Standard topological algebras: syntactic and principal congruences and profiniteness

Abstract

A topological quasi-variety Q + T (M ∼ ) := IScP + M ∼ generated by a finite algebra M ∼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 Q + T (M ∼ ) is standard provided that the algebraic quasi-variety generated by M ∼ 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 M ∼ 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. Background, motivation and overview of resultsAn algebra M = M ; F with finite underlying set M and operations F generates an (algebraic) quasi-variety Q(M) := ISP M consisting of all isomorphic copies of subalgebras of direct powers of M. Similarly a structure M ∼ = M ; G, H, R, T with finite underlying set M , operations G, partial operations H, relations R and discrete topology T generates a topological quasi-variety Q + T (M ∼ ) := IS c P + M ∼ consisting of all isomorphic copies of topologically closed substructures of non-zero direct powers, with the product topology, of M ∼ . Interest in topological quasi-varieties stems from the fact that they arise as the duals to algebraic quasi-varieties under natural dualities. The general theory of natural dualities provides methods to Presented by R. W. Quackenbush.

Boolean topological graphs of semigroups: the lack of first-order axiomatization

“…It follows from the result of Clark, Davey, Freese and Jackson [5,Theorem 8.1,Example 8.3] that the members of H are all profinite whenever H is a variety and has finitely determined syntactic congruences. In particular, it holds when H is a variety of semigroups, monoids, groups, rings or is a variety with definable principal congruences.…”

Principal and Syntactic Congruences in Congruence-Distributive and Congruence-Permutable Varieties

“…The singleton {z 1 x z 2 } is sufficient to determine principal congruences in the variety of groups for example; see [6] for many other examples. [3] Principal and syntactic congruences 61…”

“…These properties have been used by Baker et al [2] and by Baker and Wang [3] to establish finite basis theorems for the equations of finite algebras. More locally, an individual algebra has TFPC if and only if it has finitely determined syntactic congruences (FDSC), a property that arises naturally in the study of compact topological algebras [6]. For example, a Boolean topological algebra with FDSC is topologically residually finite (see [10], [13,Lemma VI.2.7] or [6,Lemma 4.2]).…”

“…More locally, an individual algebra has TFPC if and only if it has finitely determined syntactic congruences (FDSC), a property that arises naturally in the study of compact topological algebras [6]. For example, a Boolean topological algebra with FDSC is topologically residually finite (see [10], [13,Lemma VI.2.7] or [6,Lemma 4.2]). Moreover, FDSC is the most general known structural property guaranteeing topological residual finiteness (algebraic residual finiteness is not sufficient in general; see Jackson [12]).…”

Profinite semigroups and applications

“…See [32] for the precise scope of validity of the implication (5)⇒(1) in Theorem 3.1 and applications in Universal Algebra.…”