RATIO MATHEMATICA 29 (2015) 53-64 ISSN:1592-7415 Solvable groups derived from fuzzy hypergroups E. Mohammadzadeh 1, T. Nozari 2 1 Department of Mathematics, Faculty of Science, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran, mohammadzadeh.e@pnurazavi.ac.ir 2 Department of Mathematics, Golestan University, Gorgan, Iran t.nozari@gu.ac.ir Abstract In this paper we introduce the smallest equivalence relation ξ∗ on a finite fuzzy hypergroup S such that the quotient group S/ξ∗, the set of all equivalence classes, is a solvable group. The characterization of solvable groups via strongly regular relation is investigated and several results on the topic are presented. Key words: Fuzzy hypergroups, strongly regular relation, solv- able groups, fundamental relation. 2000 AMS subject classifications: 8A72; 20N20, 20F18, 20F19. doi:10.23755/rm.v29i1.23 1 Introduction In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from Abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup. All Abelian groups are trivially solvable a subnormal series being given by just the group itself and the trivial group. But non-Abelian groups may or may not be solvable. A small example of a solvable, non-nilpotent group is the symmetric group S3. In fact, as the smallest simple non-Abelian group is A5, (the alternating 53 E. Mohammadzadeh and T. Nozari group of degree 5) it follows that every group with order less than 60 is solvable. The study of fuzzy hyperstructures is an interesting research topic for fuzzy sets. There are many works on the connections between fuzzy sets and hyperstructures. This can be considered into three groups. A first group of papers studies crisp hyperoperations defined through fuzzy sets. This study was initiated by Corsini in [3, 4] and then continued by other researchers. A second group of papers concerns the fuzzy hyperalgebras. This is a direct extension of the concept of fuzzy algebras. This was initiated by Zahedi in [12]. A third group was introduced by Corsini and Tofan in [5]. The basic idea in this group of papers is the following: a multioperation assigns to every pair of elements of S a non-empty subset of S, while a fuzzy multioperation assigns to every pair of elements of S a nonzero fuzzy set on S. This idea was continuated by Sen, Ameri and Chowdhury in [10] where fuzzy semihypergroups are introduced. The fundamental relations are one of the most important and interesting concepts in fuzzy hyperstructures that ordinary algebraic structures are derived from fuzzy hyperstructures by them. Fundamental relation α∗ on fuzzy hypersemigroups is studied in [1].Also in [8], the