# Ranks and presentations of some normally ordered inverse semigroups

## TL;DR

This paper determines the rank and provides presentations for certain inverse semigroups related to $P$-stable and $P$-order preserving partial permutations, contributing to the understanding of normally ordered inverse semigroups.

## Contribution

It computes the rank and exhibits presentations for monoids of $P$-stable and $P$-order preserving partial permutations, advancing the structural knowledge of normally ordered inverse semigroups.

## Key findings

- Computed the rank of the monoids.
- Provided explicit presentations for these semigroups.
- Enhanced understanding of the generators of the pseudovariety ${
m NO}$.

## Abstract

In this paper we compute the rank and exhibit a presentation for the monoids of all $P$-stable and $P$-order preserving partial permutations on a finite set $\Omega$, with $P$ an ordered uniform partition of $\Omega$. These (inverse) semigroups constitute a natural class of generators of the pseudovariety of inverse semigroups ${\sf NO}$ of all normally ordered (finite) inverse semigroups.