Normal category of partitions of a set

TL;DR

This paper establishes that the category of partitions of a set has a normal category structure and is isomorphic to the principal right ideals of the transformation semigroup, connecting set partitions with algebraic structures.

Contribution

It introduces a normal category structure on the set partitions and proves their isomorphism to the principal right ideals of the transformation semigroup, linking combinatorial and algebraic frameworks.

Findings

The category of partitions of a set admits a normal category structure.

The category of partitions is isomorphic to the principal right ideals of the transformation semigroup.

The dual of the power-set category is isomorphic to the partition category.

Abstract

Let $T_X$ be the semigroup of all non-invertible transformations on an arbitrary set $X$. It is known that $T_X$ is a regular semigroup. The principal right(left) ideals of a regular semigroup $S$ with partial left(right) translations as morphisms form a normal category $\mathcal{R}( S )$($\mathcal{L}( S )$). Here we consider the category $\Pi(X)$ of partitions of a set $X$ and show that it admits a normal category structure and that $\Pi(X)$ is isomorphic to the category $\mathcal{R}( T_X )$. We also consider the normal dual $N^\ast \mathscr{P}(X)$ of the power-set category $\mathscr{P}(X)$ associated with $X$ and show that $N^\ast \mathscr{P}(X)$ is isomorphic to the partition category - $\Pi(X)$ of the set $X$.