Preserving self-similarity in free products of semigroups

2020-03-28

AI Summary

This paper investigates the closure of self-similar semigroups (and automaton semigroups) under free products. When is the free product of two self-similar semigroups itself self-similar—or realizable by an automaton? Method: We establish a decidable criterion based on the existence of homomorphisms between base semigroups and develop a computable algorithm for constructing free product realizations. Contributions: (1) We prove that the homomorphism condition is necessary and sufficient for preserving self-similarity under free products. (2) We show that arbitrary generators can be freely adjoined to any self-similar semigroup without violating self-similarity. (3) We characterize the boundary of the condition, disproving self-similarity for several idempotent-free semigroups. (4) We extend closure results to automaton semigroups. (5) We prove that every idempotent-containing semigroup satisfies the homomorphism condition and explicitly construct finite-generated, residually finite semigroup pairs admitting no mutual homomorphisms.

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.

Problem

• Extending self-similarity preservation in semigroup free products
• Establishing homomorphism conditions for automaton semigroup closure
• Proving limitations for idempotent-free self-similar semigroups

Innovation

• Free product closure via homomorphism condition
• Computable construction for self-similarity preservation
• Adjoining free generators while maintaining automaton properties