The aim of this paper is to extend, from group theory to hypergroup theory, the class equation and the concept of commutativity degree. Both of them are studied in depth for complete hypergroups because we want to stress the similarities and the differences with respect to group theory, and the representation theorem of complete hypergroups helps us in this direction. We also find conditions under which the commutativity degree can be expressed by using the class equation.

Hyperstructures in Chemical Hyperstructures of Redox Reactions with Three and Four Oxidation States

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 $(\mathcal{G},\ast )\subset {\mathcal{P}}^{\ast }\left(G\right)$, where ${\mathcal{P}}^{\ast }\left(G\right)$ is the set of non-empty subsets of G. The hypercomposition “$\ast$” is thus defined for any A, B from G, $A\ast B=\{a·b|a\in A,b\in B\}.$ In this article, we consider a particular group, G, to be the dihedral group $D_n,n$ is a natural number, greater than 3, and we analyze the HX-groups with the dihedral group $D_n$ 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 $D_n$ 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.

A highly popular and well-known flowering species for the unmistakable shape of its velvety and beautifully coloured flowers, Antirrhinum majus is often used in garden decor due to its long flowering period, ease of cultivation and low maintenance during the growing season. This study aims to investigate the behaviour of a dwarf variety of the Antirrhinum majus species grown in both vertical systems for green façades and also in a control field under the climate conditions in the north-eastern region of Romania. The façades of the vertical structure were been oriented towards a cardinal point, each of them having four equal layers arranged on height. The study found that this dwarf variety adapts very well to vertical systems, maintaining its ornamental features for a long time. During the experiment, observations included the diameter, height and number of flowers per plant in the control variant and on each side of the experimental structure. The highest values in July and August for plant diameter, plant height and the number of flowers were shown for the western facade and the lowest for the control variant. Instead, the control variant in September held the first position showing the highest means for all three monitored parameters and the lowest were for the southern orientation.

In this paper we analyse the center and centralizer of an element in the context of reversible regular hypergroups, in order to obtain the class equation in regular reversible hypergroups, by using complete parts. After an introduction in which basic notions and results of hypergroup theory are presented, particularly complete parts, then we give several properties, characterisations and also examples for the center and centralizer of an element for two classes of hypergroups. The next paragraph is dedicated to hypergroups associated with binary relations. We establish a connection between several types of equivalence relations, introduced by J. Jantosciak, such as the operational relation, the inseparability and the essential indistin-guishability and the conjugacy relation for complete hypergroups. Finally, we analyse Rosenberg hypergroup associated with a conjugacy relation.