  A completely simple semigroup S is a semigroup without zero which has no
proper ideals and contains a primitive idempotent. It is known that S is a
regular semigroup and any completely simple semigroup is isomorphic to the Rees
matrix semigroup. In the study of structure theory of regular semigroups,
K.S.S. Nambooripad introduced the concept of normal categories to construct the
semigroup from its principal left(right) ideals using cross-connections. A
normal category C is a small category with subobjects wherein each object of
the category has an associated idempotent normal cone and each morphism admits
a normal factorization. A cross-connection between two normal categories C and
D is a local isomorphism frm D to N*C where N*C is the normal dual of the
category C. In this paper, we identify the normal categories associated with a
completely simple semigroup S and show that the semigroup of normal cones TL(S)
is isomorphic to a semi-direct product of semigroups. We characterize the
cross-connections in this case and show that each sandwich matrix P correspond
to a cross-connection. Further we use each of these cross-connections to give a
representation of the completely simple semigroup as a cross-connection
semigroup.

Group Theory