Fuzzy covering relation and ordering: An abstract approach

Abstract

The aim of the paper is to introduce an abstract fuzzy relation generalizing "covering." These types of relations frequently appear in human activities and can be used for the construction of a fuzzy order. We start with the known connection between a crisp ordering relation and a corresponding covering relation on a same set. We also establish some new, less known connections among these. Our aim is to define a fuzzy covering relation independently, as it appears in applications, and then to define the corresponding fuzzy ordering. We consider fuzzy sets in a general way, as mappings from a set to a complete lattice. Then we define a fuzzy covering relation on a set deduced from the given partial order on the same set. We prove some properties of these. Next we start other way around: we take an abstract fuzzy (binary) relation θ, satisfying some of the mentioned properties. We prove that a fuzzy ordering relation can be defined, so that θ is precisely its fuzzy covering relation, provided that the underlying set is finite and the lattice is distributive.