On the two natural products in a Stone-Čech compactification
type: Item
instance of: scholarly article
title: On the two natural products in a Stone-Čech compactification
zbMATH Open document ID: 0796.22003
DOI: 10.1007/BF02573674
author: Ahmed El-Mabhouh
author: J. Pym
author: D. Strauss
published in: Semigroup Forum
publication date: 1994-09-26T00:00:00Z
full work available at URL: https://eudml.org/doc/135316
review text: An example of a countable discrete semigroup \(S\) distinguishing between two natural semigroup structures for the Stone-Čech compactification \(\beta S\) of \(S\) is given. If \(s,t\in\beta S\) and \((s_ \alpha)\), \((t_ \alpha)\) are nets in \(S\) with \(s_ \alpha\to s\), \(t_ \alpha\to t\), then two products \(\circ\) and \(\diamond\) in \(\beta S\) are defined by \(s\circ t =\lim_ \alpha\lim_ \beta s_ \alpha\cdot t_ \beta\), \(s\diamond t =\lim_ \beta\lim_ \alpha s_ \alpha\cdot t_ \beta\). If \(S\) is commutative, \(\circ\) and \(\diamond\) are opposite multiplications, i.e., \(s\circ t = t\diamond s\). Therefore any subsemigroup \(H\) of \((\beta S,\circ)\) is a subsemigroup of \((\beta S,\diamond)\) and vice versa. The authors show that for \(S\), the free semigroup on countably many generators, there exists a compact subsemigroup \(H\) of \((\beta S,\circ)\) such that no element of \(H\) factorizes in \((\beta S,\diamond)\) and \(H\) generates a free subsemigroup of \((\beta S,\diamond)\).
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 54D35
zbMATH DE Number: 549068
zbMATH Keywords: countable discrete semigroup
zbMATH Keywords: semigroup structures
zbMATH Keywords: Stone-Čech compactification
zbMATH Keywords: free semigroup
zbMATH Keywords: compact subsemigroup
reviewed by: Mikhail Tkachenko
MaRDI profile type: Publication
cites work: The ideal structure of the space of \(\kappa\)-uniform ultrafilters on a discrete semigroup
cites work: Oids, finite sums and the structure of the Stone-Čech compactification of a discrete semigroup
cites work: Semigroup Structure in Stone-čech Compactifications
cites work: Semigroup structures on \(\beta\mathbb{N}\)
OpenAlex ID: W2013782729
description: scientific article; zbMATH DE number 549068