Representation of lattices by fuzzy sets

Abstract

We prove that every lattice L of finite length can be represented by a fuzzy set on the collection X of meet-irreducible elements of L. A decomposition of this fuzzy set gives a family of isotone functions from X to 2 = ({0,1}, ≤), the lattice of which is isomorphic to L. More generally, conditions under which any collection of isotone functions from a finite set into 2 corresponds to a decomposition of a fuzzy set are given. As a consequence, the representation theorem for a finite distributive lattice by the lattice of all isotone functions is obtained. The collection of all lattices characterized by the same fuzzy set turns out to be a lattice with the above-mentioned distributive lattice as the greatest element.