Distributivity for 2-uninorms over semi-uninorms

Abstract

This paper is devoted to solving the distributivity equations for 2-uninorms over semi-uninorms. Our investigations are motivated by the couple of distributive logical connectives and their generalizations, such as t-norms, t-conorms, uninorms, nullnorms, and fuzzy implications,which are often used in fuzzy set theory. There are two generalizations of them. One is a 2-uninorm covering both a uninorm and a nullnorm, which forms a class of commutative, associative and increasing operators on the unit interval with an absorbing element that separates two subintervals with neutral elements. Another is a semi-uninorm, which generalizes a uninorm by omitting commutativity and associativity. In this work, all possible solutions of the distributivity equation for the three defined subclasses of 2-uninorms over semi-uninorms are characterized.