On a representation of posets by fuzzy sets

Abstract

The aim of the paper is to present an application of fuzzy sets in mathematics, namely, in the theory of ordered sets. An algorithm for the construction of P-fuzzy sets with distinct levels is given. In connection with this, every finite poset can be mapped, by an anti-isotone bijection, onto the poset of levels of a suitable fuzzy set. Moreover, it is proved that every finite partially ordered set (P, ≤) can be represented by the poset of levels of a particular fuzzy set, defined on the collection of meet-irreducible elements of (P, ≤).