a single output where inputs are ordered with respect to their value and are assigned certain weights before being aggregated.If we want to preserve the T L -transitivity of fuzzy relations the domination of an OWA operator O over T L should be checked before fusing the T L -transitive fuzzy relations by means of this OWA operator O. Therefore the characterization of OWA operators dominating the Łukasiewicz t-norm T L seemed to be important. The extension of the obtained results for some other t-norms and other dominating aggregation operators will be the final task of this contribution.
Preliminaries
2.1