The Undecidability of the Definability of Principal Subcongruences

TL;DR

This paper proves that determining whether a finite algebra has definable principal subcongruences is undecidable, linking it to the halting problem of Turing machines, and showing no algorithm can decide finite basis property.

Contribution

It establishes the undecidability of definable principal subcongruences for finite algebras by linking it to Turing machine halting behavior.

Findings

Decidable property of definable principal subcongruences is equivalent to Turing machine halting.

No algorithm can determine if a finite algebra is finitely based.

Constructs an algebra for each Turing machine to demonstrate undecidability.

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.