Ratio Mathematica Vol. 33, 2017, pp. 61-76 ISSN: 1592-7415 eISSN: 2282-8214 Some Remarks on Hyperstructures their Connections with Fuzzy Sets and Extensions to Weak Structures Piergiulio Corsini∗ †doi:10.23755/rm.v33i0.384 Abstract A brief excursus on the last results on Hyperstructures and their connec- tions with Fuzzy Sets. At the end a calculation of the Fuzzy Grade of Hv- structures of order two. Keywords: hyperstructure 2010 AMS Mathematics Subject Classification: 20N20 ∗Udine University, Italy; piergiuliocorsini@gmail.com † c©Piergiulio Corsini. Received: 31-10-2017. Accepted: 26-12-2017. Published: 31-12-2017. 61 Piergiulio Corsini 1 Introduction One knows that to every fuzzy set (H,µ0) one hypergroup can be associated (which I proved [9], is a join space) in the following way: ∀(x,y) ∈ H2, one sets x◦0 y = {z | min{µ0(x),µ0(y)}≤ µ0(z) ≤ max{µ0(x),µ0(y)}. I proved also [18] that to every hypergroupoid (H,◦) a fuzzy set corresponds, defined as you can see below: Set ∀u ∈ H, Q(z) = {(x,y) | u ∈ x◦y}, q(u) = |Q(u)|. A(u) = ∑ (x,y)∈Q(u) 1/|x◦y|, µ1(u) = A(u)/q(u). I proved that H0 = (H,◦) is a join space. So, to every hypergroupoid, a sequence of hypergroupoids and fuzzy sets is associated: (H,µ0), (H,µ1), ... If |H| < ℵ0, then the sequence is clearly finite. We call fuzzy grade [20] of (H,◦) the minimum natural number of k, such that two consecutive join spaces are isomorphic. For the Hv-structures, notion introduced by T. Vougiouklis, one can proceed in a similar way. So, one defines the fuzzy grade of a Hv-hypergroupoid as min{k | Hk ' Hk+1}. Thomas Vougiouklis is author of many papers on Hyperstructures. Just at the beginnng of his activity he invented and studied a structure, defining the following hyperoperation: given a hypergroupoid (H;∗) and a non empty subset P of H, he set x ◦ y = x ∗ P ∗ y and found several interesting results on this hyperoperation. But the most important theory that he introduced is that one of the Hv-hyperstructures. He replaced the notion of associativity with that one of ”weak associativity”. That is instead of supposing for every x,y,z ∈ H, (x∗y)∗z = x∗ (y ∗z), one supposes (x∗y)∗z ∩x∗ (y ∗z) 6= ∅. One has considered also weak rings. It is enough to set for every a,b,c in R, a◦ (b + c)∩ (a◦ b + a◦ c) 6= ∅. 62 Some Remarks on Hyperstructures their Connections with Fuzzy Sets and Extensions to Weak Structures The idea by Vougiouklis of considering weak Hyperstructures opened a new branch of Mathematics. Many significant results have been obtained in this field and probably many others will be found in the future. A theme which deserves to be considered in this context is that one of HX structures. HX-groups were born in China, invented by Li Hongxing [81], and studied by him, Wang and others, see [79], [80], [87], [117], [118], [119]. In Italy, Corsini extended this notion to Hyperstructures. He and Cristea in Italy, Fotea in Romania, Kellil and Bouaziz in Saudi Arabia worked in this direction. Given a group G and the set P∗(G) of all nonempty subsets of G, endowed with the operation ∀(A,B) ∈P∗(G)×P∗(G), A◦B = {xy | x ∈ A,y ∈ B} a subgroup of P∗(G) is called an HX-group. One has calculated the fuzzy grade for Z/nZ for n ≤ 16 and also for other structures, for instance for the multiplica- tive group Z2,22 and the direct product of some Z/nZ, see [22], [23], [24]. It