2022/09/16 by Тарас Банах, Banakh, Taras, Serhii Bardyla +1
Decision Sciences · Mathematics · #20M14 #22A15 #54B30 #54D35 #54H11 #54H12 #Advanced Topology and Set Theory #FOS: Mathematics #Fuzzy and Soft Set Theory #General Topology (math.GN) #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.2209.08013
openalex publication_date 2022/09/16 · 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 (1) \mathcal C-closed if X is closed in every topological semigroup Y∈\mathcal C containing X as a discrete subsemigroup, (2) ideally \mathcal C-closed if for any ideal I in X the quotient semigroup X/I is \mathcal C-closed; (3) 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, (4) injectively \mathcal C-closed (resp. \mathcal C-discrete) if for any injective homomorphism h:X→ Y to a topological semigroup Y∈\mathcal C, the image h[X] is closed (resp. discrete) in Y. Let \mathsfT zS be the class of Tychonoff zero-dimensional topological semigroups. For a semigroup X let V E(X) be the set of all viable idempotents of X, i.e., idempotents e such that the complement X∖\fracHee of the set \fracHee=\x∈ X:xe=ex∈ He\ is an ideal in X. We prove the following results: (i) for any ideally \mathsfT zS-closed semigroup X each subgroup of the center Z(X)=\z∈ X:∀ x∈ X (xz=zx)\ is bounded; (ii) for any \mathsfT zS-closed semigroup X, each subgroup of the ideal center I Z(X)=\z∈ Z(X):zX⊆ Z(X)\ is bounded; (iii) for any \mathsfT zS-discrete or injectively \mathsfT zS-closed semigroup X, every subgroup of Z(X) is finite, (iv) for any viable idempotent e in an ideally (and absolutely) \mathsfT zS-closed semigroup X, the maximal subgroup He is ideally (and absolutely) \mathsfT zS-closed and has bounded (and finite) center Z(He).