On the number of subdirect products involving semigroups of integers and natural numbers

Ashley Clayton; Catherine Reilly; Nik Ruskuc

doi:10.1142/S0219498826501379

On the number of subdirect products involving semigroups of integers and natural numbers

Ashley Clayton, Catherine Reilly, Nik Ruskuc^*

^*Corresponding author for this work

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

Abstract

In this paper, we extend a recent result that for the (additive) semigroup of positive integers ℕ, there are continuum many subdirect products of ℕ × ℕ up to isomorphism. We prove that for U, V each one of
ℤ (the group of integers), ℕ₀(the monoid of non-negative integers), or ℕ, the direct product U × V contains continuum many (semigroup) subdirect products up to isomorphism.

Original language	English
Journal	Journal of Algebra and Its Applications
Volume	Online Ready
Early online date	18 Feb 2025
DOIs	https://doi.org/10.1142/S0219498826501379
Publication status	E-pub ahead of print - 18 Feb 2025

Keywords

Semigroup
Natural number
Integer
Subdirect product
Indecomposable element

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10.1142/S0219498826501379

Clayton_2025_JAA_On-number-subdirect-products-semigroups_AAM_CCBY
Copyright © 2025 World Scientific / the Author(s). This work has been made available online in accordance with publisher policies or with permission. 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/10.1142/S0219498826501379
Accepted author manuscript, 325 KBLicence: CC BY

https://arxiv.org/abs/2311.04994Licence: CC BY

Cite this

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title = "On the number of subdirect products involving semigroups of integers and natural numbers",

abstract = "In this paper, we extend a recent result that for the (additive) semigroup of positive integers ℕ, there are continuum many subdirect products of ℕ × ℕ up to isomorphism. We prove that for U, V each one of ℤ (the group of integers), ℕ0 (the monoid of non-negative integers), or ℕ, the direct product U × V contains continuum many (semigroup) subdirect products up to isomorphism.",

keywords = "Semigroup, Natural number, Integer, Subdirect product, Indecomposable element",

author = "Ashley Clayton and Catherine Reilly and Nik Ruskuc",

note = "Funding: The third author acknowledges support from EPSRC EP/V003224/1.",

year = "2025",

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day = "18",

doi = "10.1142/S0219498826501379",

language = "English",

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AU - Clayton, Ashley

AU - Reilly, Catherine

AU - Ruskuc, Nik

N1 - Funding: The third author acknowledges support from EPSRC EP/V003224/1.

PY - 2025/2/18

Y1 - 2025/2/18

N2 - In this paper, we extend a recent result that for the (additive) semigroup of positive integers ℕ, there are continuum many subdirect products of ℕ × ℕ up to isomorphism. We prove that for U, V each one of ℤ (the group of integers), ℕ0 (the monoid of non-negative integers), or ℕ, the direct product U × V contains continuum many (semigroup) subdirect products up to isomorphism.

AB - In this paper, we extend a recent result that for the (additive) semigroup of positive integers ℕ, there are continuum many subdirect products of ℕ × ℕ up to isomorphism. We prove that for U, V each one of ℤ (the group of integers), ℕ0 (the monoid of non-negative integers), or ℕ, the direct product U × V contains continuum many (semigroup) subdirect products up to isomorphism.

KW - Semigroup

KW - Natural number

KW - Integer

KW - Subdirect product

KW - Indecomposable element

UR - https://www.scopus.com/pages/publications/85218790410

U2 - 10.1142/S0219498826501379

DO - 10.1142/S0219498826501379

M3 - Article

SN - 0219-4988

VL - Online Ready

JO - Journal of Algebra and Its Applications

JF - Journal of Algebra and Its Applications

ER -

On the number of subdirect products involving semigroups of integers and natural numbers

Abstract

Keywords

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Right Noetherian and coherent monoids: Right Noetherian and coherent monoids

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On the number of subdirect products involving semigroups of integers and natural numbers

Abstract

Keywords

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Right Noetherian and coherent monoids: Right Noetherian and coherent monoids

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