Counting monogenic monoids and inverse monoids

L.  Elliott; A. Levine; James David Mitchell

doi:10.1080/00927872.2023.2214821

Counting monogenic monoids and inverse monoids

L. Elliott, A. Levine, James David Mitchell^*

^*Corresponding author for this work

Pure Mathematics

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Abstract

In this short note, we show that the number of monogenic submonoids of the full transformation monoid of degree n for n>0, equals the sum of the number of cyclic subgroups of the symmetric groups on 1 to n points. We also prove an analogous statement for monogenic subsemigroups of the finite full transformation monoids, as well as monogenic inverse submonoids and subsemigroups of the finite symmetric inverse monoids.

Original language	English
Pages (from-to)	4654-4661
Number of pages	8
Journal	Communications in Algebra
Volume	51
Issue number	11
Early online date	23 May 2023
DOIs	https://doi.org/10.1080/00927872.2023.2214821
Publication status	Published - 1 Nov 2023

Keywords

Inverse semigroup
Monogenic semigroup
Transformation semigroup

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10.1080/00927872.2023.2214821Licence: CC BY

Elliott_2023_CiA_Counting-monogenic_CC
Copyright © 2023 The Author(s). Published with license by Taylor and Francis Group, LLC.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Final published version, 1.02 MBLicence: CC BY

Cite this

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note = "Funding: The second and third authors would like to thank the London Mathematical Society, the Heilbronn Institute for Mathematical Research, and the University of St Andrews, for their support of this work.",

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AU - Elliott, L.

AU - Levine, A.

AU - Mitchell, James David

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N2 - In this short note, we show that the number of monogenic submonoids of the full transformation monoid of degree n for n>0, equals the sum of the number of cyclic subgroups of the symmetric groups on 1 to n points. We also prove an analogous statement for monogenic subsemigroups of the finite full transformation monoids, as well as monogenic inverse submonoids and subsemigroups of the finite symmetric inverse monoids.

AB - In this short note, we show that the number of monogenic submonoids of the full transformation monoid of degree n for n>0, equals the sum of the number of cyclic subgroups of the symmetric groups on 1 to n points. We also prove an analogous statement for monogenic subsemigroups of the finite full transformation monoids, as well as monogenic inverse submonoids and subsemigroups of the finite symmetric inverse monoids.

KW - Inverse semigroup

KW - Monogenic semigroup

KW - Transformation semigroup

U2 - 10.1080/00927872.2023.2214821

DO - 10.1080/00927872.2023.2214821

M3 - Article

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VL - 51

SP - 4654

EP - 4661

JO - Communications in Algebra

JF - Communications in Algebra

IS - 11

ER -

Counting monogenic monoids and inverse monoids

Abstract

Keywords

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