The gradation of type A n on Lie-Santilli admissibility Pipina Nikolaidou, Thomas Vougiouklis Democritus University of Thrace, School of Education, 68 100 Alexandroupolis, Greece pnikolai@eled.duth.gr, tvougiou@eled.duth.gr Abstract The largest class of hyperstructures,called H v -structures, is the one which satisfy the weak properties. In this paper we deal with the Lie-admissible hyperstrucuttures and we present a construction of the hyperstructures used in the Lie-Santilli admissible theory on square matrices of type A n using the P-hyperstructures. Key words: hyperstructures, H v -structures, hopes, weak hopes, ∂ -hopes, e-hyperstructures, admissible Lie-algebras MSC2010: 20N20, 17B67, 17B70, 17D25. 1 Introduction The main object of this paper is the class of hyperstructures called H v - structures introduced in 1990 [15], which satisfy the weak axioms where the non-empty intersection replaces the equality. Some basic definitions are the following: Algebraic hyperstructure is called any set H equipped with at least one hyperoperation (abbreviation: hyperoperation=hope ) · : H × H → P (H ) − {∅} . We abbreviate by WASS the weak associativity :(xy)z ∩ x(yz ) = 1