Algebras from finite group actions and a question of Eilenberg and Sch\"{u}tzenberger

TL;DR

This paper investigates a longstanding question about whether finite algebraic structures can be characterized by a finite set of identities that also capture infinite models, showing certain algebras do not serve as counterexamples.

Contribution

It demonstrates that specific inherently nonfinitely based algebras from group actions do not counter Eilenberg and Schützenberger's question, and provides the first examples of such algebras constructed from group actions.

Findings

Inherently nonfinitely based algebras from group actions do not fail the finite basis property in the strange way suspected.

First known examples of inherently nonfinitely based automatic algebras constructed from group actions.

Results support the possibility that all finite algebras might be finitely based.

Abstract

In 1976 S. Eilenberg and M.-P. Sch\"{u}tzenberger posed the following diabolical question: if $\mathbf{A}$ is a finite algebraic structure, $\Sigma$ is the set of all identities true in $\mathbf{A}$, and there exists a finite subset $F$ of $\Sigma$ such that $F$ and $\Sigma$ have exactly the same finite models, must there also exist a finite subset $F'$ of $\Sigma$ such that $F'$ and $\Sigma$ have exactly the same finite and infinite models? (That is, must the identities of $\mathbf{A}$ be "finitely based"?) It is known that any counter-example to their question (if one exists) must fail to be finitely based in a particularly strange way. In this paper we show that the "inherently nonfinitely based" algebras constructed by Lawrence and Willard from group actions do not fail to be finitely based in this particularly strange way, and so do not provide a counter-example to the question of Eilenberg and Sch\"{u}tzenberger. As a corollary, we give the first known examples of inherently nonfinitely based "automatic algebras" constructed from group actions.