smallest strongly regular equivalence relation γ∗ on a fuzzy hypersemigroup S such that S/γ∗ is a commutative semigroup is studied. In this paper, we introduce and study the fundamental relation ξ∗ of a finite fuzzy hypergroup S such that S/ξ∗ is a solvable group. Finally, we introduce the concept of ξ-part of a fuzzy hypergroup and we determines necessary and sufficient conditions such that the relation ξ to be transitive. 2 Preliminary Recall that for a non-empty set S, a fuzzy subset µ of S is a function from S into the real unite interval [0, 1]. We denote the set of all nonzero fuzzy subsets of S by F∗(S). Also for fuzzy subsets µ1 and µ2 of S, then µ1 is smaller than µ2 and write µ1 ≤ µ2 iff for all x ∈ S, we have µ1(x) ≤ µ2(x). Define µ1 ∨µ2 and µ1 ∧µ2 as follows: ∀x ∈ S, (µ1 ∨µ2)(x) = max{µ1(x),µ2(x)} and (µ1 ∧µ2)(x) = min{µ1(x),µ2(x)}. A fuzzy hyperoperation on S is a mapping ◦ : S ×S 7→ F∗(S) written as (a,b) 7→ a◦ b = ab. The couple (S,◦) is called a fuzzy hypergropoid. Definition 2.1. A fuzzy hypergropoid (S,◦) is called a fuzzy hypersemigroup if for all a,b,c ∈ S, (a◦b) ◦c = a◦ (b◦c), where for any fuzzy subset µ of S (a◦µ)(r) =   ∨ t∈S ((a◦ t)(r) ∧µ(t)), µ 6= 0 0, µ = 0 54 Solvable groups derived from fuzzy hypergroups (µ◦a)(r) =   ∨ t∈S (µ(t) ∧ (t◦a)(r)), µ 6= 0 0, µ = 0 for all r ∈ S. Definition 2.2. Let µ,ν be two fuzzy subsets of a fuzzy hypergropoid (S,◦). Then we define µ ◦ ν by (µ ◦ ν)(t) = ∨ p,q∈S (µ(p) ∧ (p ◦ q)(t) ∧ ν(q)), for all t ∈ S. Definition 2.3. A fuzzy hypersemigroup (S,◦) is called fuzzy hypergroup if x◦S = S ◦x = χS, for all x ∈ S, where χS is characteristic function of S. Example 2.1. Consider a fuzzy hyperoperation ◦ on a non-empty set S by a ◦ b = χ{a,b}, for all a,b ∈ S. Then (S,◦) is a fuzzy hypersemigroup and fuzzy hypergroup as well. Theorem 2.1. Let (S,◦) be a fuzzy hypersemigroup. Then χa ◦χb = a◦ b, for all a,b ∈ S. Definition 2.4. Let ρ be an equivalence relation on a fuzzy hypersemigroup (S,◦), we define two relations ρ and ρ on F∗(S) as follows: for µ,ν ∈ F∗(S); µρν if µ(a) > 0 then there exists b ∈ S such that ν(b) > 0 and aρb, also if ν(x) > 0 then there exists y ∈ S, such that µ(y) > 0 and xρy. µρν if for all x ∈ S such that µ(x) > 0 and for all y ∈ S such that ν(y) > 0 , xρy. Definition 2.5. An equivalence relation ρ on a fuzzy hypersemigroup (S,◦) is said to be (strongly) fuzzy regular if aρb,a′ρb′ implies a◦a′ ρ b◦b′(a◦a′ ρ b◦b′). If ρ is a equivalence relation on a fuzzy hypersemigroup (S,◦), then we consider the following hyperoperation on the quotient set S/ρ as follows: for every aρ,bρ ∈ S/ρ aρ⊕ bρ = {cρ : (a′ ◦ b′)(c) > 0,aρa′,bρb′} Theorem 2.2. [2] Let (S,◦) be a fuzzy hypersemigroup and ρ be an equiva- lence relation on S. Then (i) the relation ρ is fuzzy regular on (S,◦) iff (S/ρ,⊕) is a hypersemigroup. (ii) the relation ρ is strongly fuzzy regular on (S,◦) iff (S/ρ,⊕) is a semi- group. 