On certain semigroups of full contractions of a finite chain

Abstract

Let $[n]=\{1,2,\dots ,n\}$ be a finite chain and let ${\mathcal{T}}_n$ be the semigroup of full transformations on $[n]$. Let ${\mathcal{C}\mathcal{T}}_n=\{\alpha \in {\mathcal{T}}_n:(forallx,y\in [n])\left|x\alpha -y\alpha \right|\le \left|x-y\right|\}$, then ${\mathcal{C}\mathcal{T}}_n$ 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 ${\mathcal{C}\mathcal{T}}_n$. We further show that the semigroup ${\mathcal{C}\mathcal{T}}_n$ is a left abundant semigroup.

10. arXiv admin note: text overlap with arXiv:1803.02146