Morphisms of Right Groups: Categorical Structure and Enumeration

Right groups are regular semigroups admitting a canonical decomposition as direct products G×R of group G with right–zero semigroup R. We study morphisms in the category of right groups using the equivalence RGp≃Grp×Set. This yields explicit descriptions of homomorphisms, kernels, images, quotients, and congruences and componentwise characterisations of epimorphisms and image–coimage factorisations. For finite right groups, we obtain exact counting formulas for homomorphisms, epimorphisms, and automorphisms, together with finiteness criteria and asymptotic growth estimates. We show that morphism sets typically grow exponentially or superexponentially in the right–zero component, which asymptotically dominates the group-theoretic contribution in all cases.