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

Robert D. Gray, Nik Ruskuc*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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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.
Original languageEnglish
Number of pages44
JournalJournal of the Institute of Mathematics of Jussieu
VolumeFirstView
Early online date21 Nov 2023
DOIs
Publication statusE-pub ahead of print - 21 Nov 2023

Keywords

  • One-relator monoid
  • One-relator group
  • Inverse monoid
  • Special inverse monoid
  • Units
  • Right units
  • Coherence

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