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

Abstract

For $n\in \mathbb{N}$, let $X_n=\{a_1,a_2,\dots ,a_n\}$ be an $n$ - element set and let $\text{F}=(X_n;{<}_f)$ 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 $PF{I}_n$ of all partial fence-preserving injections of $X_n$ and its subsemigroup $I{F}_n=\{\alpha \in PF{I}_n:{\alpha }^{-1}\in PF{I}_n\}$. Clearly, $I{F}_n$ is an inverse semigroup and contains all regular elements of $PF{I}_n.$ We characterize the Green's relations for the semigroup $I{F}_n$. Further, we prove that the semigroup $I{F}_n$ is generated by its elements with \rank$\ge n-2$. Moreover, for $n\in 2\mathbb{N}$ we find the least generating set and calculate the rank of $I{F}_n$.

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