RATIO MATHEMATICA 24 (2013), 11–30 ISSN: 1592-7415 On Fuzzy Gamma Hypermodules R. Ameri, R. Sadeghi School of Mathematics, Statistic and Computer Sciences, College of Science, University of Tehran, Iran Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran rameri@ut.ac.ir, razi-sadeghi@yahoo.com Abstract Let R be a Γ-hyperring and M be an Γ-hypermodule over R. We introduce and study fuzzy RΓ-hypermodules. Also, we associate a Γ- hypermodule to every fuzzy Γ-hypermodule and investigate its basic properties. Key words: Γ-hyperring, Γ-hypermodule, fundamental relation, fuzzy Γ-hypermodule. MSC2010: 20N20. 1 Introduction Hyperstructure theory was born in 1934 when Marty [13] defined hy- pergroups, began to analysis their properties and applied them to groups. Algebraic hyperstructures are a suitable generalization of classical algebraic structures. Zadeh [18] introduced the notion of a fuzzy subset of a non-empty set X, as a function from X to [0, 1]. Rosenfeld [15] defined the concept of fuzzy group. Since then many papers have been published in the field of fuzzy algebra. In [16], Sen, Ameri and Chowdhury introduced the notions of fuzzy hypersemigroups and obtained a characterization of them. Then in [10], Leoreanu-Fotea and Davvaz introduced and analyzed the fuzzy hyperring no- tion and in [11], Leoreanu-Fotea introduced the fuzzy hypermodule notion and obtained a connection between hypermodules and fuzzy hypermodules (for more information about fuzzy hypersrtuctures see [1]-[6]). The notion R. Ameri, R. Sadeghi of a Γ-ring was introduced by N. Nobusawa in [14]. Recently, W.E. Barnes [7], J. Luh [12], W.E. Coppage studied the structure of Γ-rings and obtained various generalization analogous of corresponding parts in ring theory. In [3] Ameri, Sadeghi introduced the notion of Γ-module over a Γ-ring. Now in this paper we introduced and study fuzzy Γ-hypermodules as generalization of Γ-hypermodule as well as fuzzy modules. The paper has been prepared in 5 sections. In section 2, we introduce some definitions and results of Γ-hypermodules and fuzzy sets which we need to developing our paper. In section 3, we introduced and study fuzzy Γ-hypermodules and obtain its basic results. In section 4, we study fundamental relation of fuzzy Γ-hypermodules. 2 Preliminaries In this section, we present some definitions which need to developing our paper. As it is well known a hypergroupoid is a set together with a function ◦ : H × H −→ P?(H), which is called a hyperoperation, where P?(H) denotes the set of all nonempty subsets of H. A hypergroupoid (H,◦), which is associative, that is x ◦ (y ◦ z) = (x ◦ y) ◦ z for all x,y,z ∈ H is called a semihypergroup. A hypergroup is a semihypergroup such that for all x ∈ H we have x ◦ H = H = H ◦ x (called the reproduction axiom). We say that a hypergroup H is canonical hypergroup if it is commutative, it has a scalar identity, every element has a unique inverse and it is reversible (for more details of hypergroups see [9]). Definition 2.1. The triple (R, +, .) is a hyperring (in the sense of Krasner) if the following hold: (i) (R, +) is a commutative hypergroup; (ii) (R,.) is a semihypergroup; (iii) the hyperoperation ”.” is distributive over the hyperoperation ”+”, which means that for all r,s,t of R we have: r.(s + t) = r.s + r.t and (r + s).t = r.t + s.t ( for more about hyperrings see [9] and [11]). Definition 2.2. Let (R,],◦) be a hyperring. A nonempty set M, endowed with two hyperoperations ⊕,� is called a left hypermodule over (R,],◦) if the following conditions hold: (1) (M,⊕) is a commutative hypergroup; (2) � : R×M −→ P∗(M) is such that for all a,b ∈ M and r,s ∈ R we have (i) r � (a⊕ b) = (r �a) ⊕ (r � b); (ii) (r ]s) �a = (r �a) ⊕ (s�a); (iii) (r ◦s) �a = r � (s�a). For more details about hypermodules see [8], [9], [?] and [18]). 