Chains of Archimedean semigroups (semiprimary semigroups)
A semigroup \(S\) is called Archimedean (\(r\)-Archimedean, \(t\)-Archimedean) if for all \(a,b \in S\) one has \(a \to b\) (\(a@>r>>b\), \(a@>t>> b\)). Here \(a \to b\) means \(a| b^ i\) for some \(i \in \mathbb{Z}^ +\) and \(a| b\) means \(b = xay\) for \(x,y \in S^ 1\); moreover, \(r\) means that \(b = xa\) and \(t\) means that \(b = xa\) and \(b = ay\) for some \(x,y \in S^ 1\). The authors prove that chains of Archimedean semigroups are exactly the semiprimary semigroups. In the case of \(r\)-Archimedean and \(t\)- Archimedean semigroups additional conditions are required. (A subset \(A \subseteq S\) is called semiprimary if \(\forall x,y \in S \exists n \in \mathbb{Z}^ +\) such that \(xy \in A\) implies \(x \in A\) or \(y^ n \in A\). A semigroup \(S\) is called semiprimary semigroup if all its ideals are semiprimary).
scientific article; zbMATH DE number 567156