Partial Order in the Normal Category Arising from Normal Bands

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.