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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.

20M20 Semigroups of transformations, relations, partitions, etc.
20M05 Free semigroups, generators and relations, word problems
20M17 Regular semigroups
Full Text: DOI
[1] Catarino, P.M., Higgins, P.M.: The monoid of orientation-preserving mappings on a chain. Semigroup Forum 58, 190–206 (1999) · Zbl 0919.20041 · doi:10.1007/s002339900014
[2] Gomes, G.M.S., Howie, J.M.: On the ranks of certain semigroups of order-preserving transformations. Semigroup Forum 45, 272–282 (1992) · Zbl 0769.20029 · doi:10.1007/BF03025769
[3] Xiuliang, Y.: Products of idempotents of defect 1 in certain semigroups of transformations. Commun. Algebra 27, 3557–3568 (1999) · Zbl 0943.20064 · doi:10.1080/00927879908826684
[4] Howie, J.M.: Products of idempotents in finite full transformation semigroups. Proc. R. Soc. Edinb. 86, 243–254 (1980) · Zbl 0447.20048
[5] Higgins, P.M.: Combinatorial results for semigroups of order-preserving mappings. Math. Proc. Camb. Phil. Soc. 113, 281–296 (1993) · Zbl 0781.20036 · doi:10.1017/S0305004100075964
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