Term Functions and Fundamental Relation of Fuzzy Hyperalgebras R. Ameri, T. Nozari † School of Mathematics, Statistics and Computer Science College of Sciences, University of Tehran P.O. Box 14155-6455, Teheran, Iran, e-mail:@umz.ac.ir ‡ Department of Mathematics, Faculty of Basic Science, University of Mazandaran, Babolsar, Iran Abstract We introduce and study term functions over fuzzy hyperalgebras. We start from this idea that the set of nonzero fuzzy subsets of a fuzzy hyperalgebra can be organized naturally as a universal algebra, and constructing the term functions over this algebra. We present the form of generated subfuzzy hyperalgebra of a given fuzzy hyperalgebra as a generalization of universal algebras and multialgebras. Finally, we characterize the form of the fundamental relation of a fuzzy hyperalgebra. Keywords: Hyperalgebra, Fuzzy hyperalgebra, Equivalence relation, Term function, Fundamental relation, Quotient set. Ratio Mathematica, 20, 2010 43 1 Introduction Hyperstructure theory was born in 1934 when Marty defined hypergroups, began to analysis their properties and applied them to groups, relational algebraic functions (see [15]). Now they are widely studied from theoretical point of view and for their applications to many subjects of pure and applied properties ([7]). As it is well known, in 1965 Zadeh ([28]) introduced the notion of a set µ on a nonempty set X as a function from X to the unite real interval I = [0, 1] as a fuzzy set. In 1971, Rosenfeld ([25]) introduced fuzzy sets in the context of group theory and formulated the concept of a fuzzy subgroup of a group. Since then, many researchers are engaged in extending the concepts of abstract algebra to the framework of the fuzzy setting ( for instance see [23]). The study of fuzzy hyperstructure is an interesting research topic of fuzzy sets and applied to the theory of algebraic hyperstructure. As it is known a hyperoperation assigns to every pair of elements of H a nonempty subset of H, while a fuzzy hyperoperation assigns to every pair of elements of H a nonzero fuzzy set on H. Recently, Sen, Ameri and Chowdhury introduced and analyzed fuzzy semihypergroups in [21]. This idea was followed by other researchers and extended to other branches of algebraic hyperstructures, for instance Leoreanu and Davvaz introduced and studied fuzzy hyperring notion in [13], Chowdhury in [5] studied fuzzy transposition hypergroups and Leoreanu studied fuzzy hypermodules in [15]. In this paper we follow the idea in [20] and introduced fuzzy hyperalgebras, as the largest class of fuzzy algebraic system. We introduce and study term functions over algebra of all nonzero fuzzy subsets of a fuzzy hyperalgebra, as an important tool to introduce fundamental relation on fuzzy hyperalgebra. Finally, we construct fundamental relation of fuzzy algebras and investigate its basic properties. This paper is organized in four sections. In section 2 we gather the definitions and Ratio Mathematica, 20, 2010 44 basic properties of hyperalgebras and fuzzy sets that we need to develop our paper. In section 3 we introduce term functions over the algebra of nonzero fuzzy subsets of a fuzzy hyperalgebra and we obtained some basic results on fuzzy hyperalgebras, in section 4 we will present the form of the fundamental relation of a fuzzy hyperalgebra. 