The word problem for one-relation monoids: a survey

Nyberg Brodda, Carl-Fredrik (2021) The word problem for one-relation monoids: a survey. Semigroup Forum, 103 (2). 297–355. ISSN 0037-1912

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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.

Item Type: Article
Uncontrolled Keywords: decidability,one-relation monoid,semigroups,word problem,algebra and number theory ,/dk/atira/pure/subjectarea/asjc/2600/2602
Faculty \ School: Faculty of Science > School of Mathematics
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Depositing User: LivePure Connector
Date Deposited: 12 Aug 2021 00:15
Last Modified: 07 Sep 2021 00:21
URI: https://ueaeprints.uea.ac.uk/id/eprint/81061
DOI: 10.1007/s00233-021-10216-8

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