The rank of the semigroup of order-, fence-, and parity-preserving partial injections on a finite set

TL;DR

This paper investigates the structure of a submonoid of the symmetric inverse semigroup consisting of order-, fence-, and parity-preserving transformations on a finite chain, providing its rank, generators, and normal forms.

Contribution

It characterizes the monoid of order-, fence-, and parity-preserving partial injections, determines its rank, and constructs minimal generating sets with normal forms.

Findings

The monoid has rank 3n-6.

A minimal generating set of size 3n-6 is provided.

Concrete normal forms for generators are established.

Abstract

The monoid of all partial injections on a finite set (the symmetric inverse semigroup) is of particular interest because of the well-known Wagner-Preston Theorem. In this article, we step forward the study of a submonoid of the symmetric inverse semigroup. We explore the monoid of all order-, fence-, and parity-preserving transformations on an $n$-element chain. We also characterize the transformations in that monoid and show that it has a rank $3n-6$. In particular, we provide a generating set $A_n$ of minimal size and exhibit concrete normal forms for the transformations generated by $A_n$.