2 Preliminaries In this section we present some definitions and simple properties of hyperalgebras from [2] and [3], which will be used in the next sections. In the sequel H is a fixed nonvoid set, P ∗(H) is the family of all nonvoid subsets of H, and for a positive integer n we denote for Hn the set of n-tuples over H (for more see [6] and [7]). For a positive integer n a n-ary hyperoperation β on H is a function β : Hn → P ∗(H). We say that n is the arity of β. A subset S of H is closed under the n-ary hyperoperation β if (x1, . . . , xn) ∈ Sn implies that β(x1, . . . , xn) ⊆ S. A nullary hyperoperation on H is just an element of P ∗(H); i.e. a nonvoid subset of H. A hyperalgebraic system or a hyperalgebra 〈H, (βi : i ∈ I)〉 is the set H with together a collection (βi | i ∈ I) of hyperoperations on H. A subset S of a hyperalgebra H=〈H, (βi : i ∈ I)〉 is a subhyperalgebra of H if S is closed under each hyperoperation βi, for all i ∈ I, that is βi(a1, ..., ani ) ⊆ S, whenever (a1, ..., ani ) ∈ Sni . The type of H is the map from I into the set N∗ of nonnegative integers assigning to each i ∈ I the arity of βi. In this paper we will assume that for every i ∈ I , the arity of βi is ni. For n > 0 we extend an n-ary hyperoperation β on H to an n-ary operation β on P ∗(H) by setting for all A1, ..., An ∈ P ∗(H) β(A1, ..., An) = ⋃ {β(a1, ..., an)|ai ∈ Ai(i = 1, ..., n)} Ratio Mathematica, 20, 2010 45 It is easy to see that 〈P ∗(H), (βi : i ∈ I)〉 is an algebra of the same type of H. Definition 2.1. Let H=〈H, (βi : i ∈ I)〉 and H=〈H, (βi : i ∈ I)〉 be two similar hyperalgebras. A map h from H into H is called a (i) A homomorphism if for every i ∈ I and all (a1, ..., ani ) ∈ Hni we have that h(βi((a1, ..., ani )) ⊆ βi(h(a1), ..., h(ani )); (ii) a good homomorphism if for every i ∈ I and all (a1, ..., ani ) ∈ Hni we have that h(βi((a1, ..., ani )) = βi(h(a1), ..., h(ani )), for more details about homomorphism of hyperalgebras see [12]. Let ρ be an equiva- lence relation on H. We can extend ρ on P ∗(H) in the following ways: (i) Let {A, B} ⊆ P ∗(H). We write AρB iff ∀a ∈ A, ∃b ∈ B, such that aρb and ∀b ∈ B, ∃a ∈ A, such that aρb. (ii) we write AρB iff ∀a ∈ A, ∀b ∈ B we have aρb. Definition 2.2. If H=〈H, (βi : i ∈ I)〉 be a hyperalgebra and ρ be an equivalence relation on H. Then ρ is called regular (resp. strongly regular) if for every i ∈ I, and for all a1, ..., ani , b1, ..., bni ∈ H the following implication holds: a1ρb1, ..., ani ρbni ⇒ βi(a1, ..., ani )ρβi(b1, ..., bni ) (resp. a1ρb1, ..., ani ρbni ⇒ βi(a1, ..., ani )ρβi(b1, ..., bni )). Definition 2.3. Recall that for a nonempty set H, a fuzzy subset µ of H is a function µ : H → [0, 1]. If µi is a collection of fuzzy subsets of H, then we define the fuzzy subset ⋂ i∈I µi by: ( ⋂ i∈I µi)(x) = ∧ i∈I {µi(x)}, ∀x ∈ H. Definition 2.4. Let ρ be an equivalence relation on a hyperalgebra 〈H, (βi : i ∈ I)〉 and µ and υ be two fuzzy subsets on H. We say that µρυ if the following two conditions hold: (i) µ(a) > 0 ⇒ ∃b ∈ H : υ(b) > 0 , and aρb (ii) υ(x) > 0 ⇒ ∃y ∈ H : µ(y) > 0, and xρy. Ratio Mathematica, 20, 2010 46 3 Fuzzy Hyperalgebra and Term Functions Definition 3.1. A fuzzy n-ary hyperoperation f n on S is a map f n : S×...×S −→ F ∗(S), which associated a nonzero fuzzy subset f n(a1, ..., an) with any n-tuple (a1, ..., an) of elements of S. The couple 〈S, f n〉 is called a fuzzy n-ary hypergroupoid. A fuzzy nullary hyperoperation on S is just an element of F ∗(S); i.e. a nonzero fuzzy subset of S. Definition 3.2. Let H be a nonempty set and for every i ∈ I, βi be a fuzzy ni-ary hyperoperation on H. Then H=〈H, (βi : i ∈ I)〉 is called fuzzy hyperalgebra, where (ni : i ∈ I) is the type of this fuzzy hyperalgebra. Definition 3.3. If µ1, ..., µni be ni nonzero fuzzy subsets of a fuzzy hyperalgebra H=〈H, (βi : i ∈ I)〉 , we define for all t ∈ H βi(µ1, ..., µni )(t) = ∨ (x1,...,xni )∈H ni (µ1(x1) ∧ ... ∧ µni (xni ) ∧ βi(x1, ..., xni )(t)) Finally, if A1, ..., Ank are nonempty subsets of H, for all t ∈ H βk(A1, ..., Ank )(t) = ∨ (a1,...,ank )∈H