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 $n\geq 4$.