Non-linear monotone positive maps

We study several classes of general non-linear positive maps between C*-algebras, which are not necessary completely positive maps. We characterize the class of the compositions of *-multiplicative maps and positive linear mapsas the class of non-linear maps of boundedly positive type abstractly. We consider three classes of non-linear positive maps defined only on the positive cones, which are the classes of being monotone, supercongruent or concave. Any concave maps are monotone. The intersection of the monotone maps and the supercongruent maps characterizes the class of monotone Borel functional calculus. We give many examples of non-linear positive maps, which show that there exist no other relations among these three classes in general.