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Categorically closed countable semigroups

  • Taras Banakh ORCID logo and Serhii Bardyla ORCID logo EMAIL logo
Published/Copyright: February 28, 2023

Abstract

In this paper, we establish a connection between categorical closedness and nontopologizability of semigroups. In particular, for the class 𝖳 𝟣 𝖲 of T 1 topological semigroups we prove that a countable semigroup X with finite-to-one shifts is injectively 𝖳 𝟣 𝖲 -closed if and only if X is 𝖳 𝟣 𝖲 -discrete in the sense that every T 1 semigroup topology on X is discrete. Moreover, a countable cancellative semigroup X is absolutely 𝖳 𝟣 𝖲 -closed if and only if every homomorphic image of X is 𝖳 𝟣 𝖲 -discrete. Also, we introduce and investigate a new notion of a polybounded semigroup. It is proved that a countable semigroup X with finite-to-one shifts is polybounded if and only if X is 𝖳 𝟣 𝖲 -closed if and only if X is 𝖳 𝗓 𝖲 -closed, where 𝖳 𝗓 𝖲 is the class of Tychonoff zero-dimensional topological semigroups. We show that polybounded cancellative semigroups are groups, and polybounded T 1 paratopological groups are topological groups.

MSC 2010: 22A15; 20M18

Communicated by Manfred Droste


Funding source: Austrian Science Fund

Award Identifier / Grant number: M 2967

Award Identifier / Grant number: V844

Award Identifier / Grant number: P33895

Funding statement: The second author was supported by the Austrian Science Fund FWF (Grants M 2967, V844, P33895).

Acknowledgements

The authors express their sincere thanks to the referee for many valuable suggestions which helped the authors to improve essentially the final version of the paper.

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Received: 2022-04-11
Revised: 2022-12-12
Published Online: 2023-02-28
Published in Print: 2023-05-01

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