From small-overlap conditions to automatic semigroups

TL;DR

This paper explores how small-overlap conditions relate to the automaticity of semigroups, providing a method to construct automatic structures under certain hyperbolic-like conditions and highlighting limitations of naive approaches.

Contribution

It introduces a construction of automatic structures for semigroups satisfying specific small-overlap conditions, advancing understanding of their geometric and algebraic properties.

Findings

Automatic structures can be constructed under certain small-overlap conditions.

Naive geodesic-based approaches are insufficient for these semigroups.

Conditions imply embeddability and decomposition into at least seven pieces.

Abstract

We study the connection between small-overlap conditions and automaticity of semigroups. We restrict the discussion to conditions that imply embeddability and under which each relation decomposes into at least seven pieces. For these hyperbolic-like conditions we show how to construct an automatic structure. Furthermore, we show that the naive approach of considering just geodesics fails in our case.