On algebraic and more general categories whose split epimorphisms have   underlying product projections
  We characterize those varieties of universal algebras where every split epimorphism considered as a map of sets is a product projection. In addition we obtain new characterizations of protomodular, unital and subtractive varieties as well as varieties of right omega-loops and biternary systems. 
James R. A. Gray and Nelson Martins-Ferreira
math.CT