A Categorical Duality for Semilattices and Lattices

Abstract

The main aim of this article is to develop a categorical duality between the category of semilattices with homomorphisms and a category of certain topological spaces with certain morphisms. The principal tool to achieve this goal is the notion of irreducible filter. Then, we apply this dual equivalence to obtain a topological duality for the category of bounded lattices and lattice homomorphism. We show that our topological dualities for semilattices and lattices are natural generalizations of the duality developed by Stone for distributive lattices through spectral spaces. Finally, we obtain directly the categorical equivalence between our topological spaces and those presented for Moshier and Jipsen (Algebra Univers 71(2):109–126, 2014).