On oriented alternating inverse monoids

In this paper, we consider the inverse submonoids $AO{R}_n$ of oriented transformations and $AO{P}_n$ of orientation-preserving transformations of the alternating inverse monoid $A{I}_n$ on a chain with $n$ elements. We compute the cardinalities, describe the Green’s structures and the congruences, and calculate the ranks of $AO{R}_n$ and $AO{P}_n$.