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

10 pages