A duality for involutive bisemilattices

TL;DR

This paper establishes a duality between involutive bisemilattices and semilattice inverse systems of Stone spaces, linking algebraic and topological structures through duality theories.

Contribution

It introduces a natural duality for involutive bisemilattices using Stone duality and P{}onka sums, and characterizes their dual spaces as GR spaces with involution.

Findings

Duality between involutive bisemilattices and Stone space systems

Representation of involutive bisemilattices as P{}onka sums of Boolean algebras

Dual spaces as GR spaces with involution

Abstract

We establish a natural duality between the category of involutive bisemilattices and the category of semilattice inverse systems of Stone spaces, using Stone duality from one side and the representation of involutive bisemilattices as P{\l}onka sum of Boolean algebras, from the other. Furthermore, we show that the dual space of an involutive bisemilattice can be viewed as a GR space with involution, a generalization of the spaces introduced by Gierz and Romanowska equipped with an involution as additional operation.