On transitive linear semigroups
A semigroup \({\mathcal S}\) of linear transformations acting on a finite-dimensional vector space \({\mathcal V}\) over a field \(\mathbb{F}\) acts transitively if for all \(x\neq 0\), \(y\in{\mathcal V}\) there exist \(S\in{\mathcal S}\) with \(Sx= y\). The action is sharply transitive if \(S\) is uniquely determined by \((x,y)\). The action is called pre-transitive if \({\mathcal S}(x)\supseteq{\mathcal V}\setminus\{0\}\) for all \(x\neq 0\). (Note that -- with this notation -- a group \({\mathcal S}\) cannot act transitively, but pre-transitively. And a sharply transitive semigroup \({\mathcal S}\) without zero is a group.) \({\mathcal S}\) is called left \(t\)-simple if the semigroup contains no proper transitive left ideal in \({\mathcal S}\). In the paper under review, the authors investigate transitive left \(t\)-simple semigroups (and some related classes) and characterize e.g. among others transitive semigroups which are representable as unions of groups and a zero element.
scientific article; zbMATH DE number 1419714
scientific article