On the semigroup of partial isometries of a finite chain

Abstract

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

12 pages, 4 tables, submitted to a journal