Abstract
AbstractLet $${\mathcal {C}}$$
C
be a class of topological semigroups. A semigroup X is called absolutely $${\mathcal {C}}$$
C
-closed if for any homomorphism $$h:X\rightarrow Y$$
h
:
X
→
Y
to a topological semigroup $$Y\in {\mathcal {C}}$$
Y
∈
C
, the image h[X] is closed in Y. Let $$\textsf {T}_{\!\textsf {1}}\textsf {S}$$
T
1
S
, $$\textsf {T}_{\!\textsf {2}}\textsf {S}$$
T
2
S
, and $$\textsf {T}_{\!\textsf {z}}\textsf {S}$$
T
z
S
be the classes of $$T_1$$
T
1
, Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that a commutative semigroup X is absolutely $$\textsf {T}_{\!\textsf {z}}\textsf {S}$$
T
z
S
-closed if and only if X is absolutely $$\textsf {T}_{\!\textsf {2}}\textsf {S}$$
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 $$\textsf {T}_{\!\textsf {1}}\textsf {S}$$
T
1
S
-closed if and only if X is finite. Also, for a given absolutely $${\mathcal {C}}$$
C
-closed semigroup X we detect absolutely $${\mathcal {C}}$$
C
-closed subsemigroups in the center of X.
Funder
Austrian Science Fund
Agentúra na Podporu Výskumu a Vývoja
Pavol Jozef Šafárik University in Košice
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Mathematics,Geometry and Topology,Algebra and Number Theory,Analysis
Reference73 articles.
1. Arhangelskii, A., Choban, M.: Completeness type properties of semitopological groups, and the theorems of Montgomery and Ellis. In: Topology Proceedings, vol. 37, pp. 33–60 (2011)
2. Bader, U., Leibtag, E.: Homomorhic Images of Algebraic Groups. arXiv:2212.03055
3. Banakh, T.: Categorically closed topological groups. Axioms 6(3), 23 (2017)
4. Banakh, T.: $$E$$-separated semigroups. Visnyk Lviv Univ. Ser. Mekh. Mat. 92, 17–33 (2021)
5. Banakh, T.: A Non-polybounded Absolutely Closed 36-Shelah Group. arXiv:2212.01750