Partial Order in the Normal Category Arising from Normal Bands

TL;DR

This paper explores the partial order structure within the normal category derived from normal bands, providing new characterizations and properties related to the morphism sets in this algebraic context.

Contribution

It introduces a novel characterization of the partial order in the normal category of a normal band, expanding understanding of its algebraic structure.

Findings

Derived properties of the partial order in the normal category

Established a new characterization of the partial order

Analyzed the compatibility of the partial order with morphism composition

Abstract

The notion of normal category was introduced by KSS Nambooripad in connection with the study of the structure of regular semigroups using cross connections\cite{nambooripad1994theory}. It is an abstraction of the category of principal left ideals of a regular semigroup. A normal band is a semigroup $B$ satisfying $a^2=a$ and $abca=acba$ for all $a,b,c \in B$. Since the normal bands are regular semigroups, the category $\mathcal{L}(B)$ of principal left ideals of a normal band $B$ is a normal category. One of the special properties of this category is that the morphism sets admit a partial order compatible with the composition of morphisms. In this article we derive several properties of this partial order and obtain a new characterization of this partial order.