55 E. Mohammadzadeh and T. Nozari 3 New strongly regular relation ξ∗n Now in this paper we introduce and analyze a new strongly regular re- lation ξ∗n on a fuzzy hypergroup S such that the quotient group S/ξ ∗ n is solvable. Definition 3.1. Let (S,o) be a fuzzy hypergroup. We define 1) L0(S) = S 2) Lk+1(S) = {t ∈ S | (xy)(r) > 0, (tyx)(r) > 0, in which x,y ∈ Lk(S), for some r ∈ S}. for all k ≥ 0. Suppose that n ∈ N and ξn = ∪m≥1ξm,n, where ξ1,n is the diagonal relation and for every integer m > 1,ξm,n is the relation defined as follows: aξm,nb ⇐⇒ ∃x1, ...,xm ∈ H(m ∈ N),∃σ ∈ Sm : σ(i) = i, if zi 6∈ Ln(H) : (x1o...oxm)(a) > 0 and (xσ1o...oxσm )(b) > 0. It is clear that ξn is symmetric. Define for any a ∈ S, a(a) = (χa)(a) = 1, thus ξn is reflexive. We take ξ ∗ n to be transitive closure of ξn. Then it is an equivalence relation on H. Corolary 3.1. For every n ∈ N, we have α∗ ⊆ ξ∗n ⊆ γ∗. Theorem 3.1. For every n ∈ N, the relation ξ∗n is a strongly regular relation. Proof. Suppose n ∈ N. Clearly, ξm,n is an equivalence relation. First we show that for each x,y,z ∈ S xξny ⇒ xzξnyz, zxξnzy (∗). If xξny, then there exists m ∈ N such that xξm,ny, and so there exist (z1, . . . ,zm) ∈ Sm and σ ∈ Sm such that if zi 6∈ Ln(S) then m∏ i=1 zi(x) > 0, m∏ i=1 zσ(i)(y) > 0. Let z ∈ S, for any r,s such that (xz)(r) > 0 and (yz)(s) > 0. We have (( ∏m i=1 zi)z)(r) = ∨ p{( m∏ i=1 zi)(p)∧(pz)(r)}. Let p = x, then (( m∏ i=1 zi)(z))r > 0,σ(i) = i, if zi 6∈ Ln(S), (( m∏ i=1 zσ(i))(z))(s) = ∨ q {( m∏ i=1 zσ(i))(q) ∧ (qz)(s)}. Let q = y, then (( m∏ i=1 zσ(i))(z))(s) > 0, and σ(i) = i, if zi 6∈ Ln(S). Now suppose that zm+1 = z and we define 56 Solvable groups derived from fuzzy hypergroups σ ′ ∈ Sm + 1: σ′(i) = { σ(i), ∀i ∈{1, 2, . . . ,m} m + 1, i = m + 1. Thus for all r,s ∈ S; ( m∏ i=1 zi)(r) > 0, ( m∏ i=1 z ′ σ)(s) > 0; σ ′ (i) = i if zi 6∈ Ln(S). Therefore xzξnyz. Now if xξ∗ny, then there exists k ∈ N and u0 = x,u1, . . . ,uk = y ∈ S such that u0 = xξnu1ξnu2ξn . . .ξnum = y, by the above result we have u0z = xzξnu1zξnu2zξn . . .