Banakh T, Bardyla S. Absolutely closed semigroups.
Rev R Acad Cienc Exactas Fis Nat A Mat 2023;
118:23. [PMID:
37970590 PMCID:
PMC10632307 DOI:
10.1007/s13398-023-01519-2]
[Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/02/2023] [Accepted: 10/09/2023] [Indexed: 11/17/2023]
Abstract
Let C be a class of topological semigroups. A semigroup X is called absolutely C -closed if for any homomorphism h : X → Y to a topological semigroup Y ∈ C , the image h[X] is closed in Y. Let T 1 S , T 2 S , and T z S be the classes of T 1 , Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that a commutative semigroup X is absolutely T z S -closed if and only if X is absolutely T 2 S -closed if and only if X is chain-finite, bounded, group-finite and Clifford + finite. On the other hand, a commutative semigroup X is absolutely T 1 S -closed if and only if X is finite. Also, for a given absolutely C -closed semigroup X we detect absolutely C -closed subsemigroups in the center of X.
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