On groups of units of special and one-relator inverse monoids
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We investigate the groups of units of one-relator and special inverse monoids. These are inverse monoids which are defined by presentations, where all the defining relations are of the form r=1. We develop new approaches for finding presentations for the group of units of a special inverse monoid, and apply these methods to give conditions under which the group admits a presentation with the same number of defining relations as the monoid. In particular, our results give sufficient conditions for the group of units of a one-relator inverse monoid to be a one-relator group. When these conditions are satisfied, these results give inverse semigroup theoretic analogues of classical results of Adjan for one-relator monoids, and Makanin for special monoids. In contrast, we show that in general these classical results do not hold for one-relator and special inverse monoids. In particular, we show that there exists a one-relator special inverse monoid whose group of units is not a one-relator group (with respect to any generating set), and we show that there exists a finitely presented special inverse monoid whose group of units is not finitely presented.
Gray , R D & Ruskuc , N 2023 , ' On groups of units of special and one-relator inverse monoids ' , Journal of the Institute of Mathematics of Jussieu , vol. FirstView . https://doi.org/10.1017/S1474748023000439
Journal of the Institute of Mathematics of Jussieu
Copyright © The Author(s), 2023. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
DescriptionFunding: This research of R. D. Gray was supported by the Engineering and Physical Sciences Research Council projects EP/N033353/1 “Special inverse monoids: subgroups, structure, geometry, rewriting systems and the word problem”, and EP/V032003/1 ‘’Algorithmic, topological and geometric aspects of infinite groups, monoids and inverse semigroups”.
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