Introduction to Dependence Relations and Their Links to Algebraic Hyperstructures

Abstract

The aim of this paper is to study, from an algebraic point of view, the properties of interdependencies between sets of elements (i.e., pieces of secrets, atmospheric variables, etc.) that appear in various natural models, by using the algebraic hyperstructure theory. Starting from specific examples, we first define the relation of dependence and study its properties, and then, we construct various hyperoperations based on this relation. We prove that two of the associated hypergroupoids are H v -groups, while the other two are, in some particular cases, only partial hypergroupoids. Besides, the extensivity and idempotence property are studied and related to the cyclicity. The second goal of our paper is to provide a new interpretation of the dependence relation by using elements of the theory of algebraic hyperstructures.

n-Ary Cartesian Composition of Multiautomata with Internal Link for Autonomous Control of Lane Shifting

“…By extensive hypergroup in the sense of hypercompositional structures we mean a hypergroup that {a, b} ⊆ a * b for all a, b ∈ H, i.e., that the elements are included in its "resalt of hyperoperation". For the same application on this theory in other science or real-life problems see, for example, [3][4][5][6][7].…”

On Hypergroups with a β-Class of Finite Height

“…In the last few decades, many scholars have been working in the field of algebraic hyperstructures, also called hypercompositional algebra. In fact, algebraic hyperstructures have found applications in many fields, including geometry, fuzzy/rough sets, automata, cryptography, artificial intelligence and probability [1], relational algebras [2], and sensor networks [3].…”