On certain semigroups of contraction mappings of a finite chain

Let $[n]=\{1,2,\dots,n\}$ be a finite chain and let $\mathcal{P}_{n}$ (resp., $\mathcal{T}_{n}$) be the semigroup of partial transformations on $[n]$ (resp., full transformations on $[n]$). Let $\mathcal{CP}_{n}=\{\alpha\in \mathcal{P}_{n}\colon (\text{for all }x,y\in \operatorname{Dom}\alpha)\ |x\alpha-y\alpha|\leq|x-y|\}$ (resp., $\mathcal{CT}_{n}=\{\alpha\in \mathcal{T}_{n}\colon (\text{for all }x,y\in [n])\ |x\alpha-y\alpha|\leq|x-y|\}$) be the subsemigroup of partial contraction mappings on $[n]$ (resp., subsemigroup of full contraction mappings on $[n]$). We characterize all the starred Green's relations on $\mathcal{CP}_{n}$ and it subsemigroup of order preserving and/or order reversing and subsemigroup of order preserving partial contractions on $[n]$, respectively. We show that the semigroups $\mathcal{CP}_{n}$ and $\mathcal{CT}_{n}$, and some of their subsemigroups are left abundant semigroups for all $n$ but not right abundant for $n\geq 4$. We further show that the set of regular elements of the semigroup $\mathcal{CT}_{n}$ and its subsemigroup of order preserving or order reversing full contractions on $[n]$, each forms a regular subsemigroup and an orthodox semigroup, respectively.