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Automaton monoids vs automaton semigroups that are monoids

Determine whether the class of automaton monoids coincides with the class of automaton semigroups that happen to be monoids, i.e., prove or refute that every automaton semigroup with a neutral element has that identity represented by some state sequence acting as the identity.

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Background

The authors highlight a subtle difference: an automaton monoid requires a state sequence acting as the identity, whereas an automaton semigroup that is a monoid may have a neutral element not arising from such a sequence.

Whether these two notions coincide remains unknown; resolving it would align the theory of automaton monoids with the more settled situation in automaton groups.

References

In fact, it is not known whether the two classes coincide (which contrasts the situations with automaton groups where it is known that every automaton semigroup that happens to be a group is an automaton group, see [Proposition~3.1] for the required construction).

— The Freeness Problem for Automaton Semigroups (2402.01372 - D'Angeli et al., 2 Feb 2024) in Preliminaries, Automaton Semigroups and Monoids, Remark (second)