2022/07/26 by Тарас Банах, Banakh, Taras, Serhii Bardyla +1
Computer Science · Mathematics · #20M18 #22A15 #54B30 #54D35 #54H11 #54H12 #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2207.12778
openalex publication_date 2022/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathcal C be a class of topological semigroups. A semigroup X is called absolutely \mathcal C-closed if for any homomorphism h:X→ Y to a topological semigroup Y∈\mathcal C, the image h[X] is closed in Y. Let \mathsfT 1S, \mathsfT 2S, and \mathsfT zS be the classes of T1, Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that a commutative semigroup X is absolutely \mathsfT zS-closed if and only if X is absolutely \mathsfT 2S-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 \mathsfT 1S-closed if and only if X is finite. Also, for a given absolutely \mathcal C-closed semigroup X we detect absolutely \mathcal C-closed subsemigroups in the center of X.