Does \(N^*\) contain a topological and algebraic copy of \(\beta N\)?
type: Item
instance of: scholarly article
title: Does \(N^*\) contain a topological and algebraic copy of \(\beta N\)?
zbMATH Open document ID: 0696.22006
DOI: 10.1016/0166-8641(90)90113-G
published in: Topology and its Applications
publication date: 1990-01-01T00:00:00Z
review text: 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 \(\beta\) \({\mathbb{N}}\).
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 54D40
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 4140064
zbMATH Keywords: homeomorphic and isomorphic imbeddings
zbMATH Keywords: Stone-Čech compactification
zbMATH Keywords: natural numbers
zbMATH Keywords: remainder
zbMATH Keywords: ultrafilter
zbMATH Keywords: algebraical and topological imbedding
zbMATH Keywords: AT-imbedding
zbMATH Keywords: minimal ideal
author: Neil Hindman
author: Amha Tume Lisan
reviewed by: Wolfgang A. F. Ruppert
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4140064