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

Let ${\cal I}_n$ be the symmetric inverse semigroup on $X_n = \{1, 2,..., n\}$ and let ${\cal DDP}_n$ and ${\cal ODDP}_n$ 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 ${\cal DDP}_n$ and ${\cal ODDP}_n$. We show that ${\cal ODDP}_n$ is a $0-E-unitary$ ample semigroup. We also investigate the cardinalities of some equivalences on ${\cal DDP}_n$ and ${\cal ODDP}_n$ which lead naturally to obtaining the order of the semigroups.