New perspectives on semi-primal varieties

TL;DR

This paper explores the structure of semi-primal lattice-expansions using category theory, providing new duality proofs, adjunctions, and characterizations of canonical extensions, advancing the understanding of these algebraic varieties.

Contribution

It offers a novel categorical approach to semi-primal varieties, including new duality proofs, adjunctions, and characterizations of canonical extensions.

Findings

New proof of Keimel-Werner duality for semi-primal varieties

Identification of properties of Boolean skeleton adjunctions

Characterization of canonical extensions via Boolean skeletons

Abstract

We study varieties generated by semi-primal lattice-expansions by means of category theory. We provide a new proof of the Keimel-Werner topological duality for such varieties and, using similar methods, establish its discrete version. We describe multiple adjunctions between the variety of Boolean algebras and the variety generated by a semi-primal lattice-expansion, both on the topological side and explicitly algebraic. In particular, we show that the Boolean skeleton functor has two adjoints, both defined by taking certain Boolean powers, and we identify properties of these adjunctions which fully characterize semi-primality of an algebra. Lastly, we give a new characterization of canonical extensions of algebras in semi-primal varieties in terms of their Boolean skeletons.