Optimizing HX-Group Compositions Using C++: A Computational Approach to Dihedral Group Hyperstructures

Abstract

The HX-groups represent a generalization of the group notion. The Chinese mathematicians Mi Honghai and Li Honxing analyzed this theory. Starting with a group (G,·), they constructed another group (G,∗)⊂P∗(G), where P∗(G) is the set of non-empty subsets of G. The hypercomposition “∗” is thus defined for any A, B from G, A∗B={a·b|a∈A,b∈B}. In this article, we consider a particular group, G, to be the dihedral group Dn,n is a natural number, greater than 3, and we analyze the HX-groups with the dihedral group Dn as a support. The HX-groups were studied algebraically, but the novelty of this article is that it is a computer analysis of the HX-groups by creating a program in C++. This code aims to improve the calculation time regarding the composition of the HX-groups. In the first part of the paper, we present some results from the hypergroup theory and HX-groups. We create another hyperstructure formed by reuniting all the HX-groups associated with a dihedral group Dn as a support for a natural fixed number n. In the second part, we present the C++ code created in the Microsoft Visual Studio program, and we provide concrete examples of the program’s application. We created this program because the code aims to improve the calculation time regarding the composition of HX-groups.