On inverse subsemigroups of the semigroup of orientation-preserving or orientation-reversing transformations

Paula Catarino, Peter M. Higgins, Inessa Levi

Abstract


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, transformation, order-preserving transformation, orientation-preserving transformation, orientation-reversing transformation

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References


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