Prefix monoids of groups and right units of special inverse monoids
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 redu...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2023-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509423000993/type/journal_article |
| _version_ | 1797646959353266176 |
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| author | Igor Dolinka Robert D. Gray |
| author_facet | Igor Dolinka Robert D. Gray |
| author_sort | Igor Dolinka |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-11T15:09:24Z |
| format | Article |
| id | doaj.art-ff6a1ad22d914d67b46c01e5b4564e87 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-03-11T15:09:24Z |
| publishDate | 2023-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-ff6a1ad22d914d67b46c01e5b4564e872023-10-30T01:52:26ZengCambridge University PressForum of Mathematics, Sigma2050-50942023-01-011110.1017/fms.2023.99Prefix monoids of groups and right units of special inverse monoidsIgor Dolinka0https://orcid.org/0000-0002-8644-0626Robert D. Gray1https://orcid.org/0000-0002-7200-1745Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21101 Novi Sad, Serbia; E-mail:School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, England; E-mail: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.https://www.cambridge.org/core/product/identifier/S2050509423000993/type/journal_article20M0520F0520M18 |
| spellingShingle | Igor Dolinka Robert D. Gray Prefix monoids of groups and right units of special inverse monoids Forum of Mathematics, Sigma 20M05 20F05 20M18 |
| title | Prefix monoids of groups and right units of special inverse monoids |
| title_full | Prefix monoids of groups and right units of special inverse monoids |
| title_fullStr | Prefix monoids of groups and right units of special inverse monoids |
| title_full_unstemmed | Prefix monoids of groups and right units of special inverse monoids |
| title_short | Prefix monoids of groups and right units of special inverse monoids |
| title_sort | prefix monoids of groups and right units of special inverse monoids |
| topic | 20M05 20F05 20M18 |
| url | https://www.cambridge.org/core/product/identifier/S2050509423000993/type/journal_article |
| work_keys_str_mv | AT igordolinka prefixmonoidsofgroupsandrightunitsofspecialinversemonoids AT robertdgray prefixmonoidsofgroupsandrightunitsofspecialinversemonoids |