Distributivity of the Ordinal Sum Implications Over t-Norms and t-Conorms

Recently, Su and Liu have introduced a new class of fuzzy implications, called ordinal sum implications, and discussed some of their desirable properties, such as neutrality property, consequent boundary, exchange principle, etc. In this paper, we explore the class of ordinal sum implications with respect to distributivity. Necessary and sufficient conditions, under which ordinal sum implications are distributive over t-norms and t-conorms are given.

I. Introduction

Fuzzy implication operators play an important role in the approximate reasoning and fuzzy control theory. There are mainly three ways to generate fuzzy implications in the literature, which give rise to some established families of fuzzy implications, viz., 1)

from other fuzzy logic connectives, from where we obtain, for instance, the families of -, -, -implications (see [2]– [5], [23] for details);

2)

from monotone functions, from where we obtain, for instance, the families of - and -implications proposed by Yager [38], -implication [24];

3)

From given implications, from where we obtain, for instance, the families of the convex combination of the two fuzzy implication [4], -ordinal sum implications [15], -generated implication from the two fuzzy implications [25], [26].