On residual implications derived from 2-uninorms

Abstract

Residual implications are one of the most important families of fuzzy implications, which are related to the concept of residuation in intuitionistic logic. In this paper, we focus on residual operators derived from 2-uninorms and study necessary and sufficient conditions for such operators being fuzzy implications, called RU2-implications. What is remarkable is that the RU2-implication is a generalization of residual implications generated by some popular fuzzy logic operations such as t-norms, uninorms and uni-nullnorms. Then, we obtain the block structures for RU2-implications, especially for ones generated by two special classes of conjunctive 2-uninorms. Moreover, the main properties of RU2-implications and their natural negations are discussed. Finally, we give a characterization of residual implications derived from left-continuous 2-uninorms, which preserve the residuation property.