Automatic semigroups vs automaton semigroups

TL;DR

This paper introduces a framework connecting automatic semigroups with automaton semigroups using Mealy automata, unifying various classes of well-known semigroups and exploring their automata-theoretic properties.

Contribution

It develops a natural approach to interpret automatic semigroups as automaton semigroups, establishing a novel connection between these classes through Mealy automata.

Findings

Unified framework for automatic and automaton semigroups

First known connection from automatic to automaton semigroups

Dual properties of automatic and automaton semigroups in automata theory

Abstract

We develop an effective and natural approach to interpret any semigroup admitting a special language of greedy normal forms as an automaton semigroup,namely the semigroup generated by a Mealy automaton encoding the behaviour of such a language of greedy normal forms under one-sided multiplication.The framework embraces many of the well-known classes of (automatic) semigroups: finite monoids, free semigroups, free commutative monoids, trace or divisibility monoids, braid or Artin-Tits or Krammer or Garside monoids, Baumslag-Solitar semigroups, etc.Like plactic monoids or Chinese monoids, some neither left- nor right-cancellative automatic semigroups are also investigated, as well as some residually finite variations of the bicyclic monoid. It provides what appears to be the first known connection from a class of automatic semigroupsto a class of automaton semigroups. It is worthwhile noting that, in all these cases, "being an automatic semigroup" and "being an automaton semigroup" become dual properties in a very automata-theoretical sense. Quadratic rewriting systems and associated tilings appear as a cornerstone of our construction.