ξnukz = yz and so xzξnyz. Similarly we can show that zxξnzy. Therefore ξ ∗ n is a strongly regular relation on S. 2 Proposition 3.1. For every n ∈ N, we have ξ∗n+1 ⊆ ξ∗n. Proof. Let xξn+1y so ∃(z1, ...,zm) ∈ Sm;∃δ ∈ Sm : δ(i) = i if zi 6∈ Ln+1(S), such that ( m∏ i=1 zi)(x) > 0, ( m∏ i=1 zδ(i))(y) > 0. Now let δ1 = δ, since Ln+1(S) ⊆ Ln(S) so xξny.2 The next result immediately follows from previous theorem. Corolary 3.2. If S is a commutative hypergroup, then β∗ = ξ∗n. A group G is solvable if and only if G(n) = {e} for some n ≥ 1 in which, G(0) = G, G(1) = G ′ , commutator subgroup of G, and inductively G(i) = (G(i−1)) ′ . Theorem 3.2. If S is a fuzzy hypergroup and ϕ is a strongly regular relation on S, then Lk+1(S/ϕ)) = 〈t | t ∈ Lk(S)〉 for k ∈ N. Proof. Suppose that G = S/ϕ and x = ϕ(x) for all x ∈ S. We prove the theorem by induction on k. For k = 0 we have L1(G) = 〈t | t ∈ L0(S)〉. Now suppose that a = t where t ∈ Lk+1(S) so there exist r1 ∈ S ; (xy)(r1) > 0, (tyx)(r1) > 0 in which x,y ∈ Lk(S). Then xy = z1; (xy)(z1) > 0 and so xy = r1. Also tyx = z2; (tyx)(z2) > 0 and tyx = r1 = xy which implies that t = [x,y]. By hypotheses of induction we conclude that t ∈ Lk+1(G). Hence a = [t,s] ∈ Lk+2(G). Conversely, let a ∈ Lk+2(G). Then a = [x,y], where x,y ∈ Lk+1(G), so by hypotheses of induction we have x = u and y = v, where u,v ∈ Lk(S). Let c ∈ S; (uv)(c) > 0 we show that there exists t ∈ S such that (tvu)(c) > 0. Since S ◦u = χS and c ∈ S then there exists r ∈ S such that (ru)(c) > 0 and so by r ∈ S = S ◦v there exist t ∈ S; (tv)(r) > 0. Therefore (tvu)(c) = ∨ n((tv)(n)∧(nu)(c)) ≥ (tv)(r)∧(ru)(c) > 0. Thus (uv)(c) > 0, (tvu)(c) > 0 which implies that t ∈ Lk+1(S). Now since 57 E. Mohammadzadeh and T. Nozari uv = c = tvu, then t = [u,v] = [x,y] = a and t ∈ Lk+1(S). Therefore, a = t ∈ 〈t; t ∈ Lk+1(S)〉.2 Theorem 3.3. S/ξ∗n is a solvable group of class at most n + 1. Proof. Using Theorem 3.2, Lk(S/ξ ∗ n)is an Abelian group and Lk+1(S/ξ ∗ n) = {e}. 2 4 On solvable groups derived from finite fuzzy hypergroups In this section we introduce the smallest strongly relation ξ∗ on a finite fuzzy hypergroup S such that H/ξ∗ is a solvable group. Definition 4.1. Let S be a finite fuzzy hypergroup. Then we define the relation ξ∗ on S by ξ∗ = ⋂ n≥1 ξ∗n. Theorem 4.1. The relation ξ∗ is a strongly regular relation on a finite fuzzy hypergroup S such that S/ξ∗ is a solvable group. Proof. Since ξ∗ = ⋂ n≥1 ξ ∗ n, it is easy to see that ξ ∗ is a strongly regular relation on S. By using Proposition 3.1, we conclude that there exists k ∈ N such that ξ∗k+1 = ξ ∗ k. Thus ξ∗ = ξ ∗ k for some k ∈ N. 2 Theorem 4.2. The relation ξ∗ is the smallest strongly regular relation on a finite fuzzy hypergroup S such that S/ξ∗ is a solvable group. Proof. Suppose ρ is a strongly regular relation on S such that K = S/ρ is a solvable group of class c. Suppose that xξy. Then xξny, for some n ∈ N and so there exists m ∈ N such that xξmny ⇐⇒ ∃(z1, ..zm) ∈ Sm,∃δ ∈ Sm : δ(i) = i