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

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... Show more