# Regularity and Green's relations for the semigroup of partial contractions of a finite chain

## TL;DR

This paper investigates the algebraic structure of the semigroup of partial contractions on a finite chain, providing conditions for regularity and characterizing Green's relations within this semigroup.

## Contribution

It offers a complete characterization of regular elements and Green's relations in the semigroup of partial contractions on a finite chain.

## Key findings

- Necessary and sufficient conditions for regularity in the semigroup.
- Complete characterization of Green's relations on the semigroup.
- Identification of the structure of the semigroup of partial contractions.

## Abstract

Let $[n]=\{1,2,\ldots,n\}$ be a finite chain and let $\mathcal{P}_{n}$ be the semigroup of partial transformations on $[n]$. Let $\mathcal{CP}_{n}=\{\alpha\in \mathcal{P}_{n}: (for~all ~x,y\in Dom~\alpha)~|x\alpha-y\alpha|\leq|x-y|\}$, then $\mathcal{CP}_{n}$ is a subsemigroup of $\mathcal{P}_{n}$. In this paper, we give a necessary and sufficient condition for an element in $\mathcal{P}_{n}$ to be regular and characterize all the Green's equivalences on the semigroup $\mathcal{CP}_{n}$.