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

Abstract: Let $[n]={1,2,\ldots,n}$ be a finite chain and let $\mathcal{P}{n}$ be the semigroup of partial transformations on $[n]$. Let $\mathcal{CP}{n}={\alpha\in \mathcal{P}{n}: (for~all ~x,y\in Dom~\alpha)~|x\alpha-y\alpha|\leq|x-y|}$, then $\mathcal{CP}{n}$ 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 $\mathcal{CP}_{n}$.