# The rank of the inverse semigroup of partial automorphisms on a finite fence

## TL;DR

This paper determines the minimal number of generators needed for the semigroup of all order-preserving partial injections on a finite fence, a specific partial order structure, focusing on odd-sized sets.

## Contribution

It provides the exact rank and a minimal generating set for the semigroup of order-preserving partial injections on odd-sized fences, extending previous work to this specific case.

## Key findings

- Calculated the rank of the semigroup for odd n
- Provided a minimal generating set for the semigroup
- Extended known results to a new class of partial orders

## Abstract

A fence is a particular partial order on a (finite) set, close to the linear order. In this paper, we calculate the rank of the semigroup $\mathcal{FI}_{n}$ of all order-preserving partial injections on an $n$-element fence. In particular, we provide a minimal generating set for $\mathcal{FI}_{n}$. In the present paper, $n$ is odd since this problem for even $n$ was already solved by I. Dimitrova and J. Koppitz.