Hypergroup Theory

The following sections are included:

• Preface
• Contents

The concept of algebraic hyperstructures was introduced in 1934 by Marty [115] and has been studied in the following decades and nowadays by many mathematicians. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. In this chapter we introduce the notion of a hypergroupoid and a semihypergroup. We present some examples to show that how these structures appear in natural phenomena.

This chapter contains a brief description of the ideas, constructions and examples of hypergroups. The motivating example is the quotient of a group by any, not necessary normal, subgroup. Also, we present the definition of an H_v-group as a generalization of a hypergroup, which was introduced by Vougiouklis for the first time in 1990.

In general, we are not interested in arbitrary subsets of a hypergroup H for which the hypergroups structure is not preserved. Any subsets we do consider they are subhypergroups, that preserve algebraic properties derived from those of H. The study of several types of subhypergroups was begun by Marty, Dresher, Ore, Krasner, and was continued by Sureau, Koskas, Corsini, Freni, Davvaz, Leoreanu, Račková, and many others.

Homomorphisms of hypergroups were studied by Dresher, Ore, Krasner, Kuntzmann, Koskas, Jantosciak, Corsini, Freni, Leoreanu, Davvaz and many others. In this chapter, we present several kinds of homomorphisms.

The main tools connecting the class of hyperstructures with the classical algebraic structures are the fundamental relations. The fundamental relation has an important role in the study of semihypergroups and especially of hypergroups. We study several fundamental relations which give us group, abelian group, cyclic group and solvable group structures.

In this chapter we introduce and analyze the relation ζ_e, for which the transitive closure leads to a monoid as a quotient structure. We analyze when the relation ζ_e is transitive and we give some necessary and sufficient conditions. Using the relation ${\varsigma }_e^{*}$, we characterize the derived ${\varsigma }_e^{*}$-strong semihypergroup. Then we analyze a new relation, denoted by ${\tau }_e^{*}$ which extends ${\varsigma }_e^{*}$ and for which the quotient structure is a commutative monoid. Finally, we generalize the notion of complete parts and α-parts by introducing the notion of ℜ-parts. The results of this chapter are contained in [134; 133].

Join spaces were introduced by W. Prenowitz [147; 148; 145; 146] to provide a common algebraic framework in which classical geometries could be axiomatized and studied. The underlying algebraic structure used was a hypergroup. Then this concept applied by him and J. Jantosciak both in Euclidian and in non-Euclidian geometry [149]. Using this notion, several branches of non-Euclidian geometry were rebuilt: descriptive geometry, projective geometry and spherical geometry. Then, several important examples of join spaces have been constructed in connection with binary relations, graphs, lattices, fuzzy sets and rough sets.

In this chapter, we associate a Rosenberg hypergroup with a complete hypergroup and we study it. Then the notion of weak mutually associative hypergroups is introduced and some properties of the hypergroups associated with binary relations are given. On the other hand, using some special type of Boolean matrices, the non-isomorphic Rosenberg hypergroups of order less than 7 are enumerated. Moreover, the regular and reversible Rosenberg hypergroups are identified. It is analyzed when different partial hypergroupoids associated with binary relations defined on a set H are reduced hypergroups. The cartesian product of two hypergroupoids associated with a binary relation is also analyzed.

In this chapter, a hypergroupoid (H, ⨂_ρ) is associated with an n-ary relation ρ defined on a non-empty set H. It is investigated when it is an H_v-group, a hypergroup or a join space. Then some connections between this hypergroupoid and Rosenberg’s hypergroupoid associated with a binary relation are determined.

The results of this chapter are contained in papers [41; 43].

Rough set theory was proposed by Pawlak for knowledge discovery in databases and experimental data sets. The purpose of this chapter is to introduce and discuss the concept of lower and upper approximations in a hypergroup. We consider the fundamental relation β^* defined on a hypergroup H and interpret the lower and upper approximations as subsets of the group H/β^* and give some properties of such subsets. Also, connections between hypergroups and approximation operators are studied. The results of this chapter are contained in papers [44; 46; 56].

Hypergraphs are a generalization of graphs in which an edge (hyperedge) is connected to a subset of nodes rather than just two of them. Connections between hypergraphs and hypergroups was studied by Corsini and later, they are studied by Leoreanu-Fotea, Davvaz, Farshi, Nikkhah, Maryati, and many others. In this chapter, we introduce some important and interesting results.

In this chapter we study topological hypergroupoids and topological hypergroups. Connections between topology and algebraic hyperstructures were studied by Mittas, Konstantinidou, Hošková-Mayerová, Davvaz, Al Tahan, Heidari, Modarres, Singha, Das, and many others. A topological hypergroup is a topological space with a continuous hypergroup structure.

The following sections are included:

• Bibliography
• Index