G-Hypergroups: Hypergroups with a Group-Isomorphic Heart

Far East Journal of Mathematical Sciences

The paper deals with the family of hypergroups of type U on the right. A hypergroup H is said to be ‘of type U on the right’ if there exists a right scalar identity e and for all x,y∈H, x∈xy⇒y=e. This class contains the class of type C on the right. Some general properties are proved. In such hypergroups of size 5 there are, under the cosets of the right unit, six possibilities. In this paper three of the above possibilities are determined under isomorphism.

Filomat, 2013

Let S n denote the class of hypergroups of type U on the right of size n with bilateral scalar identity. In this paper we consider the hypergroups (H, •) ∈ S 7 which own a proper and non-trivial subhypergroup h. For these hypergroups we prove that h is closed if and only if (H − h) • (H − h) = h. Moreover we consider the set E 7 of hypergroups in S 7 that own the above property. On this set, we introduce a partial ordering induced by the inclusion of hyperproducts. This partial ordering allows us to give a complete characterization of hypergroups in E 7 on the basis of a small set of minimal hypergroups, up to isomorphisms. This analysis gives a partial (negative) answer to a problem raised in [5] concerning the existence in S n of proper hypergroups having singletons as special hyperproducts.

Topology and its Applications, 2019

In this paper we define some ballean structure on the power set of a group and, in particular, we study the subballean with support the lattice of all its subgroups. If G is a group, we denote by L(G) the family of all subgroups of G. For two groups G and H, we relate their algebraic structure via the ballean structure of L(G) and L(H).

Discrete Mathematics, 1999

The class of n * -complete hypergroups is introduced. Several properties and examples are found and a geometric interpretation is given by means of hypergraphs. (G. Lo Faro) 0012-365X/99/$ -see front matter c 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 1 2 -3 6 5 X ( 9 9 ) 0 0 0 7 1 -0

Rendiconti del Circolo Matematico di Palermo, 1997

In this paper one continues the study of hypergroups with for proper pairs. In particular one considers the case when the scalars group is not empty and the set of non-scalar elements has size two or three.

Matematicki Vesnik

The hypergroups H of type U on the right can be classified in terms of the family P1 = f1 - x j x 2 Hg, where 1 2 H is the right scalar identity. If the size of H is 5, then P1 can assume only 6 possible values, three of which have been studied in (3). In this paper, we completely describe other two of the remaining possible cases: a) P1 = ff1g;f2;3g;f4g;f5gg; b) P1 = ff1g;f2;3g;f4;5gg. In these cases, P1 is a partition of H and the equivalence relation associated to it is a regular equivalence on H. We find that, apart of isomorphisms, there are exactly 41 hypergroups in case a), and 56 hypergroup in case b).

Symmetry, 2020

In every hypergroup, the equivalence classes modulo the fundamental relation β are the union of hyperproducts of element pairs. Making use of this property, we introduce the notion of height of a β -class and we analyze properties of hypergroups where the height of a β -class coincides with its cardinality. As a consequence, we obtain a new characterization of 1-hypergroups. Moreover, we define a hierarchy of classes of hypergroups where at least one β -class has height 1 or cardinality 1, and we enumerate the elements in each class when the size of the hypergroups is n ≤ 4 , apart from isomorphisms.

Annals of the Alexandru Ioan Cuza University - Mathematics, 2015

We introduce a strongly regular equivalence relation ρ*A on the hypergroup H, such that in a particular case the quotient is a cyclic group. Then by using the notion of ρ*A-parts, we investigate the transitivity condition of ρA. Finally, a characterization of the derived hypergroup Dc(H) has been considered.

Science Journal of Applied Mathematics and Statistics, 2017

The purpose of this paper is making a construction and generalization of Molaei’s generalized groups by using construction of the Rees matrix semigroup over a polygroup H and a matrix with entries in H . We call it “Molaei’s generalized hypergroups” and we give some examples.

IOCMA 2023

The aim of this paper is to review the most important properties and applications of the complete hypergroups. We will focus on the reversibility, regularity and reducibility properties, on the class equation and the commutativity degree of the complete hypergroups, as well as on the Euler's totient function defined in this kind of hypergroups.