Geodesic
On inverse subsemigroups of the semigroup of orientation-preserving or orientation-reversing transformations
Algebra and discrete mathematics, Tome 19 (2015) no. 2, pp. 162-171

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It is well-known [16] that the semigroup $\mathcal{T}_n$ of all total transformations of a given $n$-element set $X_n$ is covered by its inverse subsemigroups. This note provides a short and direct proof, based on properties of digraphs of transformations, that every inverse subsemigroup of order-preserving transformations on a finite chain $X_n$ is a semilattice of idempotents, and so the semigroup of all order-preserving transformations of $X_n$ is not covered by its inverse subsemigroups. This result is used to show that the semigroup of all orientation-preserving transformations and the semigroup of all orientation-preserving or orientation-reversing transformations of the chain $X_n$ are covered by their inverse subsemigroups precisely when $n \leq 3$.
Keywords: semigroup, semilattice, inverse subsemigroup, strong inverse, order-preserving transformation, orientation-preserving transformation, orientation-reversing transformation.
Mots-clés : transformation 
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     title = {On inverse subsemigroups of the semigroup of orientation-preserving or orientation-reversing transformations},
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Paula Catarino; Peter M. Higgins; Inessa Levi. On inverse subsemigroups of the semigroup of orientation-preserving or orientation-reversing transformations. Algebra and discrete mathematics, Tome 19 (2015) no. 2, pp. 162-171. http://geodesic.mathdoc.fr/item/ADM_2015_19_2_a1/