On finite presentability of subsemigroups of the monogenic free inverse semigroup

Jung Won Cho; Nik Ruskuc

doi:10.1017/S0017089524000314

On finite presentability of subsemigroups of the monogenic free inverse semigroup

Jung Won Cho, Nik Ruskuc^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

The monogenic free inverse semigroup FI₁ is not finitely presented as a semigroup due to the classic result by Schein (1975). We extend this result and prove that a finitely generated subsemigroup of FI₁ is finitely presented if and only if it contains only finitely many idempotents. As a consequence, we derive that an inverse subsemigroup of FI₁ is finitely presented as a semigroup if and only if it is a finite semilattice.

Original language	English
Number of pages	12
Journal	Glasgow Mathematical Journal
Volume	FirstView
Early online date	28 Nov 2024
DOIs	https://doi.org/10.1017/S0017089524000314
Publication status	E-pub ahead of print - 28 Nov 2024

Keywords

Free inverse semigroup
Monogenic
Subsemigroup
Finite presentation
Idempotents

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10.1017/S0017089524000314Licence: CC BY

Cho_2024_GMJ_On-finite-presentability-subsemigroups_CC
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Final published version, 374 KBLicence: CC BY

Cite this

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title = "On finite presentability of subsemigroups of the monogenic free inverse semigroup",

abstract = "The monogenic free inverse semigroup FI1 is not finitely presented as a semigroup due to the classic result by Schein (1975). We extend this result and prove that a finitely generated subsemigroup of FI1 is finitely presented if and only if it contains only finitely many idempotents. As a consequence, we derive that an inverse subsemigroup of FI1 is finitely presented as a semigroup if and only if it is a finite semilattice.",

keywords = "Free inverse semigroup, Monogenic, Subsemigroup, Finite presentation, Idempotents",

author = "Cho, {Jung Won} and Nik Ruskuc",

note = "Funding: The research is supported by the Engineering and Physical Sciences Research Council [EP/V003224/1].",

year = "2024",

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doi = "10.1017/S0017089524000314",

language = "English",

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T1 - On finite presentability of subsemigroups of the monogenic free inverse semigroup

AU - Cho, Jung Won

AU - Ruskuc, Nik

N1 - Funding: The research is supported by the Engineering and Physical Sciences Research Council [EP/V003224/1].

PY - 2024/11/28

Y1 - 2024/11/28

N2 - The monogenic free inverse semigroup FI1 is not finitely presented as a semigroup due to the classic result by Schein (1975). We extend this result and prove that a finitely generated subsemigroup of FI1 is finitely presented if and only if it contains only finitely many idempotents. As a consequence, we derive that an inverse subsemigroup of FI1 is finitely presented as a semigroup if and only if it is a finite semilattice.

AB - The monogenic free inverse semigroup FI1 is not finitely presented as a semigroup due to the classic result by Schein (1975). We extend this result and prove that a finitely generated subsemigroup of FI1 is finitely presented if and only if it contains only finitely many idempotents. As a consequence, we derive that an inverse subsemigroup of FI1 is finitely presented as a semigroup if and only if it is a finite semilattice.

KW - Free inverse semigroup

KW - Monogenic

KW - Subsemigroup

KW - Finite presentation

KW - Idempotents

UR - https://doi.org/10.48550/arXiv.2404.05244

U2 - 10.1017/S0017089524000314

DO - 10.1017/S0017089524000314

M3 - Article

SN - 0017-0895

VL - FirstView

JO - Glasgow Mathematical Journal

JF - Glasgow Mathematical Journal

ER -

On finite presentability of subsemigroups of the monogenic free inverse semigroup

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Keywords

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Right Noetherian and coherent monoids: Right Noetherian and coherent monoids

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On finite presentability of subsemigroups of the monogenic free inverse semigroup

Abstract

Keywords

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Right Noetherian and coherent monoids: Right Noetherian and coherent monoids

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