nk (βk(a1, ..., ank )(t)). If A is a nonempty subset of H, then we denote the characteristic function of A by χA. Note that, if f : H1 −→ H2 is a map and a ∈ H1, then f (χa) = χf (a). Example 3.4. (i) A fuzzy hypergroupoid is a fuzzy hyperalgebra of type (2), that is a set H together with a fuzzy hyperoperation ◦. A fuzzy hypergroupoid 〈H, ◦〉, which is associative, that is x ◦ (y ◦ z) = (x ◦ y) ◦ z, for all x, y, z ∈ H is called fuzzy hypersemigroup[22]. In this Ratio Mathematica, 20, 2010 47 case for any µ ∈ F ∗(H), we define (a ◦ µ)(r) = ∨ t∈H ((a ◦ t)(r) ∧ µ(t)) and (µ ◦ a)(r) =∨ t∈H (µ(t) ∧ (t ◦ a)(r)) for all r ∈ H. (ii) A fuzzy hypergroup is a fuzzy hypersemigroup such that for all x ∈ H we have x ◦ H = H ◦ x = χH (fuzzy reproduction axiom)(for more details see [22]). (iii) A fuzzy hyperring R=〈R, ⊕, �〉 ([13]) is a fuzzy hyperalgebra of type (2, 2), which in that the following axioms hold: 1) a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ c for all a, b, c ∈ R; 2) x ⊕ R = R ⊕ x = χR for all x ∈ R; 3) a ⊕ b = b ⊕ a for all a, b ∈ R; 4) a � (b � c) = (a � b) � c for all a, b, c ∈ R; 5) a � (b ⊕ c) = (a � b) ⊕ (a � c) and (a ⊕ b) � c = (a � c) ⊕ (b � c) for all a, b, c ∈ R. Example 3.5. Let H=〈H, (βi : i ∈ I)〉 be a hyperalgebra and µ be a nonzero fuzzy subset of H. Define the following fuzzy n-ary hyperoperations on H, for every i ∈ I and for all (a1, ..., ani ) ∈ Hni ; β�i (a1, ..., ani )(t) =   µ(a1) ∧ ... ∧ µ(ani ) t ∈ β(a1, ..., ani ) 0 otherwise and letting β◦i (a1, ..., ani ) = χ{a1,...,ani }. Evidently H�=〈H, (β�i : i ∈ I)〉, H◦=〈H, (β◦i : i ∈ I)〉 are fuzzy hyperalgebras. Theorem 3.6. Let H=〈H, (βi : i ∈ I)〉 be a fuzzy hyperalgebra, then for every i ∈ I and every a1, ..., ani ∈ H we have βi(χa1 , ..., χani ) = βi(a1, ..., ani ). Definition 3.7. Let H=〈H, (βi : i ∈ I)〉 be a fuzzy hyperalgebra . A nonempty subset S of H is called a subfuzzy hyperalgebra if for ∀i ∈ I, ∀a1, ..., ani ∈ S, the following condition Ratio Mathematica, 20, 2010 48 hold: βi(a1, ..., ani )(x) > 0 then x ∈ S. We denote by S(U) the set of the subfuzzy hyperalgebras of H. Definition 3.8. Consider the fuzzy hyperalgebra H=〈H, (βi : i ∈ I)〉 and φ 6= X ⊆ H be nonempty. Clearly, 〈X〉 = ⋂ {B : B ∈ S(H)| X ⊆ B} with the fuzzy hyperoperations of H form a subfuzzy hyperalgebra of H called the subfuzzy hyperalgebra of H generated by the subset X . Evidently if X is a subfuzzy hyperalgebra for H then 〈X〉 = X. Theorem 3.9. Let H=〈H, (βi : i ∈ I)〉 be a fuzzy hyperalgebra and φ 6= X ⊆ H. We consider X0 = X and for any k ∈ N, Xk+1 = Xk ∪ {a ∈ H | ∃i ∈ I, ni ∈ N, x1, ..., xni ∈ Xk; βi(x1, ..., xni )(a) > 0}. Then 〈X〉 = ⋃ k∈N Xk. Proof. Let M = ⋃ k∈N Xk, and ∀i ∈ I, consider t1, ..., tni ∈ M and βi(t1, ..., tni )(x) > 0. From X0 ⊆ X1 ⊆ ... ⊆ Xk ⊆ ... it follows the existence of m ∈ N such that t1, ..., tni ∈ Xm, which implies, according to the definition of Xm+1 that x ∈ Xm+1. Thus x ∈ M and M = ⋃ k∈N Xk is a subfuzzy hyperalgebra. From X = X0 ⊆ M , by definition of the generated subfuzzy hyperalgebra, it results 〈X〉 ⊆ 〈M〉 = M. To prove the inverse inclu- sion we will show by induction on k ∈ N that Xk ⊆ 〈X〉 for any k ∈ N, and we have X0 = X ⊆ 〈X〉. We suppose that Xk ⊆ 〈X〉. From 〈X〉 ∈ S(H) and the definition Xk+1 we can deduce that Xk+1 ⊆ 〈X〉. Hence M ⊆ 〈X〉. The two inclusion lead us to M = 〈X〉.