Cross connection compatible discrete normal categories

Abstract

The concept of normal category was introduced by Nambooripad in his theory of cross connections for describing the structure of regular semigroups. These are essentially the categories of principal left ideals and the categories of principal right ideals of regular semigroups with appropriate morphisms. These categories also have an associated partial order on the set of objects. Discrete normal categories are normal categories in which the partial order on the object set is the identity relation. A cross connection relates two normal categories [Formula: see text] and [Formula: see text] and produces a regular semigroup [Formula: see text] such that one is the category of principal left ideals and the other the category of principal right ideals of the semigroup [Formula: see text]. This paper considers the question of compatibility of two normal categories in order that they can be connected by a cross connection. It is known that given any normal category [Formula: see text] there is a normal category [Formula: see text] called the normal dual of [Formula: see text] such that [Formula: see text] is compatible with [Formula: see text]. The question of finding all normal categories [Formula: see text] that are compatible with a given normal category [Formula: see text] remains unsettled. Such categories [Formula: see text] are called compatible duals of [Formula: see text]. For the class of discrete normal categories the characterization of all compatible duals has been done here.