The algebra of rewriting for presentations of inverse monoids

N. D. Gilbert, E. A. McDougall

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We describe a formalism, using groupoids, for the study of rewriting for presentations of inverse monoids, that is based on the Squier complex construction for monoid presentations. We introduce the class of pseudoregular groupoids, an example of which now arises as the fundamental groupoid of our version of the Squier complex. A further key ingredient is the factorization of the presentation map from a free inverse monoid as the composition of an idempotent pure map and an idempotent separating map. The relation module of a presentation is then defined as the abelianized kernel of this idempotent separating map. We then use the properties of idempotent separating maps to derive a free presentation of the relation module. The construction of its kernel–the module of identities–uses further facts about pseudoregular groupoids.

Original languageEnglish
Pages (from-to)2920-2940
Number of pages21
JournalCommunications in Algebra
Issue number7
Early online date17 Feb 2020
Publication statusPublished - 2 Jul 2020


  • Crossed module
  • groupoid
  • inverse monoid
  • presentation

ASJC Scopus subject areas

  • Algebra and Number Theory


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