# Presentations for subsemigroups of $PD_n$

## TL;DR

This paper provides algebraic presentations for various semigroups of order-decreasing and order-preserving mappings on finite chains, expanding understanding of their algebraic structure.

## Contribution

It introduces explicit presentations for several classes of transformation semigroups involving order-preserving and order-decreasing mappings on finite chains.

## Key findings

- Presented generators and relations for each semigroup class
- Unified approach to partial and full mappings
- Enhanced algebraic understanding of these semigroups

## Abstract

Let $[n]=\{1,\ldots,n\}$ be the $n$-chain. We give presentations for the following transformation semigroups: the semigroup of full order-decreasing mappings of $[n]$, the semigroup of partial one-to-one order-decreasing mappings of $[n]$, the semigroup of full order-preserving and order-decreasing mappings of $[n]$, the semigroup of partial one-to-one order-preserving and order-decreasing mappings of $[n]$, and the semigroup of partial order-preserving and order-decreasing mappings of $[n]$.