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ON THE NUMBER OF SUBSEMIGROUPS OF DIRECT PRODUCTS INVOLVING THE FREE MONOGENIC SEMIGROUP

Part of: Semigroups

Published online by Cambridge University Press:  01 February 2019

ASHLEY CLAYTON Open the ORCID record for ASHLEY CLAYTON [Opens in a new window]  and
NIK RUŠKUC Open the ORCID record for NIK RUŠKUC [Opens in a new window]
ASHLEY CLAYTON
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, UK email [email protected]
NIK RUŠKUC*
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, UK email [email protected]
*
✉[email protected]
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Abstract

The direct product $\mathbb{N}\times \mathbb{N}$ of two free monogenic semigroups contains uncountably many pairwise nonisomorphic subdirect products. Furthermore, the following hold for $\mathbb{N}\times S$, where $S$ is a finite semigroup. It contains only countably many pairwise nonisomorphic subsemigroups if and only if $S$ is a union of groups. And it contains only countably many pairwise nonisomorphic subdirect products if and only if every element of $S$ has a relative left or right identity element.

Keywords

subdirect productsubsemigroupfree monogenic semigroup

MSC classification

Primary: 20M99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 109 , Issue 1 , August 2020 , pp. 24 - 35
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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