� Let H=〈H, (βi : i ∈ I)〉 be a fuzzy hyperalgebra then, the set of the nonzero fuzzy subsets of H denoted by F ∗(H), can be organized as a universal algebra with the opera- tions; βi(µ1, ..., µni )(t) = ∨ (x1,...,xni )∈H ni (µ1(x1) ∧ ... ∧ µni (xni ) ∧ βi(x1, ..., xni )(t)) Ratio Mathematica, 20, 2010 49 for every i ∈ I, µ1, ..., µni ∈ F ∗(H) and t ∈ H. We denote this algebra by F∗(H). In [13] Gratzer presents the algebra of the term functions of a universal algebra. If we consider an algebra B=〈B, (βi : i ∈ I)〉 we call n−ary term functions on B (n ∈ N) those and only those functions from Bn into B, which can be obtained by applying (i) and (ii) from bellow for finitely many times: (i) the functions eni : B n → B, eni (x1, ..., xn) = xi, i = 1, ..., n are n−ary term functions on B; (ii) if p1, ..., pni are n−ary term functions on B, then βi(p1, ..., pni ) : Bn → B, βi(p1, ..., pni )(x1, ..., xn) = βi(p1(x1, ..., xn), ..., pni (x1, ..., xn)) is also a n−ary term function on B. We can observe that (ii) organize the set of n−ary term functions over B (P (n)(B)) as a universal algebra, denoted by B(n)(B). If H is a fuzzy hyperalgebra then for any n ∈ N, we can construct the algebra of n−ary term functions on F∗(H), denoted by B(n)(F∗(H)) = 〈P (n)(F∗(H)), (βi : i ∈ I)〉. Theorem 3.10. A necessary and sufficient condition for F∗(B) to be a subalgebra of F∗(U) is that B is to be a subfuzzy hyperalgebra for U. Proof. Obvious.� The next result immediately follows from Theorem 3.10. Corollary 3.11. (i) Let H=〈H, (βi : i ∈ I)〉 be a fuzzy hyperalgebra and B a sub- fuzzy hyperalgebra of H, and p ∈ P (n)(F∗(H)),(n ∈ N). If µ1, ..., µn ∈ F ∗(B) , then p(µ1, ..., µn) ∈ F ∗(B). (ii) Let H= 〈H, (βi : i ∈ I)〉 be a fuzzy hyperalgebra and B a subfuzzy hyperalgebra of H, and p ∈ P (n)(F∗(H)),(n ∈ N). If x1, ..., xn ∈ B, then p(χx1 , ..., χxn ) ∈ F ∗(B).� Theorem 3.12. Let H=〈H, (βi : i ∈ I)〉 be a fuzzy hyperalgebra and φ 6= X ⊆ H. Then a ∈ 〈X〉 if and only if ∃n ∈ N, ∃p ∈ P (n)(F∗(H)), and ∃x1, ..., xn ∈ X, such that Ratio Mathematica, 20, 2010 50 p(χx1 , ..., χxn )(a) > 0. Proof. We denote M = {a ∈ H | ∃n ∈ N, ∃p ∈ P (n)(F∗(H)), ∃x1, ..., xn ∈ X : p(χx1 , ...χxn )(a) > 0}. For any x ∈ X we have e11(χx)(x) = χx(x) = 1, thus x ∈ X and hence X ⊆ M . Also from Corollary 3.11 (ii), it follows that p(χx1 , ..., χxn ) ∈ F∗(〈X〉), therefore M ⊆ 〈X〉. We will prove now that M is subfuzzy hyperalgebra of H. For any i ∈ I, if c1, ..., cni ∈ M and βi(c1, ..., cni )(x) > 0, we must show that x ∈ M. For c1, ..., cni ∈ M , it means that there exist mk ∈ N, pk ∈ P mk (F∗(H)), xk1, ..., xkmk ∈ X, k ∈ {1, ..., ni}, such that pk(χxk1 , ..., χxkmk )(ck) > 0, ∀k ∈ {1, ..., ni}. According to the Corollary 8.2 from [12], for any n−ary term function p over F∗(H) and for m ≥ n there exists an m−ary term function q over F∗(H), such that p(µ1, ..., µn) = q(µ1, ..., µm), for all µ1, ..., µm ∈ F ∗(H); this allows us to consider instead of p1, ..., pni the term functions q1, ..., qni all with the same arity m = m1 + ... + mni and the elements y1, ..., ym ∈ X (which are the elements x11, ..., x 1 m1 , ..., xni1 , ..., x ni mni ), such that qk(χy1 , ..., χym )(ck) > 0,∀k ∈ {1, ..., ni}. But we have βi(q1(χy1 , ..., χym ), ..., qni (χy1 , ..., χym ))(x) =∨ (a1,...,ani )∈H ni (q1(χy1 , ..., χym )(a1) ∧ ... ∧ qni (χy1 , ..., χym )(ani ) ∧ βi(a1, ..., ani )(x)), and for (a1, ..., ani ) = (c1, ..., cni ) we have (βi(q1, ..., qni )(χy1 , ..., χym ))(x) > 0 . On the other hands we have βi(q1, ..., qni ) ∈ P (m)(F∗(H)), (m ∈ N) , y1, ..., ym ∈ X which implies that x ∈ M. Therefore, M = 〈X〉 and this complete the proof.