Does \(N^*\) contain a topological and algebraic copy of \(\beta N\)?
Let \(\beta\) \({\mathbb{N}}\) denote the Stone-Čech compactification of the natural numbers, and \({\mathbb{N}}^*\) the remainder \(\beta\) \({\mathbb{N}}\setminus {\mathbb{N}}\). Furthermore, let us endow \(\beta\) \({\mathbb{N}}\) with the natural one-sided continuous multiplication which extends the usual addition on \({\mathbb{N}}\). For any point \(p\in {\mathbb{N}}^*\) define \(\sigma_ p: {\mathbb{N}}\to {\mathbb{N}}^*\), \(n\mapsto p^ n\) and write \(\sigma_ p^{\beta}: \beta {\mathbb{N}}\to {\mathbb{N}}^*\) for the canonical extension of \(\sigma_ p\) over \(\beta\) \({\mathbb{N}}\). In the present paper the authors characterize, in terms of the ultrafilter defining p, those points \(p\in {\mathbb{N}}^*\) for which \(\sigma_ p^{\beta}\) is an algebraical and topological imbedding (AT-imbedding), that is, \(\sigma_ p^{\beta}\) is a homomorphism and \(\sigma_ p^{\beta}({\mathbb{N}})\) is discrete. As a special application they show that \(\sigma_ p^{\beta}\) is not an AT-imbedding if p lies in the closure of a set B with the sum uniqueness property: if F, G are finite subsets of B with \(\sum_{f\in F}f=\sum_{g\in G}g\) then \(F=G\). On the other hand, if \(\sigma_ p^{\beta}\) is an AT-imbedding then p cannot lie in the minimal ideal M of \(\beta\) \({\mathbb{N}}\).
scientific article; zbMATH DE number 4140064