would very interesting to consider the same problems in the such general context of weak structures, that is to calculate the fuzzy grade of HX-hypergroup Zn. Given an HX-group F , one considers the set F ′ of all nonempty subsets of F . Let us suppose that K is a subgroup of F. We define the following hyperoperation x⊗y = ⋃ x∈A, y∈B, {A,B}⊆K AB in the set ∪A∈KA. The structure (H,⊗) is called an HX-hypergroupoid. One can extend the notion of HX-hypergroup to weak hyperstructures. Some open problems on weak structures: • find conditions for an HX-hypergroupoid to be a hypergroup; • the fuzzy grade of HX- weak hypergroups already considered in the classic case. 63 Piergiulio Corsini 2 Hv-hypergroupoids of order 2 The Hv-hypergroupoids of order 2, which are not associative, are 10. The following [12], [13], [15] have fuzzy grade 1. The others [9], [10], [11], [14], [16], [17], [18] have fuzzy grade 2. • H12 a b a H H b b a We have q1(a) = 3, A1(a) = 2, so µ1(a) = 2/3. q1(b) = 3, A1(b) = 2, so µ1(b) = 2/3. Then ∂H12 = 1. • H13 a b a H b b H a We have q1(a) = 3, A1(a) = 2, so µ1(a) = 2/3. q1(b) = 3, A1(b) = 2, so µ1(b) = 2/3. Then ∂H13 = 1. • H15 a b a H a b b H Indeed we find q1(a) = 3, A1(a) = 2, so µ1(a) = 2/3. q1(b) = 3, A1(b) = 2, so µ1(b) = 2/3. Hence H1 = T , the total hypergroup. Therefore ∂H15 = 1. 64 Some Remarks on Hyperstructures their Connections with Fuzzy Sets and Extensions to Weak Structures • H9 a b a H b b b a We have q1(a) = 2, A1(a) = 3/2, so µ1(a) = 3/4 = 0.75. q1(b) = 3, A1(b) = 5/2, so µ1(b) = 5/6 = 0.8333. So we obtain H19 a b a a H b H b Therefore µ2(a) = µ2(b), whence H29 is the total hypergroup, whence ∂H9 = 2. • H10 a b a a H b b a We find q1(a) = 2, A1(a) = 5/2, so µ1(a) = 0.833. q1(b) = 2, A1(b) = 3/2, so µ1(b) = 3/4 = 0.75. We obtain H110 a b a a H b H b so µ2(a) = µ2(b), whence H210 is the total hypergroup, whence ∂H10 = 2. • H11 a b a b H b a b We find q1(a) = 2, A1(a) = 3/2, so µ1(a) = 0.75. q1(b) = 3, A1(b) = 5/2, so µ1(b) = 0.833. 65 Piergiulio Corsini It follows H111 a b a a H b H b so µ2(a) = µ2(b), so ∂H11 = 2. • H14 a b a H a b a H We find q1(a) = 4, A1(a) = 3, so µ1(a) = 0.75. q1(b) = 2, A1(b) = 1, so µ1(b) = 0.50. whence we hve H114 a b a a H b H b By consequence H214 is the total hypergroup, whence ∂H14 = 2. • H16 a b a a H b H a We find q1(a) = 4, A1(a) = 3, so µ1(a) = 3/4 = 0.75. q1(b) = 2,A1(b) = 1,µ1(b) = 0.50. It follows H116 a b a a H b H b so µ2(a) = µ2(b), whence H216 is the total hypergroup, whence ∂H16 = 2. 66 Some Remarks on Hyperstructures their Connections with Fuzzy Sets and Extensions to Weak Structures • H17 a b a H H b a H We find q1(a) = 4, A1(a) = 5/2, so µ1(a) = 0.625. q1(b) = 3, A1(b) = 3/2, so µ1(b) = 0.50. By cnsequence H117 a b a a H b H b whence H217 is the total hypergroup, therefore ∂H17 = 2. • H18 a b a H H b b H We find q1(a) = 3, A1(a) = 3/2, so µ1(a) = 0.50. q1(b) = 4, A1(b) = 5/2, so µ1(b) = 0.625. We obtain H118 a b a a H b H b By consequence, H218 is the total hypergroup, whence ∂H18 = 2. References [1] Ameri R., Nozari T., A connection between categories of (fuzzy) multialge- bras and (fuzzy) algebras, Italian Journal of Pure and Applied Mathematics, 27 (2010) [2] Ameri R., Shafiyan N., Fuzzy prime and primary hyperideals in hyperrings. Advances in Fuzzy Mathematics, n. 1-2. 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