Uniform decision problems and abstract properties of small overlap monoids

TL;DR

This paper explores the structure of small overlap monoids, providing algorithms for canonical presentations, solving the isomorphism problem for C(2) monoids, and characterizing cancellativity properties with implications for complexity analysis.

Contribution

It introduces a canonical form for C(2) monoids, offers asymptotic estimates for monoid counts, and develops algorithms for presentation transformation and cancellativity decision.

Findings

Every C(2) monoid has an essentially canonical C(2) presentation.

An algorithm transforms any C(m) presentation into a canonical form.

Cancellativity properties are asymptotically detectable in generic cases.

Abstract

We study the way in which the abstract structure of a small overlap monoid is reflected in, and may be algorithmically deduced from, a small overlap presentation. We show that every C(2) monoid admits an essentially canonical C(2) presentation; by counting canonical presentations we obtain asymptotic estimates for the number of non-isomorphic monoids admitting a-generator, k-relation presentations of a given length. We demonstrate an algorithm to transform an arbitrary presentation for a C(m) monoid (m at least 2) into a canonical C(m) presentation, and a solution to the isomorphism problem for C(2) presentations. We also find a simple combinatorial condition on a C(4) presentation which is necessary and sufficient for the monoid presented to be left cancellative. We apply this to obtain algorithms to decide if a given C(4) monoid is left cancellative, right cancellative or cancellative, and to show that cancellativity properties are asymptotically visible in the sense of generic-case complexity.