Dualizability of automatic algebras

TL;DR

This paper investigates which finite automatic algebras are dualizable, providing conditions and examples, including an infinite chain demonstrating the complexity of the problem.

Contribution

It offers necessary and sufficient conditions for dualizability of automatic algebras and illustrates the problem's difficulty with a chain of examples.

Findings

Automatic algebras with abelian permutation groups are dualizable

An infinite chain of automatic algebras alternates between dualizable and non-dualizable

The dualizability problem remains complex and partially understood

Abstract

We make a start on one of George McNulty's Dozen Easy Problems: "Which finite automatic algebras are dualizable?" We give some necessary and some sufficient conditions for dualizability. For example, we prove that a finite automatic algebra is dualizable if its letters act as an abelian group of permutations on its states. To illustrate the potential difficulty of the general problem, we exhibit an infinite ascending chain $\mathbf A_1 \le \mathbf A_2 \le \mathbf A_3 \le ...b$ of finite automatic algebras that are alternately dualizable and non-dualizable.