When are dual Cayley automaton semigroups finite?

TL;DR

This paper characterizes the finiteness of dual Cayley automaton semigroups for finite semigroups, showing it depends on specific algebraic properties such as being e9-trivial and lacking non-trivial right zero subsemigroups.

Contribution

It provides a complete algebraic characterization of when dual Cayley automaton semigroups are finite for finite semigroups.

Findings

Dual Cayley automaton semigroup is finite iff the semigroup is e9-trivial and has no non-trivial right zero subsemigroups.

The paper establishes necessary and sufficient conditions for finiteness.

It advances understanding of automaton semigroup structures related to finite semigroups.

Abstract

In this note we prove that, for a finite semigroup $S$, the dual Cayley automaton semigroup $\mathbf{C^{\ast}}(S)$ is finite if and only if $S$ is $\mathcal{H}$-trivial and has no non-trivial right zero subsemigroups.