ℐ𝒢,𝒩-Implications Induced from Quasi-Grouping Functions and Negations on Bounded Lattices

Abstract

Grouping functions, as one new case of not necessarily associative particular binary aggregation functions, have been proposed in the literature for their vast applications in fuzzy community detection problems, image processing and decision making. On the other hand, due to the wide applications in fuzzy reasoning, fuzzy control and approximate reasoning, the investigations of fuzzy implications derived from specific binary aggregation functions become a natural and hot research topic. In this paper, we focus on this research area and consider the [Formula: see text]-implications induced from quasi-grouping functions and negations on bounded lattices. To be precise, firstly, by removing the continuity condition, we extend the notion of grouping functions on the unit closed interval to the so-called quasi-grouping functions on bounded lattices. Secondly, we give some basic properties and two construction methods of quasi-grouping functions on bounded lattices. Finally, we give the concept of [Formula: see text]-implications and focus on the conditions under which they can satisfy the certain algebraic properties possessed by implications on bounded lattices.