On certain semigroups of full contractions of a finite chain

Abstract: Let $[n]={1,2,\ldots,n}$ be a finite chain and let $\mathcal{T}{n}$ be the semigroup of full transformations on $[n]$. Let $\mathcal{CT}{n}={\alpha\in \mathcal{T}{n}: (for ~all~x,y\in [n])~\left|x\alpha-y\alpha\right|\leq\left|x-y\right|}$, then $\mathcal{CT}{n}$ is a subsemigroup of $\mathcal{T}{n}$. In this paper, we give a necessary and sufficient condition for an element to be regular and characterize all the Green's equivalences for the semigroup $\mathcal{CT}{n}$. We further show that the semigroup $\mathcal{CT}_{n}$ is a left abundant semigroup.