Prefix monoids of groups and right units of special inverse monoids

Dolinka, Igor and Gray, Robert D. (2023) Prefix monoids of groups and right units of special inverse monoids. Forum of Mathematics, Sigma, 11. ISSN 2050-5094

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A prefix monoid is a finitely generated submonoid of a finitely presented group generated by the prefixes of its defining relators. Important results of Guba (1997), and of Ivanov, Margolis and Meakin (2001), show how the word problem for certain one-relator monoids, and inverse monoids, can be reduced to solving the membership problem in prefix monoids of certain one-relator groups. Motivated by this, in this paper, we study the class of prefix monoids of finitely presented groups. We obtain a complete description of this class of monoids. All monoids in this family are finitely generated, recursively presented and group-embeddable. Our results show that not every finitely generated recursively presented group-embeddable monoid is a prefix monoid, but for every such monoid, if we take a free product with a suitably chosen free monoid of finite rank, then we do obtain a prefix monoid. Conversely, we prove that every prefix monoid arises in this way. Also, we show that the groups that arise as groups of units of prefix monoids are precisely the finitely generated recursively presented groups, whereas the groups that arise as Schützenberger groups of prefix monoids are exactly the recursively enumerable subgroups of finitely presented groups. We obtain an analogous result classifying the Schützenberger groups of monoids of right units of special inverse monoids. We also give some examples of right cancellative monoids arising as monoids of right units of finitely presented special inverse monoids, and we show that not all right cancellative recursively presented monoids belong to this class.

Item Type: Article
Additional Information: Funding statement: The research of the first named author is supported by the Personal Grant F-121 ‘Problems of combinatorial semigroup and group theory’ of the Serbian Academy of Sciences and Arts. The research of the second named author was supported by the EPSRC Fellowship Grant 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
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Depositing User: LivePure Connector
Date Deposited: 07 Oct 2023 01:24
Last Modified: 13 Dec 2023 01:58
DOI: 10.1017/fms.2023.99


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