The Word Problem for One-relation Monoids: A Survey

TL;DR

This survey reviews the longstanding open problem of the word problem in one-relation monoids, covering historical context, key proofs, partial solutions, recent developments, and addressing misconceptions in the literature.

Contribution

It provides a comprehensive overview of the problem, clarifies previous inaccuracies, and introduces new insights linking inverse monoids to one-relation monoids.

Findings

Reduction to the left cancellative case

Partial solutions for specific subclasses

Recent progress and connections to the Collatz conjecture

Abstract

This survey is intended to provide an overview of one of the oldest and most celebrated open problems in combinatorial algebra: the word problem for one-relation monoids. We provide a history of the problem starting in 1914, and give a detailed overview of the proofs of central results, especially those due to Adian and his student Oganesian. After showing how to reduce the problem to the left cancellative case, the second half of the survey focuses on various methods for solving partial cases in this family. We finish with some modern and very recent results pertaining to this problem, including a link to the Collatz conjecture. Along the way, we emphasise and address a number of incorrect and inaccurate statements that have appeared in the literature over the years. We also fill a gap in the proof of a theorem linking special inverse monoids to one-relation monoids, and slightly strengthen the statement of this theorem.