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Preserving self-similarity in free products of semigroups

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arXiv:2003.12810v3 Announce Type: replace-cross
Abstract: We improve on earlier results on the closure under free products of the class of automaton semigroups. We consider partial automata and show that the free product of two self-similar semigroups (or automaton semigroups) is self-similar (an automaton semigroup) if there is a homomorphism from one of the base semigroups to the other. The construction used is computable and yields further consequences. One of them is that we can adjoin a free generator to any self-similar semigroup (or automaton semigroup) and preserve the property of self-similarity (or being an automaton semigroup).
The existence of a homomorphism between two semigroups is a very lax requirement; in particular, it is satisfied if one of the semigroups contains an idempotent. To explore the limits of this requirement, we show that no simple or $0$-simple idempotent-free semigroup is a finitely generated self-similar semigroup (or an automaton semigroup). Furthermore, we give an example of a pair of residually finite semigroups without a homomorphism from one to the other.

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