Prime, minimal prime and maximal ideals spaces in residuated lattices

Abstract

In this paper, the notion of minimal prime ideal is introduced in residuated lattices and related properties are investigated. Also, new equivalent characterizations and properties for prime and maximal ideals are obtained and the relation between these ideals and minimal prime ideals is discussed for De Morgan residuated lattices. Moreover, we prove that it is possible to introduce and study, by a standard way, Zariski topology on the lattice P(L) of prime ideals of any residuated lattice L. Also, since mP(L), the set of minimal prime ideals of L, and M(L), the set of maximal ideals of L, are subsets of P(L), we endow mP(L) and M(L) with the topology induced by the Zariski topology on P(L) and we characterize these topological spaces for residuated lattices.