Discrete Normal Categories and Cross Connections

The category of principal left[or right] ideals of a regular semigroup has been abstractly characterised as a normal category by KSS Nambooripad. Cross connection is a relation connecting two normal categories C and D so that C is isomorphic to the normal category of principal left ideals and D is isomorphic to the normal category of principal right ideals of a regular semigroup. Here we describe all cross connections of certain categories with a simplified structure. The categories considered are normal categories with the property that the partial order on the set of objects is the identity relation. Such normal categories are said to be discrete normal categories.