On the existence of hyperrings associated to arithmetic functions

After introducing the definition of hyperring by Krasner, the study of hyperrings has been of great importance. In this paper, we define a new hyperoperation associated to the set G of all arithmetic functions and give a complete analysis of its properties. Also, we combine our hyperoperation (+) with an already defined hyperoperation (⋅) and define (G,+,⋅) in which we study its properties and the possibility of containing hyperring. First, we prove that (G,+,⋅) is a weak hyperring. Then we find the largest hyperring contained in (G,+,⋅).