This paper investigates compactifications of a semitopological semigroup S, focusing on the relationship between <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\psi _A(\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ψ</mi> <mi>A</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the set of limit points of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> at infinity for a non-precompact subset <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$A \subseteq S$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>⊆</mo> <mi>S</mi> </mrow> </math></EquationSource> </InlineEquation>, and the minimal ideal K(X) in a semigroup compactification <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(\psi , X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of S. It is shown that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\psi _A(\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ψ</mi> <mi>A</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> may form an ideal or subgroup under precompact conditions on A and continuity conditions on X. Consequently, a classification of semigroup compactifications that are factors of a subdirect product of the one-point and group compactifications is provided. Additionally, the paper examines topological semigroup compactifications for a broad class of subsemigroups of complete totally ordered groups, correcting and generalizing Example 3.2.4 from “Analysis on semigroups” by J. F. Berglund, D. H. Junghenn, and P. Milnes (Wiley, New York (1989)) and discussing applications to the Bohr compactification.

Limit points at infinity for semigroup compactification classification

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2026年5月19日

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Hojjatollah Samea

摘要

This paper investigates compactifications of a semitopological semigroup S, focusing on the relationship between $\psi _A(\infty )$ $ψ_{A} (\infty)$ , the set of limit points of $\psi $ $ψ$ at infinity for a non-precompact subset $A \subseteq S$ $A \subseteq S$ , and the minimal ideal K(X) in a semigroup compactification $(\psi , X)$ $(ψ, X)$ of S. It is shown that $\psi _A(\infty )$ $ψ_{A} (\infty)$ may form an ideal or subgroup under precompact conditions on A and continuity conditions on X. Consequently, a classification of semigroup compactifications that are factors of a subdirect product of the one-point and group compactifications is provided. Additionally, the paper examines topological semigroup compactifications for a broad class of subsemigroups of complete totally ordered groups, correcting and generalizing Example 3.2.4 from “Analysis on semigroups” by J. F. Berglund, D. H. Junghenn, and P. Milnes (Wiley, New York (1989)) and discussing applications to the Bohr compactification.