n-ARY Hv-MODULES WITH EXTERNAL n-ARY P -HYPEROPERATION

Hyperstructure theory was born in 1934 when Marty [19] defined hypergroups as a generalization of groups. Let H be a non-empty set and let ℘∗(H) be the set of all non-empty subsets of H. A hyperoperation on H is a map ◦ : H ×H −→ ℘∗(H) and the couple (H, ◦) is called a hypergroupoid. If A and B are non-empty subsets of H, then we denote A◦B = ∪ a∈A, b∈B a◦b, x◦A = {x}◦A and A◦x = A◦{x}. Under certain conditions, we obtain the so-called semihypergroups and hypergroups . Basic definitions and results about the hyperstructures are found in [2, 3]. Hyperrings are essentially rings with approximately modified axioms. There are several kinds of hyperrings that can be defined on a non-empty set . In 2007, Davvaz and LeoreanuFotea [9] published a book titled Hyperring Theory and Applications. Sometimes, external hyperoperation is considered. An example of a hyperstructure, endowed both with an internal hyperoperation and an external hyperoperation is the so-called hypermodule. The theory of Hv-structures has been introduced by Vougiouklis [25]. The concept of Hv-structures constitutes a generalization of the well-known algebraic hyperstructures (hypergroups, hyperrings, hypermodules). Actually, some axioms concerning the above hyperstructures are replaced by their corresponding weak axioms. Basic definitions and results about the Hv-structures are found in [6, 24]. A hypergroupoid (H, ◦) is called an Hv-semigroup if for all x, y, z of H we have x ◦ (y ◦ z) ∩ (x ◦ y) ◦ z ̸= ∅, which means that ∪ u∈x◦y u ◦ z ∩ ∪ v∈y◦z x ◦ v ̸= ∅. Professor, Department of Mathematics, Yazd University, Yazd, Iran, Email: [email protected] and [email protected] Professor, Democritus University of Thrace, School of Science of Education, 68100 Alexandroupolis, Greece, E-mail: [email protected]