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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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##### References:
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