Distributivity of N-ordinal sum fuzzy implications over t-norms and t-conorms

Abstract

Distributivity of fuzzy implications over t-norms and t-conorms provides an effective way to solve the combinatorial rule explosion problems caused by addition of inputs in fuzzy inference systems, and hence has received considerable attention in the literature. In this paper, we explore the distributivity of a newly-born class of ordinal sum fuzzy implications with respect to t-norms and t-conorms, where the involving fuzzy implications are constructed by following the structure of (S,N)-implications generated from ordinal sum t-conorms and fuzzy negations, and are called N-ordinal sum fuzzy implications with the typical feature that the summand domains of linear transformations of given fuzzy implications are rectangles displayed almost along the minor diagonal of [0,1]2. The main results of this paper are characterizations of the t-norm and t-conorm solutions to the four usual distributive equations of N-ordinal sum fuzzy implications, where, of particular interest is that the t-norm and t-conorm solutions have the ordinal sum representations with only one nontrivial summand, and then the necessary and sufficient conditions under which N-ordinal sum fuzzy implications distribute over t-norms and t-conorms in different ways are obtained.