Dualities for Plonka sums

TL;DR

This paper develops a broad topological duality theory for classes of algebras constructed via Plonka sums, linking algebraic and logical frameworks and extending their applications in logic semantics.

Contribution

It introduces a general duality framework for algebras with Plonka sum representations, connecting algebraic structures to dualizable algebras in a topological setting.

Findings

Established a topological duality for Plonka sum algebras.

Connected Plonka sums to algebraic semantics of certain logics.

Extended duality theory to classes of algebras defined by regular identities.

Abstract

Plonka sums consist of an algebraic construction similar, in some sense to direct limits, which allows to represent classes of algebras defined by means of regular identities (namely those equations where the same set of variables appears on both sides). Recently, Plonka sums have been connected to logic, as they provide algebraic semantics to logics obtained by imposing a syntactic filter to given logics. In this paper, I present a very general topological duality for classes of algebras admitting a Plonka sum representation in terms of dualisable algebras.