Some remarks on distributive semilattices

Abstract

In this paper we shall give a survey of the most important characterizations of the notion of distributivity in semilattices with greatest element and we will present some new through annihilators and relative maximal filters. We shall also simplify the topological representation for distributive semilattices given in cite{Celani} and show that the meet-relations are closed under composition. So, we obtain that the $DS$-spaces with meet-relations is a category dual to the category of distributive semilattices with homomorphisms. These results complete the topological representation presented in cite{Celani} without the use of ordered topological spaces. Finally, following the work of G. Bezhanishvili and R. Jansana in cite{Bezhanishvili-Jansana II}, we will prove a characterization of homomorphic images of a distributive semilattice $A$ by means of family of closed subsets of the dual space endowed with a lower Vitories topology.