On the semigroup of partial isometries of a finite chain

Details On the semigroup of partial isometries of a finite chain On the semigroup of partial isometries of a finite chain

2010-12-30

arxiv.org/abs/1101.0049v1

Mathematics

Group Theory

12 pages, 4 tables, submitted to a journal

Scientific paper

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.

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