On the semigroup of partial isometries of a finite chain

TL;DR

This paper studies the structure and properties of the semigroup of partial isometries on a finite chain, including cycle structures, Green's relations, and cardinalities, revealing algebraic characteristics of these semigroups.

Contribution

It characterizes Green's relations and cycle structures of partial isometries and proves that the order-preserving subsemigroup is a 0-E-unitary inverse semigroup.

Findings

Characterization of Green's relations on ${ m{DP}}_n$ and ${ m{ODP}}_n$

Proof that ${ m{ODP}}_n$ is a 0-E-unitary inverse semigroup

Determination of the order of the semigroups

Abstract

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