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Right cancellation in \(\beta S\) and \(UG\). (English) Zbl 0871.22001

Let \(\beta S\) be the Stone-Čech compactification of an infinite discrete semigroup \(S\) equipped with the right topological semigroup structure obtained by extending the multiplication of \(S\) (so that the maps \(x\mapsto xy\) are continuous). Following Rudin, a subset \(V\) of \(S\) is called a \(t\)-set if \(sV\cap tV\) is finite for any pair \(s,t\in S\) with \(s\neq t\). The author shows, among other things, that if \(S\) is cancellative and \(V\) is a countable \(t\)-set in \(S\) then every point \(q\) in the closure of \(V\) in \(\beta S\) is right cancellative in \(\beta S\). From this result he deduces that every weak \(p\)-point which lies in the closure of a countable subset of \(S\) is right cancellative in \(\beta S\). (Note that by a result of K. Kunen the set of weak \(p\)-points is dense in \(\beta S\backslash S\).) A similar result holds for the uniform compactification of a locally compact topological group.

MSC:

22A15 Structure of topological semigroups
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References:

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