On the semigroups of fence-decreasing and fence-preserving transformations on a finite fence

Abstract

For a natural number $n$, a fence $[n]=\{1\prec 2\succ 3\prec 4\succ 5\prec \cdots n\}$ is a partial ordered set. A partial transformation $\alpha$ is called fence-decreasing if $x\alpha ⪯x$ for all $x$ in the domain of $\alpha$, and fence-preserving if $x\prec y$ implies $x\alpha ⪯y\alpha$ for all $x$ and $y$ in the domain of $\alpha$. In this paper, we consider the monoids ${\mathcal{D}\mathcal{F}}_n$ (${\mathcal{PD}\mathcal{F}}_n)$ of all fence-decreasing full (partial) transformations as well as the monoid ${\mathcal{P}\mathcal{C}\mathcal{F}}_n$ of all fence-preserving transformations of ${\mathcal{PD}\mathcal{F}}_n$. For these three monoids and some of their ideals, we determine the unique minimal generating set. Moreover, we calculate the rank of ${\mathcal{D}\mathcal{F}}_n$, ${\mathcal{PD}\mathcal{F}}_n$, and ${\mathcal{P}\mathcal{C}\mathcal{F}}_n$. Additional, we provide several combinatorial results concerning these three monoids.