An Overview of the Foundations of the Hypergroup Theory

This paper is written within the framework of the Special Issue of Mathematics entitled “Hypercompositional Algebra and Applications”, and focuses on the presentation of the essential principles of the hypergroup, which is that the prominent structure of hypercompositional algebra. Within the beginning, it reveals the structural relation between two fundamental entities of abstract algebra, the group and therefore the hypergroup. Next, it presents the several sorts of hypergroups, which derive from the enrichment of the hypergroup with additional axioms besides those it had been initially equipped with, alongside their fundamental properties. Furthermore, it analyzes and studies the varied subhypergroups which will be defined in hypergroups together with their ability to decompose the hypergroups into cosets. The exploration of this far-reaching concept highlights the particularity of the hypergroup theory versus the abstract pure mathematics, and demonstrates the various techniques and special tools that has got to be developed so as to realize results on hypercompositional algebra.