Journal of Intelligent & Fuzzy Systems 32 (2017) 35–42 DOI:10.3233/JIFS-151029 IOS Press 35 A representation theorem for infinite fuzzy distributive lattices Abdelaziz Amroune ∗ and Ali Oumhani Laboratory of pure and applied Mathematics, Department of Mathematics, University Mohamed Boudiaf, M’sila, Algeria Abstract. In this paper, we show that the category of infinite fuzzy Priestley spaces is equivalent to the dual of the category of infinite fuzzy distributive lattices. A representation theorem for the infinite fuzzy distributive lattices is also given. Keywords: Fuzzy ordered relation, fuzzy ordered set, fuzzy lattice, fuzzy Priestley space, homomorphism of fuzzy lattices, homomorphism of fuzzy Priestley spaces 1. Introduction The representation theorems appeared in the thir- ties of the last century; M. Stone [12] proved that every Boolean algebra is isomorphic to a set of {I a : a ∈ A} (where I a denotes the set of prime ideals of A not containing a). The representation theorem for distributive lattices proved by Birkhoff [2]; asserts that any finite distributive lattice L is isomorphic to the lattice of the ideals of the partial order of the join-irreducible elements of L. H. Priestly developed another kind of duality for bounded distributive lattices see [9, 10]. Such rep- resentation theorems enable a deep and a concrete comprehension of the lattices as well their structures. The duality is central in making the link between syntactical and semantic approaches to logic, also in theoretical computer science this link is central as the two sides correspond to specification languages and the space of computational states. This ability to translate faithfully between algebraic specification ∗ Corresponding author. Abdelaziz Amroune, Laboratory of pure and applied Mathematics, Department of Mathematics, University Mohamed Boudiaf, P.O. Box 166, M’sila 28000, Algeria. Tel.: +213 6 64 60 23 78; Fax: +213 35 389245; E-mail: aamrounedz@yahoo.fr. and spatial dynamics has often proved itself to be a powerful theoretical tool as well as a handle for making practical problems decidable. Topological duality for Boolean algebras [11] and distributive lattices [12] is a useful tool for studying relational semantics for propositional logics. Canon- ical extensions [4–7], provide a way of looking at these semantics algebraically. Priestley’s duality for bounded distributive lattices has enjoyed growing attention and has been variously applied in the international literature since its incep- tion in 1970. After the introduction of the notion of fuzzy rela- tions by Zadeh [14], many concepts and results from the theory of ordered sets have been extended to the fuzzy sets. Venugopalan [13] introduced a definition of a fuzzy ordered set (foset), then extended it to obtain a fuzzy lattice, on which a fuzzy relation is a generalization of an equivalence relation. Another approach was proposed by Chon [3]. Hence, a fuzzy lattice is defined to be just a fuzzy set equipped with a fuzzy ordering relation, albeit the simple definition, many interesting properties of these lattices were deduced [3]. In [1], Amroune and Davvaz gave a representation theorem for finite fuzzy distributive lattices. 1064-1246/17/$35.00 © 2017 – IOS Press and the authors. All rights reserved