On three submonoids of the dihedral inverse monoid on a finite set

TL;DR

This paper investigates three specific submonoids of the dihedral inverse monoid on a finite set, analyzing their sizes, algebraic relations, and minimal generating sets.

Contribution

It provides detailed descriptions, cardinalities, Green's relations, and ranks for the submonoids of orientation-preserving, monotone, and order-preserving transformations.

Findings

Computed the cardinalities of the three submonoids.

Described Green's relations for each submonoid.

Determined the ranks of the submonoids.

Abstract

In this paper we consider three submonoids of the dihedral inverse monoid $\mathcal{DI}_n$, namely its submonoids $\mathcal{OPDI}_n$, $\mathcal{MDI}_n$ and $\mathcal{ODI}_n$ of all orientation-preserving, monotone and order-preserving transformations, respectively. For each of these three monoids, we compute the cardinal, give descriptions of Green's relations and determine the rank.