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

## TL;DR

This paper explores the structure and size of specific subsemigroups of the symmetric inverse semigroup related to order-decreasing partial isometries on finite chains, providing combinatorial insights into their cardinalities.

## Contribution

It introduces new combinatorial results for the cardinalities of certain semigroups of order-decreasing partial isometries and their order-preserving variants, advancing understanding of their algebraic structure.

## Key findings

- Cardinalities of ${
m DDP}_n$ and ${
m ODDP}_n$ computed
- Equivalence classes on these semigroups characterized
- Semigroup orders explicitly determined

## Abstract

Let ${\cal I}_n$ be the symmetric inverse semigroup on $X_n = \{1, 2, \ldots , n\}$ and let ${\cal DDP}_n$ and ${\cal ODDP}_n$ 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 ${\cal DDP}_n$ and ${\cal ODDP}_n$ which lead naturally to obtaining the order of the semigroups