Some Properties of the Set of Soft Vectors on Hypervector Spaces

Using the concept of soft sets, as well as the concepts of fuzzy sets and rough sets, is one way to solve problems that deal with uncertainty. The effect of this idea on algebraic structures has led to the different types of soft algebraic structures that have been studied by many authors. In particular, in the last decade, soft algebraic hyperstructures have been considered by some researchers, some of them have been mentioned in references.The aim of this paper is the study of algebraic structure of SE(V), the set of soft vectors over a hypervector space V. In this regard, at first by introducing a natural operation and a suitable external hyperoperation, structure of a hypervector space is given to the mentioned set. Then, using the concept of linear transformation, the relationship between hypervector spaces and their corresponding set of soft vectors is briefly studied. Finally, by presenting the concept of soft norm on the obtained hypervector space, its correlation with normal norm and soft meter on V is investigated. In particular, each parameter family of norms on hypervector space V induces a soft norm on hypervector space SE(V), as well as every norm on V induces a soft norm on SE(V). Also, any soft normed hypervector space is a soft metric space, and each soft meter on the hypervector space SE(V) induces a normal meter on it.