Categories and semigroups

The aim of this paper is to consider the correspondence between the classification of morphisms in categories and the classes of semigroups with idempotents, in particular, we establish a mutual corresponding theorem of three classes of categories and the classes of left (right) abundant, two-sidexl abundant semigroups and regular semigroups. We first apply the nine axioms (P1)-(P9) which characterize cancellation and split properties of morphisms in a category to classify categories into three subclasses: idempotent ample (JA-, for short) categories, balanced categories and normal categories. The intrinsic relationship between these categories and their cone semigroups is investigated. It is proved that the cone semigroup of an JA-category is left abundant and vice versa, the category of principal left *-ideals of a left abundant (not necessarily right abundant) semigroup is an JA-category. Similar results for balanced categories and abundant semigroups are reproved and strengthened. As a consequence, we employ categorical terms to characterize those abundant semigroups in which the regular elements form a regular subsemigroup. Therefore, the significant theory of normal categories and regular semigroups devoted by K. S. S. Nambooripad is generalized to arbitrary abundant semigroups and even to left and right abundant semigroups, respectively.