Vague Soft Hypergroups and Vague Soft Hypergroup Homomorphism

We introduce and develop the initial theory of vague soft hyperalgebra by introducing the novel concept of vague soft hypergroups, vague soft subhypergroups, and vague soft hypergroup homomorphism. The properties and structural characteristics of these concepts are also studied and discussed. 1. Introduction Vague set is a set of objects, each of which has a grade of membership whose value is a continuous subinterval of . Such a set is characterized by a truth-membership function and a false-membership function. Thus, a vague set is actually a form of fuzzy set, albeit a more accurate form of fuzzy set. Soft set theory has been regarded as an effective mathematical tool to deal with uncertainties. However, it is difficult to be used to represent the vagueness of problem parameters in problem-solving and decision-making contexts. Hence the concept of vague soft sets were introduced as an extension to the notion of soft sets, as a means to overcome the problem of assigning a suitable value for the grade of membership of an element in a set since the exact grade of membership may be unknown. Using the concept of vague soft sets, we are able to ascertain that the grade of membership of an element lies within a certain closed interval. Hyperstructure theory was first introduced in 1934 by a French mathematician, Marty [1], at the 8th Congress of Scandinavian 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. Since the introduction of the notion of hyperstructures, comprehensive research has been done on this topic and the notions of hypergroupoid, hypergroup, hyperring, and hypermodule have been introduced. A recent book by Corsini and Leoreanu-Fotea [2] expounds on the applications of hyperstructures in the areas of geometry, hypergraphs, binary relations, lattices, fuzzy sets and rough sets, automation, cryptography, combinatorics, codes, artificial intelligence, and probability theory. The concept of -structures introduced by Vougiouklis [3] constitute a generalization of the well-known algebraic hyperstructures such as hypergroups, hyperrings, and hypermodules. Some axioms pertaining to the above-mentioned hyperstructures such as the associative law and the distributive law are replaced by their corresponding weak axioms. The study of fuzzy algebraic structures started with the introduction of the concept of fuzzy subgroup of a group by Rosenfeld [4] in 1971. There is a considerable amount of work that has been done on the