Core Concepts
The free product of two self-similar or automaton semigroups is self-similar (an automaton semigroup) if there exists a homomorphism from one of the base semigroups to the other.
Abstract
The paper investigates the closure properties of self-similar and automaton semigroups under free products. The main result shows that the free product of two self-similar or automaton semigroups S and T is self-similar (an automaton semigroup) if there exists a homomorphism from one of the base semigroups to the other.
The key steps are:
The authors construct an automaton C that generates the free product S ⋆ T, where S and T are self-similar semigroups with a homomorphism between them. The construction uses a combination of marked and unmarked letters to distinguish the elements of the free product.
They prove that the action of S ⋆ T on the language generated by C is faithful, showing that C indeed generates S ⋆ T as a self-similar semigroup.
The authors also show that the constructed automaton C is finite (and complete) if the original automata generating S and T were finite (and complete), respectively. This implies that the free product of two automaton semigroups is also an automaton semigroup under the same homomorphism condition.
The existence of a homomorphism between the base semigroups is a very weak requirement, satisfied for example if one of the semigroups contains an idempotent. The authors explore the limits of this condition and show that no simple or 0-simple idempotent-free semigroup is a finitely generated self-similar or automaton semigroup.
As a byproduct, the authors show that a new free generator can be adjoined to any self-similar or automaton semigroup without losing the property of self-similarity or being an automaton semigroup.