History and new possible research directions of hyperstructures

We present a summary of the origins and current developments of the theory of algebraic hyperstructures. We also sketch some possible lines of research

An Overview of Topological and Fuzzy Topological Hypergroupoids

On a hypergroup, one can define a topology such that the hyperoperation is pseudocontinuous or continuous.This concepts can be extend to the fuzzy case and a connection between the classical and the fuzzy (pseudo)continuous hyperoperations can be given.This paper, that is his an overview of results received by S. Hoskova-Mayerova with coauthors  I. Cristea , M. Tahere and  B. Davaz, gives examples of topological hypergroupoids and show that there is no relation (in general) between pseudotopological and strongly pseudotopological hypergroupoids. In particular, it shows a topological hypergroupoid that does not depend on the pseudocontinuity nor on strongly pseudocontinuity of the hyperoperation

(Intuitionistic) Fuzzy Grade of a Hypergroupoid: A Survey of Some Recent Researches

This paper aims to present a short survey on two numerical functions determined by a hypergroupoid, called the fuzzy grade and the intuitionistic fuzzy grade of a hypergroupoid. It starts with the main construction of the sequences of join spaces and (intuitionistic) fuzzy sets associated with a hypergroupoid. After some computations of the above grades, we discuss some similarities and differences between the two grades for the complete hypergroups and for the i.p.s. hypergroups. We conclude with some open problems

FUZZY SUBSETS OF THE PHENOTYPES OF F2-OFFSPRING

This paper presents a connection between fuzzy sets, biological inheritance and hyperstructures in which we consider the set of phenotypes of the second generation $F_{2}$ in different types of inheritance, define fuzzy subsets of it and construct a sequence of join spaces associated to each of its types

Atanassov’s intuitionistic fuzzy index of hypergroupoids

In this work we introduce the concept of Atanassov’s intuitionistic fuzzy index of a hypergroupoid based on the notion of intuitionistic fuzzy grade of a hypergroupoid. We calculate it for some particular hypergroups, making evident some of its special properties

MULTIVALUED FUNCTIONS, FUZZY SUBSETS AND JOIN SPACES

One has considered the Hypergroupoid Η Γ = associated with a multivalued function Γ from H to a set D, defined as follows:∀ x ∈ H, x ο Γ x = ⎨y⏐ Γ(y) ∩ Γ(x) ≠ ∅⎬ ,∀ (y,z) ∈ H 2 , y ο Γ z = y ο Γ y ∪ z ο Γ z ,and one has calculated the fuzzy grade ∂(Η Γ ) for several functions Γ defined on sets H, such that ⎮H⎮ ∈ ⎨3, 4, 5, 6, 8, 9, 16⎬

Fuzzy hypergroups based on fuzzy relations

AbstractBased on fuzzy reasoning in fuzzy logic, this paper studies a fuzzy hyperoperation and a fuzzy hypergroupoid associated with a fuzzy relation. A sufficient and necessary condition for such a fuzzy hypergroupoid being a fuzzy hypergroup is given, and the properties of the fuzzy hypergroups associated with fuzzy relations are investigated. Furthermore, the definition of normal fuzzy hypergroups is put forward and it is shown that the category NFHG of normal fuzzy hypergroups satisfies all the axioms of topos except for the subobject classifier axiom

On Intra-Regular Semihypergroups Through Intuitionistic Fuzzy Sets

The notion of intuitionistic fuzzy sets was introduced by Atanassov as ageneralization of the notion of fuzzy sets. In this paper, using Atanassov idea, wegive some properties of intuitionistic fuzzy hyperideals and intuitionistic fuzzy bihyperidealsin a semihypergroup. We use the intuitionistic fuzzy left, right, twosidedand bi-hyperideals to characterize the intra-regular semihypergroups,generalizing some known results of intra-regular semigroups