Locally maximal idempotent-generated subsemigroups of singular orientation-preserving transformation semigroups.
The subject of the paper is the monoid of all orientation-preserving mappings on a finite chain of length \(n\). Most results concern the singular subsemigroup \(SOP_n\) of all non-permutations. It is proved that a set \(A\) of idempotents generates \(SOP_n\) if and only if it contains the set of all increasing, or the set of all decreasing idempotents in the top \(\mathcal J\)-class. The final section characterises what are called the locally maximal idempotent-generated subsemigroups of \(SOP_n\), which are those subsemigroups \(S\) generated by a set of idempotents \(E\) in the maximal \(\mathcal J\)-class of \(SOP_n\) that have the property that the set \(E\) together with any other idempotent \(e\) of the top \(\mathcal J\)-class generates all of \(SOP_n\). These are of two types, one of which consists of the so-called conjugates of the semigroup of all order-preserving mappings on the chain. This allows the authors to also characterise locally maximal idempotent-generated subsemigroups of these conjugates.
orientation-preserving transformation semigroups
maximal idempotent-generated subsemigroups
scientific article; zbMATH DE number 5377127