12 On Fuzzy Gamma Hypermodules Definition 2.3. ([7]) Let R and Γ be additive abelian groups. We say that R is a Γ − ring if there exists a mapping · : R× Γ ×R −→ R (r,γ,r′) 7−→ r.γ.r′ (= rγr′) such that for every a,b,c ∈ R and α,β ∈ Γ, the following conditions hold: (i) (a + b)αc = aαc + bαc; a(α + β)c = aαc + aβc; aα(b + c) = aαb + aαc; (ii) (aαb)βc = aα(bβc). Definition 2.4. Let R be a Γ-ring. A (left )gamma module over R is an additive abelian group M together with a mapping . : R× Γ ×M −→ M ( the image of (r,γ,m) being denoted by rγm), such that for all m,m1,m2 ∈ M and γ,γ1,γ2 ∈ Γ and r,r1,r2 ∈ R the following conditions are satisfied: (GM1) r.γ.(m1 + m2) = r.γ.m1 + r.γ.m2; (GM2) (r1 + r2).γ.m = r1.γ.m + r2.γ.m; (GM3) r.(γ1 + γ2).m = r.γ1.m + r.γ2.m; (GM4) r1.γ1.(r2.γ2.m) = (r1.γ1.r2).γ2.m. A right gamma module over R is defined in analogous manner. In this case we say that M is a left(or right) RΓ-module (for more details about gamma modules see [2]). Let (H,◦) be a hypergroupoid. If {A,B} ⊆ P∗(H) and ρ is an equivalence relation on H, then we denote Aρ̄B if ∀a ∈ A, ∃b ∈ B : aρb, and, ∀b ∈ B, ∃a ∈ A : aρb. We denote A ¯̄ρ B if ∀a ∈ A, ∀b ∈ B we have aρb. An equivalence relation ρ on H is called regular (strongly regular ) if for all a,a′,b,b′ of H. The following implication holds: aρb,a′ρb′ =⇒ (a◦a′)ρ̄(b◦ b′) (aρb,a′ρb′ =⇒ (a◦a′) ¯̄ρ(b◦ b′)). Theorem 2.1. ([17]) Let (M, +, .) be a hypermodule over a hyperring R, let δ be an equivalence relation on M and let ρ be an strongly regular relation on R. The following statements hold: (1) if δ is strongly regular on M and ∀x,y ∈ M and ∀r ∈ R the hyperopera- tions: 13 R. Ameri, R. Sadeghi δ(x) ⊕ δ(y) = {δ(z) | z ∈ x + y} and ρ(r) � δ(x) = {δ(z) | z ∈ r.x}, is define a module structure on M/δ over R/ρ; (2) if (M/δ,⊕,�) is a module over R/ρ, then δ is strongly regular on M. The relation δ∗ is the smallest strongly regular relation on the hypermod- ule (M, +, .) such that (M/δ,⊕,�) the quotient structure (M/δ,⊕,�) is a module over the ring R/ρ, and it is called the fundamental relation over hypermodule M. Hence, δ∗ is the smallest equivalence relation on M, such that M/δ∗ is a module over the ring R/ρ∗, where ρ∗ is fundamental relation on R. If we denote by U the set of all expressions consisting of finite hyperoperations either on R and M or the external hyperoperation applied on finite sets of elements of R and M, then we have xδy ⇐⇒∃u ∈U, such that {x,y}⊂ u. δ∗ is the transitive closure of δ. In the fundamental module (M/δ∗,⊕,�) over R/ρ∗, the hyperoperations ⊕ and � are defined as follows: ∀x,y ∈ M and ∀z ∈ δ∗(x) ⊕ δ∗(y), we have δ∗(x) � δ∗(y) = δ∗(z); ∀r ∈ R, ∀x ∈ M and ∀z ∈ δ∗(r).δ∗(x), we have ρ∗(r) � δ∗(x) = δ∗(z), (for more details about the fundamental relation on hyperstructures see [8] and [9]). Definition 2.5. A multivalued system (R, +, .) is a Γ-hyperring if the fol- lowing hold: (i) (R, +) and Γ are canonical hypergroups; (ii) (R,.) is semihypergroup. (iii) (.) is distributive with respect to (+), i.e., for all x,y,z in R we have x.(y + z) = (x.y) + (x.z) and (x + y).z = (x.z) + (y + z). Definition 2.6. Let (R,],◦) be a Γ-hyperring and (Γ,∗) be a canonical hy- pergroup. We say that (M, +, .) is a left Γ − hypermodule over R, if (M, +) be a canonical hypergroup and there exists a mapping · : R× Γ ×M −→ P?(M) (r,γ,m) 7−→ r ·γ ·m such that for every r,s ∈ R and α,β ∈ Γ and a,b ∈ M, the following conditions are satisfied: (GHM1) (i) (r ]s).α.a = r.α.a + s.α.a; (ii) r.(α∗β).a = r.α.a + r.β.a; (iii) r.α.(a + b) = r.α.a + r.α.b; (GHM2) (r ◦α◦s).β.a = r.α.(s.β.a). 14 On Fuzzy Gamma Hypermodules A right Γ-hypermodule of R is defined in a similar way. In this case we say that M is a RΓ-hypermodule. 