Profiniteness in finitely generated varieties is undecidable

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.