On certain semigroups of full contractions of a finite chain
Abdullahi Umar, M. M. Zubairu · arXiv (Cornell University) · 2018
Let $[n]=\{1,2,\ldots,n\}$ be a finite chain and let $\mathcal{T}_{n}$ be the semigroup of full transformations on $[n]$. Let $\mathcal{CT}_{n}=\{α\in \mathcal{T}_{n}: (for ~all~x,y\in [n])~\left|xα-yα\right|\leq\left|x-y\right|\}$, 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.