On monotone alternating inverse monoids

In this paper, we consider the inverse submonoids $A{M}_n$ of monotone transformations and $A{O}_n$ of order-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 these two submonoids of $A{I}_n$.