The Word Problem for One-relation Monoids: A Survey

Abstract

A. 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 , 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. C 4.2. Applications of A 4.3. An incorrect proof 5. Sporadic results 5.1. Normal forms 5.2. String rewriting systems 5.3. Small overlap conditions 6. Modern and future results ( -present) 6.1. Finite complete rewriting systems 6.2. Special inverse monoids 6.3. The monadic case 6.4. The Collatz conjecture Concluding remarks References

Non-finitely Generated Maximal Subgroups of Context-free Monoids

“…Such monoids were introduced by Tseitin [48], and were extensively studied by Adian [2] and Makanin [33,32]. For a more in-depth analysis of special monoids and their history, we refer the reader to [38,40,42,51]. The set {w i | i ∈ I} is called the set of defining words of M .…”