By Bijan Davvaz, Irina Cristea

This publication is meant as an creation to fuzzy algebraic hyperstructures. because the first in its style, it contains a variety of issues, so much of which replicate the authors’ prior learn and hence offers a place to begin for destiny learn instructions. The publication is equipped in 5 chapters. the 1st bankruptcy introduces readers to the fundamental notions of algebraic buildings and hyperstructures. the second one covers fuzzy units, fuzzy teams and fuzzy polygroups. the next chapters are occupied with the idea of fuzzy Hv-structures: whereas the 3rd bankruptcy offers the idea that of fuzzy Hv-subgroup of Hv-groups, the fourth covers the speculation of fuzzy Hv-ideals of Hv-rings. the ultimate bankruptcy discusses numerous connections among hypergroups and fuzzy units, and contains a learn at the organization among hypergroupoids and fuzzy units endowed with club features. as well as delivering a reference advisor to researchers, the ebook can be meant as textbook for undergraduate and graduate students.

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Deﬁne the L-fuzzy subset μ as follows: for all x ∈ X, μ(x) = μα }. Then, for all α ∈ X, we have μα = μα . 6. For any L-fuzzy subsets μ and λ on X, we have μ = λ if and only if, for any α ∈ X, we have μα = λα . 2 Groups and Fuzzy Subgroups After the introduction of fuzzy sets by Zadeh, reconsideration of the concept of classical mathematics began. On the other hand, because of the importance of group theory in mathematics, as well as its many areas of application, the notion of fuzzy subgroups was deﬁned by Rosenfeld in 1971 [160] and its structure was investigated.

Clearly, < μ > is the smallest fuzzy subgroup of G which contains μ. 10. Let μ be a fuzzy subset of a group G. Then, the following assertions are equivalent: (1) μ(xy) = μ(yx), for all x, y ∈ G; in this case, μ is called an abelian fuzzy subset of G. (2) μ(xyx−1 ) = μ(y), for all x, y ∈ G. (3) μ(xyx−1 ) ≥ μ(y), for all x, y ∈ G. (4) μ(xyx−1 ) ≤ μ(y), for all x, y ∈ G. (5) μ ◦ λ = λ ◦ μ, for every fuzzy subset λ of G. Proof. It is straightforward. 11. Let μ be a fuzzy subgroup of a group G. Then, μ is called a fuzzy normal subgroup of G if it is an abelian fuzzy subset of G.

If only the addition + is a hyperoperation and the multiplication · is a usual operation, then we say that R is an additive hyperring. A special case of this type is the hyperring introduced by Krasner [130]. 1. ) (e) z ∈ x + y implies y ∈ −x + z and x ∈ z − y. , x · 0 = 0 · x = 0. (3) The multiplication is distributive with respect to the hyperoperation +. The following elementary facts follow easily from the axioms: −(−x) = x and −(x + y) = −x − y, where −A = {−a | a ∈ A}. Also, for all a, b, c, d ∈ R we have (a + b) · (c + d) ⊆ a · c + b · c + a · d + b · d.