On the idempotents of semigroup of partial contractions of a finite chain

Let $[n]={1,2,ldots,n}$ be a finite chain. Let$mathcal{P}_{n}$ and $mathcal{T}_{n}$ be Semigroups of partialandfull transformations on $[n]$ respectively.Let $mathcal{CP}_{n}={alphain mathcal{P}_{n}: |xalpha-yalpha|leq|x-y| forall x, yin dom~alpha}$ and $mathcal{CT}_{n}={alphain mathcal{T}_{n}: |xalpha-yalpha|leq|x-y| forall x, yin [n]}$, then $mathcal{CP}_{n}$ and $mathcal{CT}_{n}$ aresubsemigroups of $mathcal{P}_{n}$ and $mathcal{T}_{n}$ respectively. In this paper, we characterize the idempotent elements and computed the number of idempotents of height, $n-1$ and $n-2$for the semigroups $mathcal{CP}_{n}$ and $mathcal{CT}_{n}$ respectively.

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