� Remark 3.13. If H has a fuzzy nullary hyperoperation then < φ >= {a ∈ H | ∃µ ∈ P 0(F∗(H)), such that µ(a) > 0}. Recall that if H=〈H, (βi : i ∈ I)〉 and B=〈B, (βi : i ∈ I)〉 are fuzzy hyperalgebras with the same type, then a map h : H → B is called a good homomorphism if for any i ∈ I we Ratio Mathematica, 20, 2010 51 have ; h(βi(a1, ..., ani )) = βi(h(a1), ..., h(ani )), ∀a1, ..., ani ∈ H. An equivalence relation on H ϕ is said to be an ideal if for any i ∈ I we have: βi(x1, ..., xni )(a) > 0 and xkϕyk(k ∈ {1, ..., ni}) ⇒ ∃b ∈ H : βi(y1, ..., yni )(b) > 0 and aϕb. For example the fuzzy regular relations on a fuzzy hypersemigroup are ideal equiva- lence. (for more details see [13, 21]) Definition 3.14. Let H=〈H, (βi : i ∈ I)〉 be a fuzzy hyperalgebra and ϕ an equivalence relation on H. Then H/ϕ can be described as a fuzzy hyperalgebra H/ϕ with the fuzzy hyperoperations: βi(ϕ(x1), ..., ϕ(xni ))(ϕ(xni+1)) = ∨ xkϕyk βi(y1, ..., yni )(yni+1). Theorem 3.15. Let h : H → B be a good homomorphism of fuzzy hyperalgebras H and B. Then the relation ϕ = {(x, y) ∈ H|h(x) = h(y)} is an ideal relation on H. Conversely, if ϕ is an ideal relation on H, then p = pϕ : H → H/ϕ is homomorphism (which is not strong). Proof. Straightforward.� Remark 3.16. Let H and B be fuzzy hyperalgebras of the same type and h be a homomorphism from H into B. We will construct the algebras F∗(H) and F∗(B). The homomorphism h induces a mapping h′ : F∗(H) → F∗(B) by h′(µ) = h(µ), for any µ ∈ F ∗(H). Let us consider H a set and F ∗(H) the set of its nonzero fuzzy subsets. Let ϕ be an equivalence on H and let us consider the relation ϕ on F ∗(H) as follows: Ratio Mathematica, 20, 2010 52 µϕν ⇔ ∀a ∈ H : µ(a) > 0 ⇒ ∃b ∈ H : ν(b) > 0 and aϕb and ∀b ∈ H : ν(b) > 0 ⇒ ∃a ∈ H : µ(a) > 0 and aϕb. It is immediate that ϕ is an equivalence relation on F ∗(H). The next result immediately follows. Theorem 3.17. An equivalence relation ϕ on a fuzzy hyperalgebra H is ideal if and only if ϕ is a congruence relation on F∗(H). Proof. Let us suppose that ϕ is an ideal equivalence on H and let us consider i ∈ I and µk, νk ∈ F ∗(H) nonzero and µkϕνk, k ∈ {1, ..., ni} . Then for any a ∈ H such that βi(µ1, ..., µni )(a) > 0, we have βi(µ1, ..., µni )(a) = ∨ (x1,...,xni )∈H ni µ1(x1) ∧ ... ∧ µni (xni ) ∧ βi(x1, ..., xni )(a). Thus there exists (x1, ..., xni ) ∈ Hni , such that µk(xk) > 0 for k ∈ {1, ..., ni} and βi(x1, ..., xni )(a) > 0. From the definition ϕ and hence there exists (y1, ..., yni ) ∈ Hni , such that νk(yk) > 0 for k ∈ {1, ..., ni} and xkϕyk, and sice ϕ is an ideal and βi(x1, ..., xni )(a) > 0, there exists b ∈ H, such that βi(y1, ..., yni )(b) > 0 and aϕb. Analogously, it can be proved that for all b ∈ H, such that βi(y1, ..., yni )(b) > 0, there exists a ∈ H, such that βi(x1, ..., xni )(a) > 0 and aϕb. Hence, it is proved that ϕ is a congruence on F∗(H). Conversely, let us take i ∈ I and a, xk, yk ∈ H, k ∈ {1, ..., ni} such that xkϕyk and βi(x1, ..., xni )(a) > 0. Obviously, χxk ϕχyk , ∀k ∈ {1, ..., ni}, and because ϕ is a congruence on F∗(H) We can write βi(χx1 , ..., χxni )ϕβi(χy1 , ..., χyni ), hence βi(x1, ..., xni )ϕβi(y1, ..., yni ), which leads us to the existence b ∈ H, such that βi(y1, ..., yni )(b) > 0 and aϕb. This com- plete the proof.� Corollary 3.18. (i) If H=〈H, (βi : i ∈ I)〉 is a fuzzy hyperalgebra, ϕ is an ideal equiv- alence relation on H and p ∈ P (n)(F∗(H)) If for any nonzero, µk, νk, such that µkϕνk Ratio Mathematica, 20, 2010 53 k ∈ {1, ..., n}, then p(µ1, ..., µn)ϕp(ν1, ..., νn). (ii) Let H=〈H, (βi : i ∈ I)〉 be a fuzzy hyperalgebra, ϕ an ideal equivalence relation on H. If xkϕyk, k ∈ {1, ..., n}, p ∈ P (n)(F∗(H)) , xk, yk ∈ H. Then have p(χx1 , ..., χxn )ϕp(χy1 , ..., χyn ). Let h be a homomorphism from H into B and take ϕ = {(x, y) ∈ H2 | h(x) = h(y)}. Then we have ϕ = {(µ, ν) ∈ (F ∗(H))2 | h′(µ) = h′(ν)}. Obviously, ϕ is an ideal of H if and only if ϕ is congruence on F∗(H). Theorem 3.19. The map h is a homomorphism ofthe universal algebras F∗(H) and F∗(B) if and only if h is a good homomorphism between H and B. Proof. Straightforward.