Stone pseudovarieties

TL;DR

This paper extends the theory of profinite algebras to Stone topological algebras, introducing Stone pseudovarieties and establishing duality and residual properties analogous to classical algebraic varieties.

Contribution

It introduces Stone pseudovarieties, characterizes their dual spaces, and proves a Birkhoff type theorem for Stone varieties, extending classical algebraic results to topological settings.

Findings

Characterization of dual spaces of Stone topological algebras

Residually in a Stone pseudovariety implies being a Stone quotient

Stone pseudovarieties are the Stone analogues of algebraic varieties

Abstract

Profinite algebras are the residually finite compact algebras; their underlying topological spaces are Stone spaces. We extend the theory of profinite algebras to a more general setting of Stone topological algebras. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed topological signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. Looking at Stone spaces as the dual spaces of Boolean algebras, we find a simple characterization of when the dual space admits a natural structure of topological algebra. This provides a new approach to duality theory which culminates in the proof that a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is also residually in it, thereby extending the corresponding result of M. Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety, and these are precisely the Stone analogues of varieties. A Birkhoff type theorem for Stone varieties is also established.