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Comptes Rendus. Mathématique
Geometry and Topology, Group theory
Free inverse monoids are not FP 2
Comptes Rendus. Mathématique, Volume 359 (2021) no. 8, pp. 1047-1057.

We give a topological proof that a free inverse monoid on one or more generators is neither of type left-FP 2 nor right-FP 2 . This strengthens a classical result of Schein that such monoids are not finitely presented as monoids.

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DOI: 10.5802/crmath.247
Classification: 20M50,  20M18,  20M05,  20J05
Robert D. Gray 1; Benjamin Steinberg 2

1 School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK
2 Department of Mathematics, City College of New York, Convent Avenue at 138th Street, New York, New York 10031, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Free inverse monoids are not ${\protect \rm FP}_2$},
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Robert D. Gray; Benjamin Steinberg. Free inverse monoids are not ${\protect \rm FP}_2$. Comptes Rendus. Mathématique, Volume 359 (2021) no. 8, pp. 1047-1057. doi : 10.5802/crmath.247. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.247/

[1] Kenneth S. Brown Cohomology of groups, Graduate Texts in Mathematics, 87, Springer, 1994

[2] Lazar M. Gluskin Elementary generalized groups, Mat. Sb., N. Ser., Volume 41 (1957) no. 83, pp. 23-36 | MR | Zbl

[3] Robert D. Gray; Benjamin Steinberg A Lyndon’s identity theorem for one-relator monoids (2019) https://arxiv.org/abs/1910.09914

[4] Peter M. Higgins Techniques of semigroup theory, Oxford Science Publications, Oxford University Press, 1992 | Zbl

[5] John M. Howie Fundamentals of semigroup theory, London Mathematical Society Monographs, 12, Clarendon Press, 1995 | MR | Zbl

[6] Sergei V. Ivanov Relation modules and relation bimodules of groups, semigroups and associative algebras, Int. J. Algebra Comput., Volume 1 (1991) no. 1, pp. 89-114 | DOI | MR

[7] Mark Lawson Inverse semigroups. The theory of partial symmetries, World Scientific, 1998 | Zbl

[8] Walter D. Munn Free inverse semigroups, Proc. Lond. Math. Soc., Volume 29 (1974), pp. 385-404 | DOI | MR

[9] Friedrich Otto; Yuji Kobayashi Properties of monoids that are presented by finite convergent string-rewriting systems—a survey, Advances in algorithms, languages, and complexity, Kluwer Academic Publishers, 1997, pp. 225-266 | DOI | Zbl

[10] Mario Petrich Inverse semigroups, Pure and Applied Mathematics, John Wiley & Sons, 1984 | Zbl

[11] Stephen J. Pride Homological finiteness conditions for groups, monoids, and algebras, Commun. Algebra, Volume 34 (2006) no. 10, pp. 3525-3536 | DOI | MR | Zbl

[12] H. E. Scheiblich Free inverse semigroups, Proc. Am. Math. Soc., Volume 38 (1973), pp. 1-7 | DOI | MR | Zbl

[13] Boris M. Schein Free inverse semigroups are not finitely presentable, Acta Math. Acad. Sci. Hung., Volume 26 (1975), pp. 41-52 | DOI | MR | Zbl

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