The Hv-matrix representations

The Theory of Representations of Hyperstructures was started in mid ۸۰’s but that time there was not any general definition of hyperfield. The Hv-structures, were introduced in ۴th AHA Congress ۱۹۹۰, and at the same time, the general definition of the hyperfield, was given. Since then the Theory of Representations is refereed mainly on Hv-groups by Hv-matrices. In Hv-structures the weak axioms replace the ”equality” by the ”non empty intersection”. The characteristic property of Hv-structures, is that a partial order on Hv-structures on the same underline set, is defined. The weak properties increase extremely the number of hyperstructures defined in the same set. In representation theory the researchers have to treat well almost all the classical algebraic structures from semigroups to Lie-algebras. We present the problems, some new results and we give to researchers open problems in mathematics from hyperstructures.