Perfect semigroups and aliens (Q1865683)

Let us call a topological semigroup \(S\) \textit{perfect} if its multiplication is a closed mapping such that the inverse image of each point is compact. The second condition implies in particular that the divisors of each element form a compact set, i.e., each element is \textit{compactly divided}. An element \(s\) of a topological semigroup \(S\) is called \textit{purely alien} if it is not compactly divided. It is called \textit{alien} if for each compact subset \(K \subseteq S\) the element \(s\) lies in the closure of \((S\setminus K)S \cup S(S \setminus K)\). In the first half of the paper under review the authors study these concepts for general topological semigroups. Of particular interest are the non-alien elements and the structure of their divisor set, which is studied in particular for closed subsemigroups of locally compact groups. In this case the determination of the alien elements is crucial for the structure of semigroup compactifications of \(S\). In the second half of the paper the general machinery is applied to closed subsemigroups of \(SL_2(\mathbb R)\). For these semigroups detailed structural information is available, which makes them quite accessible to a detailed analysis.