Cancellation and embedding theorems for compact uniquely divisible semigroups
The author discusses conditions on a compact connected uniquely divisible semigroup \(S\) with trivial groups to have the property that the complement of the minimal ideal is cancellative. If \(x\in S\), then \([x]\) is the minimal uniquely divisible subsemigroup of \(S\) containing \(x\). For such a semigroup \(S\), these are known to have a trivial minimal ideal (zero). The zero of \([q]\) is denoted \(e_q\) and the core of an idempotent \(e\in S\) is defined \(\text{core}(e)=\{ x\in S: xe=e=ex\}\). The author defines a map \(\psi: S\rightarrow \text{core}(e_q)\) and proves that it is a continuous homomorphism. Consequently he selects \(u,v,q\in S\) which yields an important structural embedding of \(S\) into \([u][v]\times \text{core}(e_q)\).
scientific article; zbMATH DE number 1309710
scientific article