On certain semigroups 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 $\mathcal{CP}{n}={\alpha\in \mathcal{P}{n}: (for ~all~x,y\in Dom~\alpha)~|x\alpha-y\alpha|\leq|x-y|}$ be the subsemigroup of partial contraction mappings on $[n]$. We have shown that the semigroup $\mathcal{CP}{n}$ and some of its subsemigroups are nonregular left abundant semigroups for all $n$ but not right abundant for $n\geq 4$.