The scarcity of products in \(\beta S\)
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.
scientific article; zbMATH DE number 6695380