[Paper Review] Automaton semigroups: new construction results and examples of non-automaton semigroups

Tara Brough, Alan J. Cain|arXiv (Cornell University)|Jan 6, 2016

semigroups and automata theory10 references3 citations

TL;DR

This paper establishes that free products of finite semigroups and wreath products of automaton monoids with finite monoids are always automaton semigroups, resolving open questions about closure under these constructions. It further proves that no nontrivial subsemigroup of (N, +) or (N₀, +) is an automaton semigroup, providing the first known examples of finitely generated, residually finite semigroups with solvable word problem that are not automaton semigroups.

ABSTRACT

This paper studies the class of automaton semigroups from two perspectives: closure under constructions, and examples of semigroups that are not automaton semigroups. We prove that (semigroup) free products of finite semigroups always arise as automaton semigroups, and that the class of automaton monoids is closed under forming wreath products with finite monoids. We also consider closure under certain kinds of Rees matrix constructions, strong semilattices, and small extensions. Finally, we prove that no subsemigroup of $(\mathbb{N}, +)$ arises as an automaton semigroup. (Previously, $(\mathbb{N},+)$ itself was the unique example of a finitely generated residually finite semigroup that was known not to arise as an automaton semigroup.)

Motivation & Objective

• To resolve open questions about whether automaton semigroups are closed under free products and wreath products with finite monoids.
• To investigate whether certain semigroup constructions—such as Rees matrix constructions, strong semilattices, and small extensions—preserve the automaton semigroup property.
• To identify new examples of finitely generated, residually finite semigroups with solvable word problem that are not automaton semigroups, addressing a long-standing gap in the theory.
• To provide a general technique for proving that certain semigroups are not automaton semigroups, particularly focusing on subsemigroups of (N, +).

Proposed method

• Constructing automaton realizations for free products of finite semigroups using state mappings and wreath recursion techniques.
• Proving closure under wreath products by showing that S ≀ T, where S is an automaton monoid and T is finite, embeds as an automaton monoid via recursive state behavior.
• Using wreath recursion analysis to detect periodic elements in hypothetical automaton realizations of subsemigroups of (N₀, +).
• Applying Lemma 14, which states that if a state recurses only to itself and a zero state, it represents a periodic element.
• Analyzing the structure of wreath recursions for powers of states (e.g., qₖˡ and qₗᵏ) to derive contradictions when assuming the existence of a finite automaton for a nontrivial subsemigroup of (N₀, +).
• Using the fact that subsemigroups of (N, +) are finitely generated and large subsemigroups of (N, +), to show that such semigroups cannot be automaton semigroups due to periodicity constraints in automaton realizations.

Experimental results

Research questions

• RQ1 Are free products of finite semigroups always automaton semigroups?
• RQ2 Is the class of automaton monoids closed under wreath products with finite monoids?
• RQ3 Can Rees matrix constructions over automaton semigroups yield automaton semigroups under certain conditions?
• RQ4 Are strong semilattices of automaton semigroups themselves automaton semigroups?
• RQ5 Are small extensions of automaton semigroups necessarily automaton semigroups? In particular, can a non-automaton semigroup become an automaton semigroup upon adjoining a zero?

Key findings

• Free products of finite semigroups are always automaton semigroups, resolving an open problem and extending the result that free products of finite groups are automaton groups.
• Wreath products of automaton monoids with finite monoids are always automaton monoids, providing a complete answer to a previously partial result.
• No nontrivial subsemigroup of (N₀, +) — including any subsemigroup of (N, +) — is an automaton semigroup, establishing the first known examples of finitely generated, residually finite semigroups with solvable word problem that are not automaton semigroups.
• The proof relies on showing that any hypothetical automaton realizing a nontrivial subsemigroup of (N₀, +) would require a state to represent a periodic element, contradicting the fact that such semigroups have no nontrivial periodic elements.
• The result implies that subsemigroups of (N, +) cannot become automaton semigroups even upon adjoining a zero, ruling out a potential positive answer to an open question about such extensions.
• A general technique based on wreath recursion and periodicity detection is proposed as a tool for proving that certain semigroups are not automaton semigroups, suggesting a possible 'pumping lemma'-like criterion for future work.