The Undecidability of the Definability of Principal Subcongruences

Abstract

For each Turing machine T, we construct an algebra A'(T) such that the variety generated by A'(T) has definable principal subcongruences if and only if T halts, thus proving that the property of having definable principal subcongruences is undecidable for a finite algebra. A consequence of this is that there is no algorithm that takes as input a finite algebra a decides whether that algebra is finitely based.