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

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 $n \leq 3$.