The Variety Generated by A(T) -- Two Counterexamples

TL;DR

This paper presents two counterexamples in algebraic variety theory, showing that certain properties like definable principal subcongruences and bounded Maltsev depth do not always hold, especially in finitely generated semilattice-based varieties.

Contribution

It provides the first known examples of finitely generated semilattice-based varieties with finite subdirectly irreducible members lacking these properties.

Findings

V(A(T)) lacks definable principal subcongruences.

V(A(T)) does not have bounded Maltsev depth.

When T halts, V(A(T)) has finitely many finite subdirectly irreducible members.

Abstract

We show that V(A(T)) does not have definable principal subcongruences or bounded Maltsev depth. When the Turing machine T halts, V(A(T)) is an example of a finitely generated semilattice based (and hence congruence meet-semidistributive) variety with only finitely many subdirectly irreducible members, all finite. This is the only known example of a variety with these properties that does not have definable principal subcongruences or bounded Maltsev depth.