The scarcity of products in \(\beta S\)
type: Item
instance of: scholarly article
title: The scarcity of products in \(\beta S\)
zbMATH Open document ID: 1364.54019
author: Neil Hindman
author: D. Strauss
published in: Topology and its Applications
publication date: 2017-03-16T00:00:00Z
review text: Let \((S,\cdot)\) be a discrete semigroup, \(\beta S\) its Stone-Čech compactification, and \(S^{\ast}= \beta S \setminus S\). It is known that the operation on \(S\) extends uniquely to \(\beta S\) and that \((\beta S,\cdot)\) is a right topological semigroup. The positions of the sets \(S^{\ast}\cdot S^{\ast}\) and \(S^{\ast} \setminus S^{\ast} \cdot S^{\ast}\) in \(S^{\ast}\) are investigated. Extending some results on countable discrete semigroups, the authors prove that \(S^{\ast}\cdot S^{\ast}\) is nowhere dense in \(S^{\ast}\) whenever \(S\) is infinite and embeddable in a group, and that there is a cancellative semigroup \(S\) of size continuum so that \(S^{\ast}\cdot S^{\ast}\) is not nowhere dense in \(S^{\ast}\). If \(p \in \beta S\), then the norm of \(p\) is the smallest cardinality of \(A\in p\), and for a cardinal \(\kappa\), \(U_{\kappa}\) denotes the set of all \(p\in \beta S\) having norm \(\kappa\). If \(S\) is right cancellative and very weakly left cancellative and the cardinality of \(S\) is \(\kappa \geq \omega\), then \(S^{\ast}\cdot U_{\kappa}\) is nowhere dense in \(U_{\kappa}\). Conditions guaranteeing that \(S^{\ast} \setminus S^{\ast}\cdot S^{\ast}\) is dense in \(S^{\ast}\) are established. Several other interesting results are obtained.
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 22A15
zbMATH DE Number: 6695380
zbMATH Keywords: Stone-Čech compactification
zbMATH Keywords: nowhere dense
zbMATH Keywords: weak cancellative
zbMATH Keywords: set of products
reviewed by: Ljubiša D. R. Kočinac
MaRDI profile type: Publication
full work available at URL: j.topol.2017.01.028
OpenAlex ID: W2593412794
Wikidata QID: Q59889431
cites work: Q4373417
cites work: \(t\)-sets and some algebraic properties in \(\beta S\) and in \(\ell_\infty(S)^*\)
cites work: Q3281123
cites work: Minimal ideals and cancellation in beta(N)
cites work: Algebra in the Stone-Čech compactification. Theory and applications
cites work: Cancellation in the Stone–Čech compactification of a discrete semigroup
cites work: Semigroup structures on \(\beta\mathbb{N}\)
DOI: 10.1016/J.TOPOL.2017.01.028
description: scientific article; zbMATH DE number 6695380