if zi 6∈ Ln(S) such that ( ∏m i=1 zi)(x) > 0, ( ∏m i=1 zδ(i))(y) > 0, Lc+1(S/ρ) = 〈ρ(t); t ∈ Lc(S)〉 = {ρ(e)}, and so ρ(zi) = ρ(e), for every zi ∈ Lc(S). Therefore ρ(x) = ρ(y), which implies that xρy.2 58 Solvable groups derived from fuzzy hypergroups 5 Transitivity of ξ∗ In this section we introduce the concept of ξ-part of a fuzzy hypergroup and we determine necessary and sufficient condition such that the relation ξ to be transitive. Definition 5.1. Let X be a non-empty subset of S. Then we say that X is a ξ-part of S if the following condition holds: for every k ∈ N and (z1, ...,zm) ∈ Hm and for every σ ∈ Sk such that σ(i) = i if zi 6∈ ∪n≥1Ln(S), and there exists x ∈ X such that ( m∏ i=1 zi)(x) > 0, then for all y ∈ S\X, ( m∏ i=1 zσ(i))(y) = 0. Theorem 5.1. Let X be a non-empty subset of a fuzzy hypergroup S. Then the following conditions are equivalent: 1) X is a ξ-part of S, 2) x ∈ X, xξy =⇒ y ∈ X, 3) x ∈ X, xξ∗y =⇒ y ∈ X. Proof. (1) =⇒ (2) if (x,y) ∈ S2 is a pair such that x ∈ X and xξy, then there exist (z1, ...,zi) ∈ Sk; ( m∏ i=1 zi)(x) > 0, ( m∏ i=1 zσ(i))(y) > 0 and σ(i) = i if zi 6∈ ∪n≥1Ln(S). Since X is a ξ-part of S, we have y ∈ X. (2) =⇒ (3) Suppose that (x,y) ∈ S2 is a part such that x ∈ X and xξ∗y. Then there is (z1, ...,zi) ∈ Sk such that x = z0ξz1ξ...ξzk = y. Now by using (2) k-times we obtain y ∈ X. (3) =⇒ (1) For every k ∈ N and (z1, ...,zi) ∈ Sk and for every σ ∈ Sk such that σ(i) = i if zi 6∈ ∪n≥1Ln(S), then there exists x ∈ X; ( m∏ i=1 zi)(x) > 0 and there exist y ∈ S\X ; ( ∏ i=1 zσ(i))(y) > 0, then xξny and so xξy. Therefore by (3) we have y ∈ X which is a contradiction.2 Theorem 5.2. The following conditions are equivalent: 1) for every a ∈ H, ξ(a) is a ξ-part of S, 2) ξ is transitive. Proof. (1) =⇒ (2) Suppose that xξ∗y. Then there is (z1, ...,zi) ∈ Sk such that x = z0ξz1ξ...ξzk = y, since ξ(zi) for all 0 ≤ i ≤ k, is a ξ-part, we have zi ∈ ξ(zi−1), for all 1 ≤ i ≤ k. Thus y ∈ ξ(x), which means that xξy. (2) =⇒ (1) Suppose that x ∈ S, z ∈ ξ(x) and zξy. By transitivity of ξ, we have y ∈ ξ(x). Now according to the last theorem, ξ(x) is a ξ-part of S.2 59 E. Mohammadzadeh and T. Nozari Definition 5.2. The intersection of all ξ-parts which contain A is called ξ-closure of A in S and it will be denoted by K(A). In what follows, we determine the set W(A), where A is a non-empty subset of S. We set 1) W1(A) = A and 2) Wn+1(A) = {x ∈ S | ∃(z1, ...,zi) ∈ Sk : ( m∏ i=1 z(i))(x) > 0,∃σ ∈ Sk such that σ(i) = i, if zi 6∈ ∪n≥1Ln(S) and there exists a ∈ Wn(A); ( m∏ i=1 zσ(i))(a) > 0}. We denote W(A) = ⋃ n≥1 Wn(A). Theorem 5.3. For any non-empty subset of S, the following statements hold: 1) W(A) = K(A), 2) K(A) = ∪a∈AK(a). Proof. 