A topological duality for mildly distributive meet-semilattices

Sergio A. Celani,Luciano J. González

doi:10.33044/revuma.v59n2a04

Abstract

We develop a topological duality for the category of mildly distributive meet-semilattices with a top element and certain morphisms between them. Then, we use this duality to characterize topologically the lattices of Frink ideals and filters, and we also obtain a topological representation for some congruences on mildly distributive meet-semilattices.

Highlights

Duality theory for ordered algebraic structures goes back to Stone’s pioneering work [15]
He proved that the category of Boolean algebras and Boolean homomorphisms is dually equivalent to the category of Boolean spaces and continuous maps. This duality was generalized to distributive lattices by Stone himself [16]. He showed a duality between the category of distributive lattices and lattice homomorphisms and the category of spectral spaces and spectral functions
In the present article, we develop a dual equivalence between the category of mildly distributive meet-semilattices and strong homomorphisms and the category of certain topological spaces and particular continuous maps

Summary

Introduction

Duality theory for ordered algebraic structures goes back to Stone’s pioneering work [15] He proved that the category of Boolean algebras and Boolean homomorphisms is dually equivalent to the category of Boolean spaces (compact and totally disconnected spaces) and continuous maps. In the present article, we develop a dual equivalence between the category of mildly distributive meet-semilattices and strong homomorphisms and the category of certain topological spaces and particular continuous maps. We follow the topological approach of Stone to achieve our aim We use this topological duality to obtain topological representations for the lattices of filters and Frink ideals of mildly distributive meet-semilattices and we get a topological representation for some congruences.

Preliminaries

Mildly distributive meet-semilattices

Representation theorem for md-semilattices

A duality for md-semilattices

Subbasic saturation and the connection between filters and closed sets

Topological representation of congruences