3 Fuzzy Gamma Subhypermodules In the sequel R is a Γ-hyperring and all gamma hypermodules are con- sidered over R. In [16] M.K. Sen, R. Ameri, G. Chowdhury introduced the notion of fuzzy semihypergroups, in [10] V. Leoreanu-Fotea, B. Davvaz study fuzzy hyperrings and V. Leoreanu-Fotea in [11] studied fuzzy hypermodules. Now in this section we follows these and introduce and studied fuzzy gamma hypermodules. Let S and Γ be two nonempty sets. F∗(S) denotes the set H of all nonzero fuzzy subset of S. A Fuzzy Γ − hyperoperation on S is a map ◦ : S × Γ × S −→ F∗(S), which associates a nonzero subset a ◦ γ ◦ b for all a,b ∈ S and γ ∈ Γ. (S,◦) is called a Fuzzy Γ − hypergroupoid . A fuzzy Γ-hypergroupoid (S,◦) is called a fuzzy Γ-hypersemigroup if for all a,b,c ∈ S and α,β ∈ Γ, we have a ◦ α ◦ (b ◦ β ◦ c) = (a ◦ α ◦ b) ◦ β ◦ c, where for any µ ∈ F∗(S), we have (a◦γ ◦µ)(r) = ∨ t∈S((a◦γ ◦ t)(r) ∧µ(t)) and (µ◦γ ◦a)(r) = ∨ t∈S(µ(t) ∧ (t◦γ ◦a)(r)) for all r ∈ S,γ ∈ Γ. If A is a nonempty subset of S and x ∈ S, then for all r ∈ S,γ ∈ Γ we have: (x◦γ ◦A)(r) = ∨ a∈A (x◦γ ◦a)(r), and (A◦γ ◦x)(r) = ∨ a∈A (a◦γ ◦x)(r). A fuzzy Γ-hypersemigroup (S,◦) is called a fuzzy Γ-hypergroup if for all a ∈ S and γ ∈ Γ, we have a◦γ◦S = S◦γ◦a = χS. We say that an element e of (S,◦) is identity (resp. scalar identity) if for all s,r ∈ S,γ ∈ Γ, we have (e◦γ ◦ r)(r) > 0, and (r ◦γ ◦e)(r) > 0, ((e◦γ ◦ r)(s) > 0, and (r ◦γ ◦e)(s) > 0 itfollowsr = s). Let (S,◦) be a fuzzy hypergroup, endowed with at least an identity. An element a′ ∈ S is called an inverse of a ∈ S if there is an identity e ∈ S, such that 15 R. Ameri, R. Sadeghi (a◦a′)(e) > 0, and (a′ ◦a)(e) > 0. Definition 3.1. A fuzzy hypergroup S is regular if it has at least one identity and each element has at least one inverse. A regular fuzzy hypergroup (S,◦) is called reversible if for any x,y,a ∈ S, it satisfies the following conditions: (1) if (a◦x)(y) > 0, then there exists an inverse a1 of a, such that (a1◦y)(x) > 0; (2) if (x◦a)(y) > 0, then there exists an inverse a2 of a, such that (y◦a2)(x) > 0. Definition 3.2. We say that a fuzzy hypergroup S is a fuzzy canonical if (1) it is commutative; (2) it has an scalar identity; (3) every element has a unique inverse; (4) it is reversible. Let µ and ν be two nonzero fuzzy subsets of a fuzzy Γ-hypergroupoid (S,◦). We define (µ◦γ ◦ν)(t) = ∨ p,q∈S (µ(p) ∧ (p◦γ ◦ q)(t) ∧ν(q),∀t ∈ S,γ ∈ Γ. In the following we introduce and study fuzzy gamma hyperrings . Definition 3.3. Let R, Γ be two nonempty sets and �,� be two fuzzy hy- peroperations on R and ⊗ be a fuzzy hyperoperation on Γ. Let (R,�) and (Γ,⊗) be two canonical fuzzy hypergroups. R is called a fuzzy Γ-hyperring if there exists the mapping: � : R× Γ ×R −→ F∗(R) (r,γ,s) 7−→ r �γ �s, such that for all r,s,t ∈ R,α,β ∈ Γ, the following conditions are satisfied: (i) r �α� (s� t) = (r �α�s) � (r �α� t); (ii) r � (α⊗β) �s = (r �α�s) � (r �β �s); (iii) (r �s) �α� t = (r �α� t) � (s�α� t); (iv) r �α� (s�β � t) = (r �α�s) �β � t. Definition 3.4. Let (Γ,⊗) be a fuzzy canonical hypergroups. Let (R,�,�) be a fuzzy Γ-hyperring. A nonempty set M, endowed with two fuzzy Γ- hyperoperation ⊕,� is called a left fuzzy Γ-hypermodule over (R,�.�) if the following conditions hold: 16 On Fuzzy Gamma Hypermodules (1) (M,⊕) is a canonical fuzzy Γ-hypergroup; (2) � : R × Γ × M −→ F∗(M) is such that for all a,b ∈ M,r,s ∈ R and α,β ∈ Γ we have (i) r �α� (a⊕ b) = (r �α�a) ⊕ (r �α� b); (ii) (r �s) �α�a = (r �α�a) ⊕ (s�α�a); (iii) r � (α⊗β) �a = (r �α�a) ⊕ (r �β �a); (iv) r �α� (s�β �a) = (r ·α ·s) �β �a. If both (R,�), (Γ,⊗) and (M,⊕) have scaler identities, denoted by 0R, 0Γ and 0M , then the fuzzy Γ-hypermodule (M,⊕,�) also satisfies the condition: 0R �γ �a = χ0M , r � 0Γ �a = χ0Γ, r �γ � 0M = χ0M , for all r ∈ R,γ ∈ Γ,a ∈ A. Moreover, if (R,�) has an identity, say 1, then the fuzzy Γ-hypermodule (M,⊕,�) is called unitary if it satisfies the condition: for all a of M, we have 1 �γ �a = χa. Clearly, any fuzzy Γ-hyperring is a fuzzy Γ-hypermodule over itself. Proposition 3.5. Let (M, +, .) be a module over a ring (R,],◦) and Γ = R. We define the following fuzzy Γ-hyperoperations: for a,b of M, a⊕ b = χ{a,b}, for all a of M and r ∈ R,γ ∈ Γ, r �γ �a = χ{r.