Combinatorial results for certain semigroups of order-decreasing partial isometries of a finite chain

Abstract

Let ${\mathcal{I}}_n$ be the symmetric inverse semigroup on $X_n=\{1,2,\dots ,n\}$ and let ${\mathcal{DDP}}_n$ and ${\mathcal{O}\mathcal{DDP}}_n$ be its subsemigroups of order-decreasing partial isometries and of order-preserving and order-decreasing partial isometries of $X_n$, respectively. In this paper we 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

14 pages, 2 tables