Stone-type representations and dualities for varieties of bisemilattices

TL;DR

This paper extends representation theorems for various bisemilattice varieties using duality theory, establishing categorical equivalences similar to Stone duality for Boolean algebras.

Contribution

It generalizes Balbes' representation theorem to bounded, De Morgan, and involutive bisemilattices and introduces dual categories of 2spaces, proving their dual equivalence.

Findings

Extended Balbes' theorem to new bisemilattice varieties

Established dual equivalences with categories of 2spaces

Unified duality framework for distributive bisemilattices

Abstract

In this article we will focus our attention on the variety of distributive bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and involutive bisemilattices. After extending Balbes' representation theorem to bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas-Dunn duality and introduce the categories of 2spaces and 2spaces$^{\star}$. The categories of 2spaces and 2spaces$^{\star}$ will play with respect to the categories of distributive bisemilattices and De Morgan bisemilattices, respectively, a role analogous to the category of Stone spaces with respect to the category of Boolean algebras. Actually, the aim of this work is to show that these categories are, in fact, dually equivalent.