On the two natural products in a Stone-Čech compactification
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 \((\beta S,\diamond)\).
scientific article; zbMATH DE number 549068