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.

This paper presents a series of important results from the theory of n-hypergroups. Connections with binary relations and with lattices are presented. Special attention is paid to the fundamental relation and to the commutative fundamental relation. In particular, join n-spaces are analyzed.

The first use of the largest class of hyper-structures, the Hv-structures, and their fundamental relations was to define the general hyper-field. Moreover, the enormous number Hv-structures defined on the same set, admits a partial order and has a lot of applications in pure mathematics and other sciences. We present ways to obtain appropriate Hv-fields from given well-known classical rings.

The class of Hv-structures is the largest class of hyperstructures defined on the same set. For this reason, they have applications in mathematics and in other sciences, which range from biology, hadronic physics, leptons, linguistics, sociology, to mention but a few. They satisfy the weak axioms where the non-empty intersection replaces equality. The fundamental relations connect, by quotients, the Hv-structures with the classical ones. In order to specify the appropriate hyperstructure as a model for an application which fulfill a number of properties, the researcher can start from the basic ones. Thus, the researcher must know the minimal hyperstructures. Hv-numbers are elements of Hv-field, and they are used in representation theory. In this presentation we focus on minimal Hv-fields derived from rings.

The Lie-Santilli admissible theory gives in physics applications mainly in producing energy. The mathematical models need to be multivalued and the class of Hv-structures is the most appropriate, since this is the largest class of hyper-structures. The Hv-numbers, weak hyper-numbers, used satisfy all needed axioms. We focus on small and very-thin Hv-fields with P-hyper-operations on the product.