On the subsemigroup generated by idempotents of the semigroup of order preserving and decreasing contraction mappings of a finite chain

On the subsemigroup generated by idempotents

Abstract

Denote $[n]$ to be a finite chain $\{1,2,\ldots,n\}$ and let $\mathcal{ODP}_{n}$ be the semigroup of order preserving and order decreasing partial transformations on $[n]$. Let $\mathcal{CP}_{n}=\{\alpha\in \mathcal{P}_{n}: (\textnormal{for all}~x,y\in \dom~\alpha)~|x\alpha-y\alpha|\leq|x-y|\}$ be the subsemigroup of partial contraction mappings on $[n]$. Now let $\mathcal{ODCP}_{n}=\mathcal{ODP}_{n}\cap \mathcal{CP}_{n}$. Then $\mathcal{ODCP}_{n}$ is a subsemigroup of $\mathcal{ODP}_{n}$ In this paper, we identify the subsemigroup generated by the idempotents in the semigroup of order-preserving and order-decreasing partial contractions $\mathcal{ODCP}_n$. In particular, we characterize the idempotents in the semigroup and study factorization in the subsemigroup generated by the idempotents in $\mathcal{ODCP}_n$. We give a necessary and sufficient condition for product of two idempotents to be an idempotent and otherwise.