� The next result immediately follows from Theorem 3.12. Corollary 3.20. (i) Let H=〈H, (βi : i ∈ I)〉 and B=〈B, (βi : i ∈ I)〉 be fuzzy hyperalge- bras of the same type, h : H → B a homomorphism and p ∈ P (n)(F∗(H)). Then for all µ1, ..., µn ∈ F ∗(H) we have h′(p(µ1, ..., µn)) = p(h′(µ1), ..., h′(µn)). (ii) Let H=〈H, (βi : i ∈ I)〉 and B=〈B, (βi : i ∈ I)〉 be fuzzy hyperalgebras of the same type, h : H → B a homomorphism and p ∈ P (n)(F∗(H)). Then for all a1, ..., an ∈ H, we have h′(p(χa1 , ..., χan )) = p(h ′(χa1 ), ..., h ′(χan )).� 4 Fundamental Relation of Fuzzy Hyperalgebra As it is known that if R is an strongly regular equivalence relation on a given hyper- group (resp. hypergroupoid, semihypergroup) H, then we can define a binary operation ⊗ on the quotient set H/R, the set of all equivalence classes of H with respect to R, such that (H/R, ⊗) consists a group (resp. groupoid, semigroup). In fact the relation β∗ is the Ratio Mathematica, 20, 2010 54 smallest equivalences relation such that the quotient H/β∗ is a group (resp. groupoid, semigroup) and it is called fundamental relation of H. The equivalence relation β∗ was studied on hypergroups by many authors( for more details see [6]). As the fundamental relation plays an important role in the theory of algebraic hyperstructure it extended to other classes of algebraic hyperstructure, such as hyperrings, hypermodules, hypervec- torspaces( for more details see [25], [26] and [27]). In [20] Pelea introduced and studied the fundamental relation of a multialgebra based on term functions. In the sequel we extend fundamental relation on fuzzy hyperalgebras and investigate its basic properties. Let B=〈B, (βi : i ∈ I)〉 be an universal algebra. If we add to the set of the operations of B the nullary operations corresponding to the elements of B, the n−ary term functions of this new algebra are called the n−ary polynomial functions of B. The n−ary polynomial functions P n(B) of B form a universal algebra with the operations (βi : i ∈ I), denoted by P(n)(B), P(n)(B)=〈P n(B), (βi : i ∈ I)〉. Let H=〈H, (βi : i ∈ I)〉 be a fuzzy hyperalgebra. For any n ∈ N, we can construct the algebra P(n)(F∗(H)) of n−ary polynomial functions on F∗(H), ( P(n)(F∗(H)) = 〈P n(F∗(H)), (βi : i ∈ I)〉) . Consider the subalgebra P (n) H (F ∗(H)) of P(n)(F∗(H)) obtained by adding to the operations (βi : i ∈ I) of F∗(H) only the nullary operations associated to the characteristic functions of the elements of H. Thus the elements of P(n)H (F ∗(H)) (n ∈ N) are those and only those functions from (F ∗(H))n into F ∗(H) which can obtained by applying (i), (ii), (iii) from bellow for finitely many times: (i) the functions Cnχa : (F ∗(H))n → F ∗(H), defined by setting Cnχa (µ1, ..., µn) = χa, for all µ1, ..., µn ∈ F ∗(H) are elements of P (n) H (F ∗(H)), for every a ∈ H; (ii) the functions eni : (F ∗(H))n → F ∗(H), eni (µ1, ..., µn) = µi, for all µ1, ..., µn ∈ F ∗(H), i = 1, ..., n are elements of P(n)H (F ∗(H)); (iii) if p1, ..., pni are elements of P (n) H (F ∗(H)), and i ∈ I then βi(p1, ..., pni ) : (F ∗(H))n → Ratio Mathematica, 20, 2010 55 F ∗(H), defined by setting for all µ1, ..., µn ∈ F ∗(H), (βi(p1, ..., pni ))(µ1, ..., µn) = βi(p1(µ1, ..., µn), ..., pni (µ1, ..., µn)) is also an element of P (n) H (F ∗(H)). In the continue, we will use only polynomial functions from P(n)H (F ∗(H)). Thus we will drop the subscript with no danger of confusion. Definition 4.1. Let α be the relation defined on H for x, y ∈ H set xαy follows: xαy ⇐⇒ p(χa1 , ..., χan )(x) > 0and p(χa1 , ..., χan )(y) > 0, for some p ∈ P n(F∗(H)), a1, ..., an ∈ H. It is clear that α is symmetric. Because for any a ∈ H, e11(χa)(a) > 0, the relation α is also reflexive. We take α∗ to be the transitive closure of α. Then α∗ is an equivalence relation on H. Lemma 4.2. If f ∈ P 1(F∗(H)) and a, b ∈ H satisfy aα∗b then f (χa)α∗f (χb). Proof. By the definition of α∗ : a = y1αy2α...