Small overlap monoids II: automatic structures and normal forms

TL;DR

This paper proves that monoids satisfying the small overlap condition C(4) have efficiently computable normal forms, are automatic, and possess decidable membership problems, extending the understanding of their algebraic and automata-theoretic properties.

Contribution

It establishes that C(4) monoids have regular normal forms, are rational and automatic, and provides linear-time algorithms for computing these forms, advancing the theory of small overlap monoids.

Findings

Normal forms form a regular language

Normal forms can be computed in linear time

C(4) monoids are rational and automatic

Abstract

We show that any finite monoid or semigroup presentation satisfying the small overlap condition C(4) has word problem which is a deterministic rational relation. It follows that the set of lexicographically minimal words forms a regular language of normal forms, and that these normal forms can be computed in linear time. We also deduce that C(4) monoids and semigroups are rational (in the sense of Sakarovitch), asynchronous automatic, and word hyperbolic (in the sense of Duncan and Gilman). From this it follows that C(4) monoids satisfy analogues of Kleene's theorem, and admit decision algorithms for the rational subset and finitely generated submonoid membership problems. We also prove some automata-theoretic results which may be of independent interest.