Small overlap monoids: the word problem

TL;DR

This paper introduces a combinatorial method for solving the word problem in small overlap monoids and semigroups, providing efficient algorithms that outperform previous geometric approaches.

Contribution

It presents a new combinatorial approach that enables linear and polynomial time solutions for the word problem under small overlap conditions, improving computational efficiency.

Findings

Linear time solution for the word problem under condition C(4)

Polynomial time solution for the uniform word problem under the same condition

Practical algorithms outperform geometric methods

Abstract

We develop a combinatorial approach to the study of semigroups and monoids with finite presentations satisfying small overlap conditions. In contrast to existing geometric methods, our approach facilitates a sequential left-right analysis of words which lends itself to the development of practical, efficient computational algorithms. In particular, we obtain a highly practical linear time solution to the word problem for monoids and semigroups with finite presentations satisfying the condition C(4), and a polynomial time solution to the uniform word problem for presentations satisfying the same condition.