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The smallest ideal of \((\beta \mathbb N,\cdot)\). (English) Zbl 1169.22005

Summary: We are concerned with the semigroup \((\beta\mathbb N,+)\). As with any compact (Hausdorff) right topological semigroup, \((\beta\mathbb N,+)\) has a smallest two sided ideal \(K(\beta\mathbb N,+)\). Known results about this ideal are presented in Section 2.
By way of contrast, not much has been known about the smallest ideal of \((\beta\mathbb N,\cdot)\). In Theorem 3.2 in Section 3, we present one such result. In particular, in this theorem, we show that each maximal subgroup of \(K(\beta\mathbb N,\cdot)\) contains a copy of the free group on \(2^{\mathfrak c}\) generators.

MSC:

22A15 Structure of topological semigroups
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
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