Representing ordered structures by fuzzy sets: An overview

Abstract

We present a survey on representations of ordered structures by fuzzy sets. Any poset satisfying some finiteness condition, semilattice, lattice belonging to a special class, e.g., distributive, Noetherian, complete and others - can be represented by a single function, i.e., by a fuzzy set. Its domain and co-domain are particular subsets of the same structure, and consist of irreducible elements. The representation is minimal in the sense that another representation could not be obtained by replacing the domain of the former by its proper subset. By this approach, the structure itself is uniquely represented by the collection of cuts ordered dually to inclusion. © 2002 Elsevier Science B.V. All rights reserved.