fundamental relations in hv-structures. the ’judging from the results’ proof

The largest class of hyperstructures is the one which satisfy the weak axioms. these are called hv-structures introduced in 1990 and they proved to have a lot of applications on several sciences. the main tool in the study of hv-structures is the ’fundamental structure’ which is based on the ’fundamental relations’. these relations connect the hyperstructures with the corresponding classical structures. one cannot find the fundamental classes in an analytic way since they depend on the results of hyperoperations used. in this paper we focus on the fact that the fundamental classes depend on the results which gives new proofs and a lot of new important, for applications, large classes of hyperstructures.