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Stone pseudovarieties

Jorge Almeida, Ondřej Klíma

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.

Stone pseudovarieties

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.

Paper Structure

This paper contains 15 sections, 44 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stone pseudovarieties and relatively free Stone topological algebras
  4. Relatively free Stone topological algebras
  5. Free Stone topological algebras are not profinite
  6. Finite quotients
  7. Preliminaries on Boolean algebra
  8. Boolean algebras of sets
  9. An algebraized continuity condition
  10. A relaxed continuity condition
  11. A simplified version of the continuity condition
  12. Duality
  13. From Boolean algebras to Stone topological algebras
  14. From Stone topological algebras to Boolean algebras
  15. Stone varieties

Key Result

Theorem 2.1

A Stone topological algebra $S$ is profinite if and only if, for every clopen subset $L\subseteq S$, there exists a continuous homomorphism $\varphi:S\to F$ onto a finite algebra $F$ such that $L=\varphi^{-1}(\varphi(L))$.

Theorems & Definitions (89)

  • Theorem 2.1: AlmeidaKlimaGoulet-Ouellet:2023
  • Theorem 2.2
  • Proposition 2.3
  • proof
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • ...and 79 more