Abstract
Let J be an infinite set and let \(I={\cal P}_{f}( J)\), i.e., I is the collection of all non empty finite subsets of J. Let \(\beta I\) denote the collection of all ultrafilters on the set I and let \(( \beta I,\uplus )\) be the compact (Hausdorff) right topological semigroup that is the Stone-Cech Compactification of the semigroup \(( I,\cup )\) equipped with the discrete topology. This paper continues the study of \(( \beta I,\uplus )\) that was started in [3] and [5]. In [5], Koppelberg established that \(K( \beta I) =\beta _{J}( I)\) (where K( S) is the smallest ideal of a semigroup S) and for non empty \(A\subseteq J,\) she established \(K( \widehat{{\cal P}_{f}(A) }) =\beta _{A}( I) \cap \widehat{{\cal P}_{f}( A) }\). In this note, we show that for \(A\subseteq J\) such that \(J\backslash A\) is infinite, \(\overline{K( {\cal V}_{A}) }\) is a proper subset of \(\beta _{A}( I)\) and \(K( {\cal V}_{A}) =K( \beta _{A}( I)) =K( \beta _{0}^{( A) }( I) )\), where \({\cal V}_{A}=\bigcup \{ \beta _{B}( I) \mid B\subseteq A\}\).
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Grainger, A. Ideals of Ultrafilters on the Collection of Finite Subsets of an Infinite Set. Semigroup Forum 73, 234–242 (2006). https://doi.org/10.1007/s00233-006-0632-3
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DOI: https://doi.org/10.1007/s00233-006-0632-3