αym = b for some m ∈ N and y2, ..., ym−1 ∈ H. Let f (χyi )(ui) > 0, i = 1, ..., m. Consider 1 ≤ j < m. Clearly yjαyj+1 means that pj(χa1 , ..., χan )(yj) > 0 and pj(χa1 , ..., χan )(yj+1) > 0, for some nj ∈ N, pj ∈ P nj (F∗(H)), a1, ..., an ∈ H. Define the nj−ary hyperoperation qj on F ∗(H) by setting qj(χx1 , ..., χxnj ) = ∨ {f (χt) : pj(χx1 , ..., χxnj )(t) > 0} for all x1, ..., xnj ∈ H. Clearly qj ∈ P nj (F∗(H)) and for x ∈ H; qj(χa1 , ..., χan )(x) = ∨ pj (χa1 ,...,χan )(z)>0 f (χz)(x). From pj(χa1 , ..., χan )(yj) > 0 and pj(χa1 , ..., χan )(yj+1) > 0 we get 0 < f (χyj )(uj) ≤ qj(χa1 , ..., χan )(uj) and 0 < f (χyj+1 )(uj+1) ≤ qj(χa1 , ..., χan )(uj+1) proving ujαuj+1. Thus u1α ∗um. Since f (χa)(u1) = f (χy1 )(u1) > 0 and f (χb)(um) = f (χym )(um) > 0 were arbitrary, we obtain f (χa)α ∗f (χb).� Remark 4.3. For a given fuzzy hyperalgebra H and equivalence relation ρ on H, the set H/ρ can be considered as a hyperalgebra with the hyperoperations Ratio Mathematica, 20, 2010 56 βi(ρ(a1), ..., ρ(ani )) = {ρ(z) | βi(b1, ..., bni )(z) > 0, bk ∈ ρ(ak), ∀k ∈ {1, ..., ni}} (1) for all i ∈ I. Lemma 4.4. Let ρ be an equivalence relation on H such that H/ρ be an universal algebra . Then for any n ∈ N, p ∈ P n(F∗(H)) and a1, ..., an ∈ H the following gold: p(χa1 , ..., χan )(x) > 0 and p(χa1 , ..., χan )(y) > 0 =⇒ xρy. Proof. We will prove this statement by induction over the steps of construction of an n−ary polynomial function( for n ∈ N arbitrary). If p = Cnχa , from C n χa (χa1 , ..., χan )(x) > 0 and C n χa (χa1 , ..., χan )(y) > 0 we deduce that x = y = a, thus xρy. If p = eni with i ∈ {1, ..., n}, from eni (χa1 , ..., χan )(x) > 0 and eni (χa1 , ..., χan )(y) > 0 we deduce that x = y = ai, , and hence xρy. We suppose that the statement holds for the n−ary polynomial functions p1, ..., pnk and we will prove it for the n−ary polynomial function βk(p1, ..., pnk ). If 0 < βk(p1, ..., pnk )(χa1 , ..., χan )(x) = βk(p1(χa1 , ..., χan ), ..., pnk (χa1 , ..., χan ))(x) =∨ (x1,...,xnk )∈H nk (p1(χa1 , ..., χan )(x1) ∧ ... ∧ pnk (χa1 , ..., χan )(xnk ) ∧ βk(x1, ..., xnk )(x)) and if we set y instead of x, above statement is true. Thus there exist x1, ..., xnk , y1, ..., ynk ∈ H, such that pi(χa1 , ..., χan )(xi) > 0 and pi(χa1 , ..., χan )(yi) > 0, for i ∈ {1, ..., nk} and βk(x1, ..., xnk )(x) > 0 and βk(y1, ..., ynk )(y) > 0. Obviously, xiρyi for all i ∈ {1, ..., nk} and according to (1) and by the hypothesis we obtain that ρ(x) = ρ(y), i.e., xρy, as desired.� The next result immediately follows from previous two lemmas. Theorem 4.5. The relation α∗ is the smallest equivalence relation on fuzzy hyperalgebra H such that H/ρ is an universal algebra. Ratio Mathematica, 20, 2010 57 We call H/ρ, fundamental universal algebra of fuzzy hyperalgebra H such that H/ρ. Proof. At the first, we show that H/ρ is a universal algebra. For this we take any x, y ∈ H, such that α∗(x), α∗(y) ∈ βk(α∗(a1), ..., α∗(ank )) for k ∈ I and a1, ..., ank ∈ H. This means that there exist x1, ..., xnk , y1, ..., ynk ∈ H, such that βk(x1, ..., xnk )(x) > 0 and βk(y1, ..., ynk )(y) > 0 and xiα ∗aiα ∗yi for all i ∈ {1, ..., nk}. Applying Lemma 4.2 to the unary polynomial functions βi(z, C n χx2 , ..., Cnχxnk ), βi(C n χy1 , z, Cnχx3 , ..., Cnχxnk ), ..., , βi(C n χy1 , ..., Cnχynk−1 , z), we obtain the following relations: βi(χx1 , ..., χxnk )α ∗β(χy1 , χx2 , ..., χxnk ) βi(χy1 , χx2 , ..., χxnk )α ∗βi(χy1 , χ22 , χx3 ..., χxnk ) ... βi(χy1 , χy2 , ..., χxnk−1 )α ∗βi(χy1 , χy2 , ..., χynk ), which leads us to xα∗y (from definition α∗), i.e. α∗(x) = α∗(y). Clearly, βi in (1) is an operation on H/α∗, for any i ∈ I, and H/α∗ is a universal algebra. Now we prove that α∗ is smallest. If ρ is an arbitrary equivalence relation on H such that H/ρ is a universal algebra, we can show that α∗ ⊆ ρ. If xαy then there exist n ∈ N, p ∈ P n(F∗(H)) and a1, ..., an ∈ H for which p(χa1 , ..., χan )(x) > 0 and p(χa1 , ..., χan )(y) > 0, and hence by Lemma 4.4 we have xρy, hence α ⊆ ρ, which implies α∗ ⊆ ρ.