Profiniteness in finitely generated varieties is undecidable

TL;DR

This paper proves that it is impossible to algorithmically determine whether a variety generated by a finite algebra is standard, by linking properties like definable principal subcongruences to standardness.

Contribution

It establishes the undecidability of standardness in finitely generated varieties, solving a problem posed by Clark, Davey, Freese, and Jackson.

Findings

No algorithm can decide standardness of varieties generated by finite algebras.

Standardness relates to properties like definable principal subcongruences.

The proof combines results on definable principal subcongruences and syntactic congruences.

Abstract

Profinite algebras are exactly those that are isomorphic to inverse limits of finite algebras. Such algebras are naturally equipped with Boolean topologies. A variety $\mathcal V$ is standard if every Boolean topological algebra with the algebraic reduct in $\mathcal V$ is profinite. We show that there is no algorithm which takes as input a finite algebra $\mathbf A$ of a finite type and decide whether the variety ${\mathcal V}({\mathbf A})$ generated by ${\mathbf A}$ is standard. We also show the undecidability of some related properties. In particular, we solve a problem posed by Clark, Davey, Freese and Jackson. We accomplish this by combining two results. The first one is Moore's result saying that there is no algorithm which takes as input a finite algebra $\mathbf A$ of a finite type and decides whether ${\mathcal V}({\mathbf A})$ has definable principal subcongruences. The second is our result saying that possessing definable principal subcongruences yields possessing finitely determined syntactic congruences for varieties. The latter property is known to yield standardness.