Presentations for three remarkable submonoids of the dihedral inverse monoid on a finite set

TL;DR

This paper studies three specific submonoids of the dihedral inverse monoid on a finite set, providing presentations for each to understand their algebraic structure better.

Contribution

It introduces presentations for the submonoids of orientation-preserving, monotone, and order-preserving transformations within the dihedral inverse monoid.

Findings

Presented algebraic structures of the submonoids.

Derived generators and relations for each submonoid.

Enhanced understanding of the algebraic properties of these submonoids.

Abstract

In this paper we consider the submonoids $\mathcal{OPDI}_n$, $\mathcal{MDI}_n$ and $\mathcal{ODI}_n$ of the dihedral inverse monoid $\mathcal{DI}_n$ of all orientation-preserving, monotone and order-preserving transformations, respectively. Our goal is to exhibit presentations for each of these three monoids.