γ.a}, for all r,s of R, r �s = χ{r,s} and r �γ �s = χ{r◦γ◦s}. Then (M,⊕,�) is a fuzzy Γ-hypermodule over the fuzzy Γ-hyperring (R,�,�). Note that the last theorem is satisfied, when M is a Γ-module over a Γ-ring R, such that Γ 6= R. Proposition 3.6. Let (R,◦) and (S,•) be two fuzzy Γ-hyperrings. Let (M,⊕,�) be a left fuzzy Γ-hypermodule over R and a right fuzzy Γ-hypermodule over S. Then A = { ( r m 0 s ) | r ∈ R,s ∈ S,m ∈ M} is a fuzzy Γ-hyperring and fuzzy Γ-hypermodule over A, under the mappings ? : A× Γ ×A −→ F∗(A) ( ( r m 0 s ) ,γ, ( r1 m1 0 s1 ) ) 7−→( r ◦γ ◦ r1 r �γ �m1 ⊕m�γ �s1 0 s•γ •s1 ) . 17 R. Ameri, R. Sadeghi such that( r ◦γ ◦ r1 r �γ �m1 ⊕m�γ �s1 0 s•γ •s1 )( r2 m2 0 s2 ) =( (r ◦γ ◦ r1)(r2) (r �γ �m1 ⊕m�γ �s1)(m2) 0 (s•γ •s1)(s2) ) ={ 1, r2,m2,s2 6= 0 0, otherwise. . Proof. Straightforward.2 Example 3.7. Let R be a Γ-ring and (M, +, .) a Γ-module. Consider the mapping α : M −→ R. Then M is an fuzzy Γ-hypermodule over M, under the following operations: m⊕n = m+n. and ◦ : M×Γ×M −→ F∗(M)(m,γ,n) 7−→ m◦γ◦n = χα(m).γ.n, for all m,n ∈ M,γ ∈ Γ. Proposition 3.8. Let (M, +, .) be a Γ-module over Γ-ring R and ν be a nonzero fuzzy Γ-semigroup on M. Let µ and ρ be two nonzero fuzzy Γ-semigroups on R. For r ∈ R, a,b ∈ M and γ ∈ Γ, define a fuzzy Γ- hyperoperation � on M by (r �γ �a)(t) = { µ(r) ∧ρ(γ) ∧ν(a), if t = r.γ.a 0 , otherwise. Also, a ⊕ b = χ{a+b}. It is easy to verify that (M,⊕,�) is a fuzzy Γ- hypermodule. Let S, Γ be nonempty sets, and S endowed with a fuzzy Γ-hyperoperation ◦. For all a,b ∈ S,γ ∈ Γ and p ∈ [0, 1] consider the p-cuts: (a◦γ ◦ b)p = {t ∈ S : (a◦γ ◦ b)(t) ≥ p} of a◦γ ◦ b, where p ∈ [0, 1]. For all p ∈ [0, 1], we define the following crisp Γ-hyperoperation on S: a◦p γ ◦p b = (a◦γ ◦ b)p. Example 3.9. Let R = Γ = Z and M = Zn for n ∈ N. Define following fuzzy Γ-hyperoperations for all a,b ∈ M,γ ∈ Γ: a⊕ b = χ{a,b},∀a ∈ M,∀r ∈ R,γ ∈ Γ, 18 On Fuzzy Gamma Hypermodules r �γ �a = χ{rγa}, ∀r,s ∈ R,∀γ ∈ Γ, r.γ.s = χ{rγs} and r + s = χ{r,s}, for all α,β ∈ Γ, and α�β = χ{α,β}, such that x is denote a typical element in Zn. Then it is easy to verify that (M,⊕,�) is a fuzzy Γ-hypermodule over fuzzy Γ-hyperring R and canonical fuzzy hypergroup (Γ,�). Proposition 3.10. Let (M,◦) be a fuzzy Γ-hyperoperation. For all a,b,c,u ∈ M and α,β ∈ Γ and for all p ∈ [0, 1] the following equivalence holds: (a◦α◦ (b◦β ◦ c)) ≥ p ⇐⇒ u ∈ a◦p α◦p (b◦p β ◦p c). ((a◦α◦ b) ◦β ◦ c) ≥ p ⇐⇒ u ∈ (a◦p α◦p b) ◦p β ◦p c.) Proof. Clearly, (a◦α◦ (b◦β ◦ c))(u) = ∨ t∈M (a◦α◦ t)(u) ∧ (b◦β ◦ c)(t) ≥ p, if and only if there exists t0 ∈ M, such that (a ◦ α ◦ t0)(u) ≥ p and (b◦β◦c)(t0) ≥ p, which means that u ∈ a◦pα◦pt0, t0 ∈ b◦pβ◦pc. Therefore, u ∈ a◦p α◦p (b◦p β ◦p c).2 Proposition 3.11. Let (M,⊕,�) be a fuzzy Γ-hypermodule over a fuzzy Γ-hyperring (R,�,�). Then for all a ∈ M,r ∈ R,γ ∈ Γ, conditions are equivalence: (1) a⊕M = χM ⇐⇒∀p ∈ [0, 1], a⊕P M = M; (2) r �γ �M = χM ⇐⇒∀p ∈ [0, 1], r �p γ �p M = M. Proof. We only proof (2). Let r � γ � M = χM . Then for all t ∈ M and p ∈ [0, 1], we have ∨ u∈M (r�γ �u)(t) = 1 ≥ p, whence there exists m ∈ M, such that (r � γ � m)(t) ≥ p, which means that t ∈ r �p γ �p m. Hence, ∀p ∈ [0, 1], r�pγ�pM = M. Conversely, for p = 1 we have r�1γ�1M = M, whence for all t ∈ M, there exists u ∈ M, such that t ∈ r �1 γ �1 u, which means that (r �γ �u)(t) = 1. In other words, r �γ �M = χM .2 Proposition 3.12. The structure (M,⊕,�) is a fuzzy Γ-hypermodule over a fuzzy Γ-hyperring (R,�,�) if and only if ∀p ∈ [0, 1], (M,⊕p,�p) is a Γ-hypermodule over the hyperring (R,�p,�p). Proof. It is straightforward.2 19 R. Ameri, R. Sadeghi Consider (M,⊕,�) as a fuzzy Γ-hypermodule over a fuzzy Γ-hyperring (R,�,�) and canonical fuzzy hypergroup (Γ,⊗). Now we follow [8], and define a new types of Γ-hyperoperations on M,R, Γ, as follows: ∀a,b ∈ M, a + b = {x ∈ M|(a⊕ b)(x) > 0}, ∀r,s ∈ R, r ]s = {t ∈ R | (r �s)(t) > 0},forallα,β ∈ Γ, α∗β = {γ ∈ Γ | (α∗β)(γ) > 0}, ∀a ∈ M, ∀r ∈ R,∀γ ∈ Γ, r.