On semigroups of orientation-preserving partial permutations with restricted range
De Biao Li, Vítor H. Fernandes
TL;DR
This paper studies the semigroup $\mathcal{POPI}_{n}(Y)$ of orientation-preserving injective partial transformations on a finite chain $\Omega_n$ with restricted range $Y$. It provides a complete description of regular elements and Green's relations, showing regularity occurs iff $Y=\Omega_n$ and detailing $\mathscr{L}$, $\mathscr{R}$, $\mathscr{H}$, and $\mathscr{D}$ classes. It then determines the size of $\mathcal{POPI}_{n}(Y)$, proves an isomorphism theorem governed by dihedral symmetries, and derives explicit rank formulas, with $\operatorname{rank}(\mathcal{POPI}_{n}(Y))=\binom{n}{r}$ for $1\le r\le n-1$ and $2$ for $r=n$. Overall, the results advance structural understanding and classification of restricted-range orientation-preserving transformation semigroups, linking regularity, Green's relations, isomorphisms, and generating sets.
Abstract
Let $Ω_n$ be a finite chain with $n$ elements $(n\in\mathbb{N})$, and let $\mathcal{POPI}_{n}$ be the semigroup of all injective orientation-preserving partial transformations of $Ω_n$. In this paper, for any nonempty subset $Y$ of $Ω_n$, we consider the subsemigroup $\mathcal{POPI}_{n}(Y)$ of $\mathcal{POPI}_{n}$ of all transformations with range contained in $Y$. We describe the Green's relations and study the regularity of $\mathcal{POPI}_{n}(Y)$. Moreover, we calculate the rank of $\mathcal{POPI}_{n}(Y)$ and determine when two semigroups of this type are isomorphic.