1) It is enough to prove: a) W(A) i a ξ-part, b) if A ⊆ B and B is a ξ-part, then W(A) ⊆ B. In order to prove (a), suppose that a ∈ W(A) such that ( ∏ i=1 zi)(a) > 0 and σ ∈ Sk such that σ(i) = i, if zi 6∈ ∪n≥1Ln(S). Therefore, there exists n ∈ N such that ( m∏ i=1 zi)(a) > 0 a ∈ Wn(A). Now if there exists t ∈ S such that ( ∏ i=1 zσ(i))(t) > 0 we obtain t ∈ Wn+1(A). Therefore, t ∈ W(A) which is a contradiction. Thus ( m∏ i=1 zσ(i))(t) = 0 and so W(A) is a ξ-part. Now we prove (b) by induction on n. We have W1(A) = A ⊆ B. Suppose that Wn(A) ⊆ B. We prove that Wn+1(A) ⊆ B. If z ∈ Wn+1(A), then there exists k ∈ N; (z1, ...,zk) ∈ Sk; ( m∏ i=1 zi)(z) > 0 and there exists σ ∈ Sk such that σ(i) = i,if zi 6∈ ∪t≥1Lt(S) and there exists t ∈ Wn(A) ; ( m∏ i=1 zσi )(t) > 0, since Wn(A) ⊆ B we have t ∈ B and ( m∏ i=1 zσi )(t) > 0. Now since B is ξ-part , ( m∏ i=1 zi)(z) > 0 then z ∈ B. 60 Solvable groups derived from fuzzy hypergroups 2) It is clear that for all a ∈ A, K(a) ⊆ K(A). By part 1), we have K(A) = ∪n≥1Wn(A) and W1(A) = A = ∪a∈A{a}. It is enough to prove that Wn(A) = ∪a∈AWn(a), for all n ∈ N. We follow by induction on n. Suppose it is true for n. We prove that Wn+1(A) = ∪a∈AWn+1(a). If z ∈ Wn+1(A), then there exists k ∈ N, (z1, ...,zk) ∈ Sk; ( m∏ i=1 zi)z > 0 and there exists σ ∈ Sk such that σ(i) = i, if zi 6∈ ∪t≥1Lt(S) and there exist a ∈ Wn(A); ( m∏ i=1 zσ(i))(a) > 0. By the hypotheses of induction there exists a ∈ Wn(A) = ∪b∈AWn(b); ( m∏ i=1 zσ(i))(a ′ ) > 0 for some a ′ ∈ Wn(b) in which b ∈ A. Therefore, z ∈ Wn+1(b), and so Wn+1(A) ⊆∪b∈AWn+1(b). Hence K(A) = ∪a∈AK(a).2 Theorem 5.4. The following relation is equivalence relation on H. xWy ⇐⇒ x ∈ W(y), for every (x,y) ∈ S2, where W(y) = W({y}). Proof. It is easy to see that W is reflexive and transitive. We prove that W is symmetric. To this, we check that: 1) for all n ≥ 2 and x ∈ S, Wn(W2(x)) = Wn+1(x), 2) x ∈ Wn(y) if and only if y ∈ Wn(x). We prove (1) by induction on n. W2(W2(x)) = {z | ∃q ∈ N, (a1, ...,aq) ∈ Sq; ( ∏ i=1 ai)(z) > 0 and ∃σ ∈ Sk such that σ(i) = i, if zi 6∈ ∪s≥1Ls(S) and ∃y ∈ W2(x); ( m∏ i=1 aσ(i))(y) > 0} = W3(x). Now we proceed by induction on n. Suppose Wn(W2(x)) = Wn+1(x) then Wn+1(W2(x)) = {z | ∃q ∈ N, (a1, ...,aq) ∈ Sq; ( m∏ i=1 ai)(z) > 0 and ∃σ ∈ Sk such that σ(i) = i, if zi 6∈ ∪s≥1Ls(S) and ∃t ∈ Wn(W2(x)); ( ∏ i=1 aσ(i))(t) > 0} = Wn+2(x). Now we prove (2) by induction on n, too. It is clear that x ∈ W2(y) if and only if y ∈ W2(x). Suppose x ∈ Wn(y) if and only if y ∈ Wn(x). Let x ∈ Wn+1(y), then there exists q ∈ N, (a1, ...,aq) ∈ Sq; ( m∏ i=1 ai)(x) > 0 and ∃σ ∈ Sk such that σ(i) = i, if ai 6∈ 61 E. Mohammadzadeh and T. Nozari ∪s≥1Ls(S) and ∃t ∈ Wn(y); ( m∏ i=1 aσ(i))t > 0. Now, ( m∏ i=1 ai)(x) > 0, x ∈ W1(x) and ( m∏ i=1 aσ(i))(t) > 0 implies that t ∈ W2(x). Since t ∈ Wn(y), then by hypotheses of induction y ∈ Wn(t) and we see that t ∈ W2(x), therefore y ∈ Wn(W2(x)) = Wn+1(x). 