γ.a = {b ∈ M | (r �γ �a)(b) > 0}, ∀r,s ∈ R, ∀γ ∈ Γ, r ◦γ ◦s = {t ∈ R | (r �γ �s)(t) > 0}. Proposition 3.13. If (M,⊕,�) is a fuzzy Γ-hypermodule over a fuzzy Γ- hyperring (R,�,�) and canonical fuzzy hypergroup (Γ,⊗), then (M, +, .) is a Γ-hypermodule over the Γ-hyperring (R,],◦) and canonical hypergroup (Γ,?). Proof. By [10], it is obtained that (R,]), (Γ,∗) and (M, +) are canonical hypergroups. It is sufficient to verify (M,.) is a Γ-hypermodule. We consider the following cases: Case: (i) (r ]s).γ.a = (r.γ.a) + (s.γ.a), for all r,s ∈ R,γ ∈ Γ,a ∈ M. Suppose that x ∈ (r ] s).γ.a = ⋃ y∈r]s y �γ �a. Then (y �γ �a)(x) > 0 and (r � s)(y) > 0, for some y ∈ r ] s, and hence ∨p∈M ((r � s)(p) ∧ (p � γ � a)(x) > 0. Thus ((r � s) � γ � a)(x) > 0, which implies that ((r � γ � a) ⊕ (s � γ � a))(x) > 0. Thus there exist z,t ∈ M, such that (z⊕t)(x) > 0, (r�γ�a)(z) > 0 and (s�γ�a)(t) > 0 i.e., x ∈ z+t,z ∈ r.γ.a and t ∈ s.γ.a and hence x ∈ (r.γ.a) + (s.γ.a). Therefore, (r ] s).γ.a ⊆ (r.γ.a) + (s.γ.a). Similarly, we can show that (r.γ.a) + (s.γ.a)t ⊆ (r]s).γ.a. Therefore, (r ] s).γ.a = (r.γ.a) + (s.γ.a). The other conditions are verified similarly and omitted. 2 20 On Fuzzy Gamma Hypermodules On the other hands, if (M, +, .) is a Γ-hypermodule over a Γ-hyperring (R,],◦), then we define the following fuzzy Γ-hyperoperations: a⊕ b = χ{a+b},∀a,b ∈ M,r �s = χ{r]s},∀r,s ∈ R,γ ∈ Γ,r �γ �a = χ{r.γ.a},∀a ∈ M,r ∈ R,r �γ �s = χ{r◦γ◦s},∀r,s ∈ R,∀γ ∈ Γ,β = χ{α∗β}∀α,β ∈ Γ,α⊗β. The next result is immediately follows from above discussion and [14]. Proposition 3.14. For every hypergroup (M, +), there is an associated fuzzy hypergroup. Proposition 3.15. Let (M, +, .) be a Γ-hypermodule over a Γ-hyperring. Let (R,],◦) be a canonical hypergroup (Γ,?). Then (M,⊕,�) is a fuzzy Γ- hypermodule over a fuzzy Γ-hyperring (R,�,�) and canonical fuzzy hyper- group (Γ,⊗), where the fuzzy hyperoperations ⊕,�,�,� and ⊗ are defined above. Proof. By Proposition 3.14, (M,⊕) is a commutative fuzzy Γ-hypergroup. We show that (M,⊕) is canonical. Since (M, +) is canonical Γ-hypergroup, then there exists e ∈ M,∀a ∈ M, a = e + a = a + e =⇒ (e ⊕ a)(a) = χ{e+a}(a) > 0, (a ⊕ e)(a) = χ{e+a}(a) > 0 and because for all a ∈ M there exists b ∈ M, such that e ∈ a + b∩ b + a, b) is the inverse of a with respect to +). Then (a⊕ b)(e) = χ{a+b}(e) = χ{b+a}(e) = (b⊕a)(e) > 0. Let (a⊕x)(y) = χ{a+x}(y) > 0 =⇒ y ∈ a + x =⇒ ∃ b ( the inverse of a such that x ∈ b + y =⇒ (b⊕y)(x) = χ{b+y}(x) > 0. The other cases is can be proved in a similar way and omitted. Then (M,⊕) is a canonical fuzzy Γ-hypergroup. Now, we show that (M,⊕,�) is a fuzzy Γ-hypermodule. We investigate only the condition (iv) of Definition 3.4. First , we show that that for all r,s ∈ R,α,β ∈ Γ,a ∈ M, we have (r �α� (s�β �a)) = (r �α�s) �β �a, ∀t ∈ M. Then (r �α� (s�β �a))(t) = ∨ p∈M , [(r �α�p)(t) ∧ (s�β �a)(p)] = ∨ p∈M [χr.α.p(t) ∧χs.βa(p)] = 21 R. Ameri, R. Sadeghi { 1, t ∈ r.α.(s.β.a) 0, otherwise = { 1, t ∈ (r.α.s).β.a 0, otherwise = ((r �α�s) �β �a)(t), for all t ∈ M. It is easy to verify that the other conditions of Definition 3.4 can be obtained in a similar way.2 Proposition 3.16. Let M an RΓ-module and µ be a fuzzy Γ-module of M. Then the set M will be a fuzzy Γ-hypermodule. Proof. Let (Γ,∗) be an abelian group and (M, +, .) be a Γ-module over Γ-ring (R,],◦). We define fuzzy Γ-hyperoperations on M as follows: (a⊕ b)(t) = χ{a+b}, (r �γ �a)(t) = µ(r.γ.a− t), (α⊗β)(γ) = χ{α∗β}(γ) = χ{r]s}r �s)(z)(r �α�s)(z) = χ{r◦α◦s}(z), ∀a,b,t ∈ M,r,s,z ∈ R,α,β,γ ∈ Γ. It is easy to verify that (M,⊕) is a canonical fuzzy hypergroup. Now, we show (M,⊕,�) is a fuzzy Γ-hypermodule with µ(0) = 1. (i) ((r �s) �γ �a)(t) = ∨p∈R(r �s)(p) ∧ (p�γ �a)(t) = ∨p∈Rχr]s(p) ∧µ(p.