Thomas Vougiouklis

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.

The helix-sum and helix-product overcome restrictions which ordinary sum and product of the classical non-square matrices have. They, in fact, are weak hyperstructures and can be used, among the other applications, in the representation theory of Hv-groups, as well. However, we focus on low dimensions and small cardinality of the underline sets. In the study of helix-products, new interesting Hv-groups, appeared.

The number of Hv-structures, defined on a set is extremely big and gives the chance to applied sciences to work in an opposite to usual direction. Thus one asks from applied sciences to give all restrictions they have, because they reduce the number of possible Hv-structures expressing their problem, by a mathematical model. Moreover, the way of treating Hv-structures gives some new proving methods which can be used in applications. This is the main reason that the Hv-structures have so many applications. Here we present an overview on applications of Hv-structures on other sciences as in Hadronic Mechanics, Lie-Santilli admissible, leptons, biology, teaching, education, and mathematics itself, as well.

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 helix-hyperoperations, hyper-sum and hyperproduct, are defined on any type of ordinary matrices. Thus, they overcome restrictions which ordinary sum and product on matrices have. We focus on the representation theory of hyperstructures, where the helix-product on any type of matrices, can be used. In fact, the helixproduct gives a hyper- semi-hypergroup structure or its generalization the Hv-semigroup. We restrict on the case of two m × n, with m < n, matrices, cases. The crucial point is that all entries of the original matrices are used, so the helix-product do not lose any information. For the case of n × (n + 1) we present subsets closed under the helix-product. Finite or infinite fields and Hv- fields, are used, as well.

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 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 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.

Hyperstructure theory can overcome restrictions which ordinary algebraic structures have. A hyperproduct on non-square ordinary matrices can be defined by using the so called helix-hyperoperations. We define and study the helix-hyperstructures on the representations and we extend our study up to Lie-Santilli theory by using ordinary fields. Therefore the related theory can be faced by defining the hyperproduct on the extended set of non square matrices. The obtained hyperstructure is an Hv-algebra or an Hv-Lie-alebra

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