A characterisation of semigroups with only countably many subdirect products with Z

Ashley Michael William Clayton; Catherine Reilly; Nik Ruskuc

doi:10.1017/S0004972724000613

A characterisation of semigroups with only countably many subdirect products with Z

Ashley Michael William Clayton
, Catherine Reilly
, Nik Ruskuc^*

^*Corresponding author for this work

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

Abstract

Let Z be the additive (semi)group of integers. We prove that for a finite semigroup S the direct product Z × S contains only countably many subdirect products (up to isomorphism) if and only if S is regular. As a corollary we show that Z × S has only countably many subsemigroups (up to isomorphism) if and only if S is completely regular.

Original language	English
Number of pages	10
Journal	Bulletin of the Australian Mathematical Society
Volume	First View
Early online date	4 Oct 2024
DOIs	https://doi.org/10.1017/S0004972724000613
Publication status	E-pub ahead of print - 4 Oct 2024

Keywords

Direct product
Subdirect product
Subsemigroup
Regular semigroup
Completely regular semigroup

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10.1017/S0004972724000613Licence: CC BY

Clayton_2024_BoAMS_Characterisation-semigroups-with-only-countably_AAM_CCBY
Copyright © 2024 the Authors. This work has been made available online in accordance with the University of St Andrews Open Access policy. This accepted manuscript is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. The final published version of this work is available at https://doi.org/10.1017/S0004972724000613.
Accepted author manuscript, 487 KBLicence: CC BY
Clayton_2024_BoAMS_Characterisation-semigroups-with-only-countably-many-subdirect-products-Z_CC
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc. 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, 181 KBLicence: CC BY

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title = "A characterisation of semigroups with only countably many subdirect products with Z",

abstract = "Let Z be the additive (semi)group of integers. We prove that for a finite semigroup S the direct product Z × S contains only countably many subdirect products (up to isomorphism) if and only if S is regular. As a corollary we show that Z × S has only countably many subsemigroups (up to isomorphism) if and only if S is completely regular.",

keywords = "Direct product, Subdirect product, Subsemigroup, Regular semigroup, Completely regular semigroup",

author = "Clayton, \{Ashley Michael William\} and Catherine Reilly and Nik Ruskuc",

year = "2024",

month = oct,

day = "4",

doi = "10.1017/S0004972724000613",

language = "English",

volume = "First View",

journal = "Bulletin of the Australian Mathematical Society",

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TY - JOUR

T1 - A characterisation of semigroups with only countably many subdirect products with Z

AU - Clayton, Ashley Michael William

AU - Reilly, Catherine

AU - Ruskuc, Nik

PY - 2024/10/4

Y1 - 2024/10/4

N2 - Let Z be the additive (semi)group of integers. We prove that for a finite semigroup S the direct product Z × S contains only countably many subdirect products (up to isomorphism) if and only if S is regular. As a corollary we show that Z × S has only countably many subsemigroups (up to isomorphism) if and only if S is completely regular.

AB - Let Z be the additive (semi)group of integers. We prove that for a finite semigroup S the direct product Z × S contains only countably many subdirect products (up to isomorphism) if and only if S is regular. As a corollary we show that Z × S has only countably many subsemigroups (up to isomorphism) if and only if S is completely regular.

KW - Direct product

KW - Subdirect product

KW - Subsemigroup

KW - Regular semigroup

KW - Completely regular semigroup

UR - https://doi.org/10.48550/arXiv.2404.18122

U2 - 10.1017/S0004972724000613

DO - 10.1017/S0004972724000613

M3 - Article

SN - 0004-9727

VL - First View

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

ER -

A characterisation of semigroups with only countably many subdirect products with Z

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Keywords

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