γ.a− t) = µ((r ]s).γ.a− t) if p = r ]s. Also, ((r �γ �a) ⊕ (s�γ �a))(t) = = ∨p,q∈M (r �γ �a)(p) ∧ (p⊕ q)(t) ∧ (s�γ �a)(q) = ∨p,q∈Mµ(r.γ.a−p) ∧χ{p+q}(t) ∧µ(s.γ.a− q) = ∨p,q∈M,t=p+qµ(r.γ.a−p) ∧µ(s.γ.a− q) ≤ µ(r.γ.a−p + s.γ.a− q) = µ((r ]s).γ.a− (p + q)), On the other hands, if q = s.γ.a, p = t−s.γ.a., then ∨p,q∈M,t=p+qµ(r.γ.a−p) ∧µ(r.γ.a− q) ≥ ∨p∈Mµ(r.γ.a−p) ≥ µ(r.γ.a− t + s.γ.a) = µ((r ]s).γ.q − t). (ii) (r � (α⊗β) �a)(t) = ∨γ∈Γ[(r �γ �a)(t) ∧ (α⊗β)(γ)] = ∨µ(r.γ.a− t) ∧χ{α∗β}(γ) = µ(r.(α∗β).a− t). 22 On Fuzzy Gamma Hypermodules Also, ((r �α�a) ⊕ (r �β �a))(t) = = ∨p,q∈M [(r �α�a)(p) ∧ (p⊕ q)(t) ∧ (r �β �a)(q) = ∨p,q∈M [µ(r.α.a−p) ∧χ{p+q}(t) ∧µ(r.β.a− q)] = ∨t=p+q µ(r.α.a−p) ∧µ(r.βa− q) ≤ µ(r.αa−p + r.βa− q) = µ(r.(α∗β).a− (p + q)). On the other hands, suppose that q = r.β.a, then for p = t − r.β.a we have ∨t=p+qµ(r.α.a−p) ∧µ(r.βa− q) = ∨p∈Mµ(r.αa−p) ≥ µ(r.αa− (t− rβa)) = µ(r.(α∗β).a− (p + q)), (iii) r �γ � (a⊕ b) = ∨p∈M (r �γ �p)(t) ∧ (a⊕ b)(p) = ∨p∈Mµ(r.γ.p− t) ∧χ{a+b}(p) = µ(r.γ.(a + b) − t) and ((r �γ �a) ⊕ (r �γ � b))(t) = ∨p,q∈M (r �γ �a)(p) ∧ (p⊕ q)(t) ∧ (r �γ � b)(q) = ∨p,q∈Mµ(r.γ.a−p) ∧χ{p+q}(t) ∧µ(r.γ.b− q) = ∨p,q∈M,t=p+qµ(r.γ.a−p) ∧µ(r.γ.b− q) ≤ µ(r.γ.a−p + r.γ.b− q) = µ(r.γ.(a + b) − t). On the other hands, for q = r.γ.b,p = t− r.γ.b. we have ∨p,q∈M,t=p+qµ(r.γ.a−p) ∧µ(r.γ.b− q) ≥ ∨p∈Mµ(r.γ.a−p) ≥ µ(r.γ(a + b) − t). (iv) (r �α� (s�β �a))(t) = ∨p∈M (r �α�p)(t) ∧ (s�β �a)(p) = ∨p∈Mµ((r.α.p) − t) ∧µ((s.β.a) −p) = µ(r.α.(s.β.a) − t), and ((r �α�s) �β �a)(t) = ∨p∈R(r �α�s)(p) ∧ (p�β �a)(t) = ∨p∈Rχ{r◦α◦s}(p) ∧µ(p.β.a− t) = µ(r ◦α◦s · (β ·a) − t) if p = r ◦α◦s. 23 R. Ameri, R. Sadeghi 2 Remark. Let H = 〈H, (βi : i ∈ I)〉 be a fuzzy hyperalgebra. Denote by F∗(H) the set of the nonzero fuzzy subsets of H. Then H can be organized as a universal algebra under the following operations: βi(µ1, ...,µni )(t) = ∨ (x1,...,xni )∈H ni [(µ1(x1) ∧ ...µni (xni ) ∧ βi(x1, ...,xni )(t))], for every i ∈ I,µ1, ...,µni ∈ F ∗ (H) and t ∈ H. We denote this algebra by F∗(H). Proposition 3.17. If (M,⊕,♦) is a fuzzy Γ-hypermodule, then (F∗(M),∗,©) is a Γ-module. Proof. We define operations ∗,♦ on F∗(M) by µ∗ν = µ⊕ν, and r♦γ♦µ = r�γ�µ for all µ,ν ∈ F∗(M),r ∈ R,γ ∈ Γ. It is easy to see that (F∗(M),∗) is a group. Clearly, (F∗(M),⊕) is a semigroup. (i) Identity: we must prove that there exists a ν ∈ F∗(M) such that ,µ∗ν = µ. We have (µ∗ν)(t) = (µ⊕ν)(t) = ∨p,q∈Mµ(p) ∧ (p⊕ q)(t) ∧ν(q) = ∨p∈Mµ(p) ∧ (p⊕e)(t) = µ(t) ⊕ if q = e; ν(q) = 1,p = t. Thus it is enough we choose ν = χe. (ii) Inverse: it must prove that for µ ∈ F∗(M), there exists a ν ∈ F∗(M), such that µ∗ν = χe. It is sufficient to consider ν = −µ, then we have (µ∗ν)(t) = (µ⊕ν)(t) = ∨p,q∈Mµ(p) ∧ (p⊕ q)(t) ∧ (−µ)(q) = ∨p,q∈Mµ(p) ∧ (p⊕ q)(t) ∧µ(−q) ≤ µ(p− (−q)) ∧ (p⊕ q)(t) ≤ (p⊕ q)(t) = χe(t) where, p is inverse of q. On the other hands, we have ∨p,q∈Mµ(p) ∧ (p⊕ q)(t) ∧µ(−q) ≥ ∨p∈Mµ(p) ∧ (p⊕−p)(t) ≥ (p⊕−p)(t) = χe(t). 24 On Fuzzy Gamma Hypermodules Other cases are easy and omitted. 2 Definition 3.18. Let (M,⊕,�) be a fuzzy Γ-hypermodule over a fuzzy Γ-hyperring (R,�,�). A nonempty subset N of M is called a subfuzzy Γ- hypermodule if for all x,y ∈ N,r ∈ R and γ ∈ Γ, the following conditions hold: (1) (x⊕y)(t) > 0 ⇒ t ∈ N; (2) x⊕N = χN ; (3) (r �γ �x)(t) > 0 ⇒ t ∈ N. Proposition 3.19. (i) If (N,⊕,�) is a subfuzzy Γ-hypermodule of (M,⊕,�) over a fuzzy Γ-hyperring (R,�,�), then the associated Γ-hypermodule (N, +, .) is a Γ-hypersubmodule of (M, +, .) over (R,],◦); (ii) (N, +, .) is a Γ-hypersubmodule of (M, +, .) over (R,],◦) if and only if (N,⊕,�) is a subfuzzy Γ-hypermodule of (M,⊕,�) over (R,�,�). 