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Absolutely closed semigroups.
Taras Banakh1,2, Serhii Bardyla3,4
1Ivan Franko National University of Lviv, Lviv, Ukraine.
Revista De La Real Academia De Ciencias Exactas, Fisicas Y Naturales. Serie A, Matematicas
|November 16, 2023
Summary
This study defines and characterizes absolutely closed topological semigroups. Commutative semigroups are absolutely closed if and only if they are finite or satisfy specific algebraic finiteness conditions.
Area of Science:
- Topology
- Abstract Algebra
- Semigroup Theory
Background:
- Topological semigroups are algebraic structures with a topology.
- Absolutely closed semigroups are a specific class with implications for homomorphisms.
Purpose of the Study:
- To define and investigate absolutely closed topological semigroups.
- To establish characterizations for these semigroups within specific classes.
Main Methods:
- Exploration of homomorphism properties in topological semigroups.
- Analysis of algebraic finiteness conditions (chain-finite, bounded, group-finite, Clifford+finite).
Main Results:
- A commutative semigroup is absolutely -closed iff it's absolutely -closed iff it's chain-finite, bounded, group-finite, and Clifford+finite.
- A commutative semigroup is absolutely -closed iff it is finite.
- Identification of absolutely closed subsemigroups within the center of a given absolutely closed semigroup.
Conclusions:
- Provides distinct characterizations for different classes of absolutely closed semigroups.
- Establishes a strong link between topological closure properties and algebraic finiteness.
- Offers insights into the structure of absolutely closed semigroups and their subsemigroups.
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