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Locally maximal idempotent-generated subsemigroups of singular orientation-preserving transformation semigroups. (English) Zbl 1157.20039
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.

MSC:
20M20 Semigroups of transformations, relations, partitions, etc.
20M05 Free semigroups, generators and relations, word problems
20M17 Regular semigroups
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