4 Fundamental Relation of Fuzzy Γ-hypermodule In [14], fuzzy regular relations are introduced in the context of fuzzy hyper- semigroups. In this section we extend this notion to fuzzy Γ-hypermodules. Let ρ be an equivalence relation on a fuzzy Γ-hypersemigroup (M,◦) and µ,ν be two fuzzy subsets on M. We say that µρν if the following conditions hold: (1) if µ(a) > 0, then there exists b ∈ M, such that ν(b) > 0 and aρb and; (2) if ν(x) > 0, then there exists y ∈ M, such that µ(y) > 0 and xρy. An equivalence relation ρ on a fuzzy Γ-hypersemigroup (M,◦) is called a fuzzy regular relation (or fuzzy hypercongruence) on (M,◦) if, for all a,b,c ∈ M,γ ∈ Γ, the following implication holds: aρb =⇒ (a◦γ ◦ c) ρ (b◦γ ◦ c) and (c◦γ ◦a) ρ (c◦γ ◦ b). This condition is equivalent to aρa′,bρb′ ⇒ (a◦γ ◦ b)ρ(a′ ◦γ ◦ b′),∀a,b,a′,b′ ∈ M,γ ∈ Γ. Definition 4.1. An equivalence relation ρ on a fuzzy Γ-hypermodule (M,⊕,�) over a fuzzy Γ-hyperring (R,�,�) and a canonical fuzzy hypergroup (Γ,⊗) is called a fuzzy regular relation on (M,⊕,�) if it is a fuzzy regular relation on (M,⊕) and for all x,y ∈ M,r ∈ R,γ ∈ Γ, the following implication holds: xρy =⇒ (r �γ �x)ρ(r �γ �y). 25 R. Ameri, R. Sadeghi Let (M,⊕,�) be a fuzzy Γ-hypermodule over a fuzzy Γ-hyperring (R,�,�) and a canonical fuzzy hypergroup (Γ,⊗). Suppose (M, +, .) is the associated Γ-hypermodule over the Γ-hyperring (R,],◦) and the canonical hypergroup (Γ,∗). Then we have the next result. Theorem 4.2. An equivalence relation ρ is a fuzzy regular relation on (M,⊕,�) over a fuzzy Γ-hyperring (R,�,�) and canonical fuzzy hypergroup (Γ,⊗) if and only if ρ is a regular relation on (M, +, .) over the Γ-hyperring (R,],◦) and canonical hypergroup (Γ,∗). Proof. Letting xρy and x′ρy′, where x,x′,y,y′ ∈ M. We have (x⊕x′)ρ(y+y′) if and only if the following conditions hold: (x⊕x′)(u) > 0,⇒∃v ∈ M : (y ⊕y′)(v) > 0 and uρv, and (y ⊕y′)(t) > 0 ⇒ ∃w ∈ M : (x⊕x′)(w) > 0 and atρw. These are equivalent to: if u ∈ x + x′, then there exists v ∈ y + y′, such that uρv; if t ∈ y + y′, then there exists w ∈ x + x′, such that tρw; which mean that (x + x′)ρ̄(y + y′). Hence ρ is fuzzy regular on (M,⊕) if and only if ρ is regular on (M, +). On the other hands, if xρy and r ∈ R,γ ∈ Γ. We have (r�γ�x)ρ(r�γ�y) if and only if the next conditions hold: if (r �γ �x)(u) > 0, then there exists v ∈ M, such that (r �γ �y)(v) > 0 and uρv; if (r�γ �y)(t) > 0, then there exists w ∈ M, such that (r�γ �x)(w) > 0 and tρw. These are equivalent to: if u ∈ r.γ.x, then there exists v ∈ r.γ.y, such that uρv; if t ∈ r.γ.y, then there exists w ∈ r.γ.x, such that tρw; which means that (r.γ.x)ρ(r.γ.y).2 Definition 4.3. An equivalence relation ρ on a fuzzy Γ-hypersemigroup (M,◦) is called a fuzzy strongly regular relation on (M,◦) if, for all a,a′,b,b′ of M and for all γ ∈ Γ, such that aρb and a′ρb′, the following condition holds: (a◦γ ◦ c)(x) > 0, (b◦γ ◦d)(y) > 0 ⇒ xρy, for all x,y ∈ M. Note that if ρ is a fuzzy strongly relation on a fuzzy Γ- hypersemigroup (M,◦), then it is a fuzzy regular on (M,◦). An equivalence relation ρ on a fuzzy Γ-hyperring (R,�,�) is called a fuzzy strongly regular 26 On Fuzzy Gamma Hypermodules relation on (R,�,�) if it is a fuzzy strongly regular relation both on (R,�) and on (R,�). Definition 4.4. Let ρ be a fuzzy strongly regular relation on a fuzzy Γ- hyperring (R,�,�) and θ be a fuzzy strongly regular relation on a canon- ical fuzzy Γ-hypergroup (Γ,∗). An equivalence relation δ on a fuzzy Γ- hypermodule (M,⊕,�) over a fuzzy Γ-hyperring (R,�,�) and canonical fuzzy Γ-hypergroup (Γ,⊗) is called a fuzzy strongly regular relation on (M,⊕,�) if it is a fuzzy strongly regular relation on (M,⊕) and if xδy, rρs and αθβ, then the next condition holds: for all u ∈ M, such that (r � α � x)(u) > 0 and for all v ∈ M, such that (s�β �y)(v) > 0, we have uδv. Theorem 4.5. An equivalence relation δ is a fuzzy strongly regular relation on (M,⊕,�) if and only if δ is a strongly regular relation on (M, +, .). Proof. Set xδy and x′δy′, where x,x′,y,y′ ∈ M and set rρs, where r,s ∈ R and αθβ, where α,β ∈ Γ. The relation δ is strongly regular on (M,⊕,�) if and only if the following conditions are satisfied: ∀u ∈ M, such that (x⊕x′)(u) > 0 and ∀v ∈ M, such that (y ⊕ y′)(v) > 0, we have uδv; ∀t ∈ M, such that (r�α�x)(t) > 0 and ∀w ∈ M, such that (s�β�y)(w) > 0, we have tδw. These conditions are equivalent to the following ones: ∀u ∈ M, such that u ∈ x + x′ and ∀v ∈ M, such that v ∈ y + y′, we have uδv; ∀t ∈ M, such that t ∈ r.