Regularity and Green's relations for the semigroup of partial contractions of a finite chain

Abstract

Let $[n]=\{1,2,\dots ,n\}$ be a finite chain and let ${\mathcal{P}}_n$ be the semigroup of partial transformations on $[n]$. Let ${\mathcal{C}\mathcal{P}}_n=\{\alpha \in {\mathcal{P}}_n:(forallx,y\in Dom\alpha )|x\alpha -y\alpha |\le |x-y|\}$, then ${\mathcal{C}\mathcal{P}}_n$ is a subsemigroup of ${\mathcal{P}}_n$. In this paper, we give a necessary and sufficient condition for an element in ${\mathcal{P}}_n$ to be regular and characterize all the Green's equivalences on the semigroup ${\mathcal{C}\mathcal{P}}_n$.

15 pages