2 Remark 5.1. If S is a fuzzy hypergroup, then S/ξ∗ is a group. We define ωS = φ −1(1S/ξ∗), in which φ : S → S/ξ∗ is the canonical projection. Lemma 5.1. If S is a fuzzy hypergroup and M is a non-empty subset of S, then (i) φ−1(φ(M)) = {x ∈ S : (ωSM)(x) > 0} = {x ∈ S : (MωS)(x) > 0} (ii) If M is a ξ part of S, then φ−1(φ(M)) = M. Proof. (i) Let x ∈ S and (t,y) ∈ ωS × M such that (ty)(x) > 0, so φ(x) = φ(t) ⊕ φ(y) = 1S/ξ∗ ⊕ φ(y) = φ(y), therefore x ∈ φ−1(φ(y)) ⊂ φ−1(φ(M)). Conversely, for every x ∈ φ−1(φ(M)), there exists b ∈ M such that φ(x) = φ(b). By reproducibility, a ∈ S exists such that (ab)(x) > 0, so φ(b) = φ(x) = φ(a) ⊕φ(b). This implies φ(a) = 1S/ξ∗ and a ∈ φ−1(1S/ξ∗) = ωS. Therefore (ωSM)(x) > 0. In the same way, we can prove that φ−1(φ(M)) = {x ∈ S : (MωS)(x) > 0}. (ii) We know M ⊆ φ−1(φ(M)). If x ∈ φ−1(φ(M)), then there exists b ∈ M such that φ(x) = φ(b). Therefore x ∈ ξ∗(x) = ξ∗(b). Since M is a ξ part of S and b ∈ M, by Lemma 5.1, we conclude ξ∗(b) ⊆ M and x ∈ M. 2 Definition 5.3. Let (S, ·) be a fuzzy hypergroup. K ⊆ S is called a fuzzy subhypergroup of S if i) (a · b) · c = a · (b · c), for all a,b,c ∈ S ii) a ·K = χK, for all a ∈ K. Theorem 5.5. ωS is a fuzzy subhypergroup of S, which is also a ξ-part of S. Proof. Clearly, ωS ⊆ S and so (a·b)·c = a·(b·c), for all a,b,c ∈ ωS. Now we show that ωSy = χωS for all y ∈ ωS. Let x,y ∈ ωS, then there exists u ∈ S such that (uy)(x) > 0. Therefore, uy = x, which implies that u = 1. Thus u ∈ ωS. Consequently, ωSy = χωS . Hence, ωS is a fuzzy subhypergroup of S. Now we prove that K(y) = φ−1(φ({y})) = {x ∈ S : (ωSy)(x) > 0} = ωS. 62 Solvable groups derived from fuzzy hypergroups z ∈ φ−1(φ({y})) ⇐⇒ ϕ(z) = ϕ(y) ⇐⇒ ξ∗(z) = ξ∗(y) ⇐⇒ zξ∗y ⇐⇒ z ∈ ξ∗(z) = ω({y}) = K(y). Also since y ∈ ωS, then {x ∈ S : (ωSy)(x) > 0} = {x ∈ S : (χωS )(x) > 0} = ωS. Therefore K(y) = ωS and so ωS is ξ part. 2 Acknowledgment The first outer was supported by a grant from Payame Noor University. 63 E. Mohammadzadeh and T. Nozari References [1] R. Ameri, T. Nozari, Complete parts and fundamental relation on fuzzy hypersemigroups, J. of Mult.-Valued Logic. Soft Computing, Vol. 19 (2011) 451-460. [2] R. Ameri, T. Nozari, Fuzzy sets, join spaces and factor spaces, Pure Math. Appll. 11 (3) (2000) 439-446. [3] P. 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