α.x and ∀w ∈ M, such that w ∈ s.β.y, we have tδw, which mean that (x + x′)¯̄δ(y + y′) and (r.α.x)¯̄δ(s.β.y). Hence δ is strongly regular on (M,⊕,�) if and only if δ is strongly regular on (M, +, .). Now, Let δ be a fuzzy regular relation on a fuzzy Γ-hypermodule (M,⊕,�) over a fuzzy Γ-hyperring (R,�,�) and canonical fuzzy Γ-hypergroup (Γ,⊗) and ρ,θ be fuzzy strongly regular relations on the Γ-hyperring (R,�,�) and canonical fuzzy Γ-hypergroup. (Γ,⊗). We consider the following Γ-hyperoperations on the quotient set M/δ: x̄ ? ȳ = {z̄ | z ∈ x + y} = {z̄ | (x⊕y)(z) > 0}, r̄ } ᾱ} x̄ = {z̄ | z ∈ r.α.x} = {z̄ | (r �α�x)(z) > 0}. Theorem 4.6. Let (M,⊕,�) be a fuzzy Γ-hypermodule over a fuzzy Γ- hyperring (R,�,�) and canonical fuzzy hypergroup (Γ,∗). Let (M, +, .) be the associated Γ-hypermodule over the corresponding Γ-hypergroup (R,],◦) and canonical hypergroup (Γ,∗). Then we have: 27 R. Ameri, R. Sadeghi (i) The relation δ is a fuzzy regular relation on (M,⊕,�) if and only if (M/δ,?,}) is a Γ-hypermodule over (R,],◦) and (Γ,∗). (ii) The relation δ is a fuzzy strongly regular relation on (M,⊕,�) over (R,�,�) and (Γ,⊗) if and only if (M/δ,?,}) is a Γ-module over R/ρ and Γ/θ. If we denote by U the set of all expressions consisting of finite fuzzy Γ- hyperoperations either on R, Γ and M or the external fuzzy Γ-hyperoperations applied on finite sets of elements of R, Γ and M, then we have x�y ⇐⇒∃u ∈ U : {x,y}⊂ u. Now, we introduced fundamental relation on fuzzy Γ-hypemodules. Definition 4.7. An equivalence relation �∗ is called fundamental relation on a fuzzy Γ-hypermodule (M,⊕,�) if �∗ is fundamental relation on the associated Γ-hypermodule (M, +, .). Hence, �∗ is fundamental relation on a fuzzy Γ-hypermodule (M,⊕,�) if and only if �∗ is the smallest fuzzy strongly equivalence relation on (M,⊕,�). Denote by UF the set of all expressions consisting of finite fuzzy Γ-hyperoperations either on R, Γ and M or the external fuzzy Γ-hyperoperation applied on finite sets of elements of R, Γ and M. We obtain x�y ⇐⇒∃ µf ∈ UF : {x,y}⊆ µfγ ⇐⇒ µfγ(x) > 0 and µfγ(y) > 0. The relation �∗ is the transitive closure of �. Denote by ∑∗ ⊕ any finite fuzzy hypersum and by ∏∗ � any finite fuzzy Γ- hyperproduct of the fuzzy Γ-hypemodule (M,⊕,�). As above, we obtain that ( ∑∗ i⊕ ∏∗ j � aji)(p) > 0 if and only if p ∈ ∑∗ i⊕ ∏∗ j� aji. Hence, {x,y} ⊂ ∑∗ i⊕ ∏∗ j � aji if and only if ( ∑∗ i⊕ ∏∗ j � aji)(x) > 0 and ( ∑∗ i⊕ ∏∗ j � aji)(y) > 0. Therefore, we obtain x�y ⇐⇒ ∃µfγ ∈ UF such that µfγ(x) > 0 and µfγ(y) > 0. So, in order to obtain a fuzzy Γ-module starting from a fuzzy Γ-hypermodule, we consider first the relation �, then the transitive closure �∗ of � and finally the quotient structure (M/�∗,?,}) of the fuzzy Γ-hypermodule (M,⊕,�). Acknowledgements The first author partially has been supported by the ”Research Center in Algebraic Hyperstructures and Fuzzy Mathematics, University of Mazan- daran, Babolsar, Iran” and ”Algebraic Hyperstructure Excellence, Tarbiat Modares University, Tehran, Iran”. 28 On Fuzzy Gamma Hypermodules References [1] Ameri, R., Norouzi, M., Prime and primary hyperideals in Krasner (m,n)-ary hyperrings, European J. Combin. 34 (2013) 379-390. [2] R. Ameri, T. Nozari, Complete Parts and Fundamental Relation of Fuzzy Hypersemigroups, J. of Mult.-Valued Logic and Soft Computing, Vol. 19, (2012), 451460. [3] R. Ameri, R. Sadeghi, Gamma Modules, Ratio Mathematica, No. 20,(2010), 1-14. [4] R. Ameri, T. 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