Pseudo-t-norms and implication operators on a complete Brouwerian lattice

Abstract

In this paper, we introduce and discuss the concept of pseudo-t-norms on a complete Brouwerian lattice L, study the relation between the set of all infinitely ∨-distributive pseudo-t-norms and the set of all infinitely ∧-distributive implications on L in detail, and point out that three classes fundamental fuzzy implications and Yager's implication on [0,1] are all infinitely ∧-distributive implications that can be induced by infinitely ∨-distributive pseudo-t-norms on L when $\text{L}\text{=}\text{[0,1]}$. We also give out a method for generating pseudo-t-norms and implications on L.