Normal categories of semigroup of order-preserving transformations on a finite chain

TL;DR

This paper characterizes the ideal categories of the regular semigroup of order-preserving transformations on a finite chain, revealing their structure and isomorphisms with related categories.

Contribution

It describes the ideal categories of the semigroup of order-preserving transformations and establishes their isomorphisms with power set and ordered partition categories.

Findings

Principal left ideal category is the power set category of the chain.

Principal right ideal category is the category of ordered partitions.

The cone semigroup is isomorphic to the original semigroup.

Abstract

K. S. S. Nambooripad intoduced nornal categories to enable to describe the structure of regular semigroups fully. In this paper we describe the ideal categories of the regular semigroup $OX_n,$ of non-invertible order-preserving transformations on a finite chain $X_n=\{1\leq 2\leq \cdots \leq n \}$ which are normal categories. Further it is shown that the principal left ideal category of $OX_n$ as the power set category $P_o(X_n)$ of $OX_n$ and the principal right ideal category as $\prod_o(X_n)$ category of ordered partitions of $X_n$ and described the cone semigroup $T\mathscr L(OX_n)$ and prove that it is isomorphic to $OX_n.$