Abstract In this paper, we consider the inverse submonoids AORₙ AOR n of oriented transformations and AOPₙ AOP n of orientation-preserving transformations of the alternating inverse monoid AIₙ 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 AORₙ AOR n and AOPₙ. AOP n.
Vítor Hugo Fernandes (2026) studied this question.