On groups of units of special and one-relator inverse monoids

Gray, Robert D. and Ruškuc, Nik (2023) On groups of units of special and one-relator inverse monoids. Journal of the Institute of Mathematics of Jussieu. ISSN 1474-7480

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Abstract

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.

Item Type:	Article
Additional Information:	Funding information: 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”.
Faculty \ School:	Faculty of Science > School of Mathematics
UEA Research Groups:	Faculty of Science > Research Groups > Logic Faculty of Science > Research Groups > Algebra and Combinatorics
Depositing User:	LivePure Connector
Date Deposited:	11 Oct 2023 01:16
Last Modified:	25 Nov 2023 03:18
URI:	https://ueaeprints.uea.ac.uk/id/eprint/93241
DOI:	10.1017/S1474748023000439

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