Standard topological algebras: syntactic and principal congruences and profiniteness

Abstract

A topological quasi-variety \(\mathcal{Q}_\mathcal{J}^ + (\mathop {\text{M}}\limits_ \sim ): = \mathbb{I}\mathbb{S}_{\text{c}} \mathbb{P}^ + \mathop {\text{M}}\limits_ \sim \) generated by a finite algebra \(\mathop {\text{M}}\limits_ \sim \) with the discrete topology is said to be standard if it admits a canonical axiomatic description. Drawing on the formal language notion of syntactic congruences, we prove that \(\mathcal{Q}_\mathcal{J}^ + (\mathop {\text{M}}\limits_ \sim )\) is standard provided that the algebraic quasi-variety generated by \(\mathop {\text{M}}\limits_ \sim \) is a variety, and that syntactic congruences in that variety are determined by a finite set of terms. We give equivalent semantic and syntactic conditions for a variety to have Finitely Determined Syntactic Congruences (FDSC), show that FDSC is equivalent to a natural generalisation of Definable Principle Congruences (DPC) which we call Term Finite Principle Congruences (TFPC), and exhibit many familiar algebras \(\mathop {\text{M}}\limits_ \sim \) that our method reveals to be standard. As an application of our results we show, for example, that every Boolean topological lattice belonging to a finitely generated variety of lattices is profinite and that every Boolean topological group, semigroup, and ring is profinite. While the latter results are well known, the result on lattices was previously known only in the distributive case.