On the semigroup of all partial fence-preserving injections on a finite set

TL;DR

This paper investigates the algebraic structure of the semigroup of all partial injections that preserve a fence order on a finite set, characterizing its Green's relations, generators, and rank.

Contribution

It introduces and analyzes the semigroup of fence-preserving injections, providing new characterizations, generating sets, and rank calculations for this algebraic structure.

Findings

Characterized Green's relations for the inverse semigroup

Proved the semigroup is generated by elements with rank ≥ n-2

Calculated the rank and least generating set for even n

Abstract

For $n \in \mathbb{N}$, let $X_n = \{a_1, a_2, \ldots, a_n\}$ be an $n$ - element set and let $\textbf{F} = (X_n; <_f)$ be a fence, also called a zigzag poset. As usual, we denote by $I_n$ the symmetric inverse semigroup on $X_n$. We say that a transformation $\alpha \in I_n$ is \textit{fence-preserving} if $x <_f y$ implies that $x\alpha <_f y\alpha$, for all $x, y$ in the domain of $\alpha$. In this paper, we study the semigroup $PFI_n$ of all partial fence-preserving injections of $X_n$ and its subsemigroup $IF_n = \{\alpha \in PFI_n : \alpha^{-1}\in PFI_n\}$. Clearly, $IF_n$ is an inverse semigroup and contains all regular elements of $PFI_n.$ We characterize the Green's relations for the semigroup $IF_n$. Further, we prove that the semigroup $IF_n$ is generated by its elements with \rank$\geq n-2$. Moreover, for $n \in 2\mathbb{N}$ we find the least generating set and calculate the rank of $IF_n$.