Uninorms which are neither conjunctive nor disjunctive in interval-valued fuzzy set theory

Uninorms are a generalization of t-norms and t-conorms for which the neutral element is an element of [0, 1] which is not necessarily equal to 0 (as for t-norms) or 1 (as for t-conorms). Uninorms on the unit interval are either conjunctive or disjunctive, i.e. they aggregate the pair (0, 1) to either 0 or 1. In real-life applications, this kind of aggregation may be counter-intuitive. Interval-valued fuzzy set theory and Atanassov’s intuitionistic fuzzy set theory are extensions of fuzzy set theory which allows to model uncertainty about the membership degrees. In these theories there exist uninorms which are neither conjunctive nor disjunctive. In this paper we study such uninorms more deeply and we investigate the structure of these uninorms. We also give several examples of uninorms which are neither conjunctive nor disjunctive.