We study the Euler's totient function (called also the Euler's phi function) in the framework of finite complete hypergroups. These are algebraic hypercompositional structures constructed with the help of groups, and endowed with a multivalued operation, called hyperoperation. On them the Euler's phi function is multiplicative and not injective. In the second part of the article we find a relationship between the subhypergroups of a complete hypergroup and the subgroups of the group involved in the construction of the considered complete hypergroup. As sample application of this connection, we state a formula that relates the Euler's totient function defined on a complete hypergroup to the same function applied to its subhypergroups.

In this paper we consider the cases of simple dominance for dihybrid cross, trihybrid cross and the 4- hybrid cross, in order to notice that the distribution of phenotypes forms a lattice. Moreover, the lattice has an interesting property: it is isomorphic with the lattice of the divisors of 2(k), k being a nonzero natural number.

This article aims to create commutative hyperstructures, starting with a non-commutative group. Therefore, we consider the starting group to be the dihedral group Dn, where n is a natural number, n>1, and we determine the HX groups associated with the dihedral group. For a fixed number n, we note Gn=Gp2p1HX−groups,foranyp1,p2∈N*suchthatn=p1p2 as the set of all HX groups. This paper analyses this new structure’s properties for particular cases when the dihedral group D4 is the support group.

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.

The chapter aims to introduce the concept of NeutroHX-Group. Starting with HX-groups and using the newly defined concept of NeutroAlgebra, the authors define the NeutroHX-Group. Using the HX-groups associated with the dihedral group, they obtain the Chinese hypergroup associated with them. They present some examples to illustrate situations when the NeutroHX-Groups and AntiHX-Groups are obtained. Also, they determine a way to obtain commutative hyperstructure, considering the set of all HX-Groups with the dihedral group as support. The important result of this chapter is given by Theorem 1, where it is establishing a connection between NeutroHX-Groups and NeutroSemihypergroups.

The main goal of this paper is to introduce the Euler totient function in canonical and i.p.s. hypergroup theory. Also, we determine a way to construct finite i.p.s. hypergroups using the concept of the extension of i.p.s. hypergroup by i.p.s. hypergroup.

Abstract The aim of this paper is to compute the commutativity degree in polygroup’s theory, more exactly for the polygroup PG and for extension of polygroups by polygroups, obtaining boundaries for them. Also, we have analyzed the nilpotencitiy of 𝒜[𝒝], meaning the extension of polygroups 𝒜 and 𝒝.

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.

In this paper, we determine the hypergroups associated with the HX-groups of dihedral group D-n. Also we calculate the fuzzy grade of these hypergroups and the commutativity degree for the corresponding HX-groups.