On the semigroup of order-decreasing partial isometries of a finite chain

Abstract

Let ${\mathcal{I}}_n$ be the symmetric inverse semigroup on $X_n=\{1,2,...,n\}$ and let ${\mathcal{DDP}}_n$ and ${\mathcal{O}\mathcal{DDP}}_n$ be its subsemigroups of order-decreasing partial isometries and of order-preserving order-decreasing partial isometries of $X_n$, respectively. In this paper we investigate the cycle structure of order-decreasing partial isometry and characterize the Green's relations on ${\mathcal{DDP}}_n$ and ${\mathcal{O}\mathcal{DDP}}_n$. We show that ${\mathcal{O}\mathcal{DDP}}_n$ is a $0-E-unitary$ ample semigroup. We also investigate the cardinalities of some equivalences on ${\mathcal{DDP}}_n$ and ${\mathcal{O}\mathcal{DDP}}_n$ which lead naturally to obtaining the order of the semigroups.

11 pages