On permutations dependent operators

Abstract

In this paper, we introduce permutations dependent operators. The motivation for studying such a concept arises from standard fuzzy integrals, where permutations play a crucial role. In contrast to standard fuzzy integrals, our construction allows any permutation of the basic set in a formula to be considered, rather than limiting it to permutations that reorder the components of the input vector monotonically. We herein present an approach to integration with respect to sets of permutation pairs, i.e., databases in which each vector has a preselected permutation. This new operator generalizes several concepts known in the literature. We investigate the properties of this new concept and highlight its practical utility in image processing.