A topological duality for monotone expansions of semilattices

TL;DR

This paper develops a Stone-style duality for monotone semilattices using topological methods, providing a new way to understand their structure and congruences through duality and topological descriptions.

Contribution

It introduces a novel duality framework for monotone semilattices and characterizes their congruences via monotone lower-Vietoris-type topologies.

Findings

Established a topological duality for monotone semilattices.

Characterized congruences using monotone lower-Vietoris topologies.

Connected duality theory with canonical extensions.

Abstract

In this paper we provide a Stone style duality for monotone semilattices by using the topological duality developed in \cite{Celani2020} for semilattices together with a topological description of their canonical extension. As an application of this duality we obtain a characterization of the congruences of monotone semilattices by means of monotone lower-Vietoris-type topologies.