� Remark 4.6. For a given fuzzy hyperalgebra H and equivalence relation α∗ on H. Let us define the operations of the universal algebra H/α∗ as follows : βi(α ∗(a1), ..., α ∗(ani )) = {α∗(b) | βi(a1, ..., ani )(b) > 0}. Moreover, we can write βi(α ∗(a1), ..., α ∗(ani )) = α ∗(b) βi(a1, ..., ani )(b) > 0. Example 4.7. Let H=〈H, ◦〉 be a fuzzy hypersemigroup, i.e. a fuzzy hyperalgebra with one binary fuzzy hyperoperation ◦, which is associative, that is x ◦ (y ◦ z) = (x ◦ y) ◦ z, Ratio Mathematica, 20, 2010 58 for all x, y, z ∈ H ( for more details see [21]). Let F∗(H)=〈F ∗(H), ◦〉 be the universal algebra with one binary operation defined as follows: (µ ◦ ν)(r) = ∨ x,y∈H µ(x) ∧ ν(y) ∧ (x ◦ y)(r) ∀ µ, ν ∈ F ∗(H),r ∈ H. By distributivity of the lattice ([0, 1], ∨, ∧) and associativity of ◦ in H, we will prove that the operation ◦ in F∗(H) is associative. So for every µ, ν, η ∈ F ∗(H) and r ∈ H we have ((µ ◦ ν) ◦ η)(r) = ∨ x,y∈H [(µ ◦ ν)(x) ∧ η(y) ∧ (x ◦ y)(r)] =∨ x,y∈H [( ∨ p,q∈H µ(p) ∧ ν(q) ∧ (p ◦ q)(x)) ∧ η(y) ∧ (x ◦ y)(r)] =∨ p,q,y∈H [µ(p) ∧ ν(q) ∧ η(y) ∧ ( ∨ x∈H (p ◦ q)(x) ∧ (x ◦ y)(r))] =∨ p,q,y∈H [µ(p) ∧ ν(q) ∧ η(y) ∧ ( ∨ x∈H (p ◦ x)(r) ∧ (q ◦ y)(x))] = ∨ p,x∈H [µ(p) ∧ (p ◦ x)(r) ∧ ( ∨ q,y∈H ν(q) ∧ η(y) ∧ (q ◦ y)(x))] =∨ p,x∈H [µ(p) ∧ (p ◦ x)(r) ∧ (ν ◦ η)(x)] = (µ ◦ (ν ◦ η))(r). Consider now the universal algebra of polynomial functions of 〈F ∗(H), ◦〉. The images of the elements of this algebra are the sums of nonzero fuzzy subsets of H. Thus we can define α on H by: aαb ⇐⇒ ∃x1, ..., xn ∈ H(n ∈ N): (χx1 ◦ ... ◦ χxn )(a) > 0 and (χx1 ◦ ... ◦ χxn )(b) > 0. Consider the quotient set H/α∗ with the hyperoperation α∗(a) ◦ α∗(b) = {α∗(c) | (a′ ◦ b′)(c) > 0, a′α∗a, b′α∗b}. Really ◦ is an operation, because α∗ is the fundamental relation on H. Also α∗(x) ◦ α∗(y) ◦ α∗(z)) = α∗(x) ◦ α∗(k) = α∗(l), where (y ◦ z)(k) > 0 and (x ◦ k)(l) > 0. Therefore, 0 < (x ◦ (y ◦ z))(l) = ((x ◦ y) ◦ z)(l) = ∨ p∈H [(x ◦ y)(p) ∧ (p ◦ z)(l)]. Thus Ratio Mathematica, 20, 2010 59 there exists p ∈ H, such that α∗(l) = α∗(p) ◦ α∗(z) = (α∗(x) ◦ α∗(y)) ◦ α∗(z), that the operation ◦ in H/α∗ is associative. Moreover, if H=〈H, ◦〉 be a fuzzy hypergroup, that is x ◦ H = H ◦ x = χH , for every x ∈ H, since for every α∗(a), α∗(b) ∈ H/α∗, there exist α∗(t), α∗(s) ∈ H/α∗, such that, α∗(a) ◦ α∗(t) = α∗(b) and α∗(s) ◦ α∗(a) = α∗(b), it is concluded that H/α∗=〈H/α∗, ◦〉 is a group. Example 4.8. Let R=〈R, ⊕, �〉 be a fuzzy hyperring. This means that 〈R, ⊕〉 is a commutative fuzzy hypergroup, 〈R, �〉 is a fuzzy hypersemigroup and for all x, y, z ∈ R satisfies: x�(y⊕z) = (x�y)⊕(x�z) and (x⊕y)�z = (x�z)⊕(y�z) ( for more details see [13]). Let F∗(R)=〈F ∗(R), ⊕, �〉 be the universal algebra with two binary operations defined as follows: (µ ⊕ ν)(r) = ∨ x,y∈H [µ(x) ∧ ν(y) ∧ (x ⊕ y)(r)], (µ � ν)(r) = ∨ x,y∈H [µ(x) ∧ ν(y) ∧ (x � y)(r)], for all µ, ν ∈ F ∗(R), r ∈ R. Obviously, the operation ⊕ in F ∗(R) is commutative, and ⊕ and � in F ∗(R) are associative. By distributivity of the lattice [0, 1] and distributivity � with respect to ⊕ in R, we will prove that the operation � in F ∗(R) is distributive with respect to the operation ⊕, too. For every µ, ν, eta ∈ F ∗(R) and r ∈ R we have: (µ � (ν ⊕ η))(r) = ∨ x,y∈R [µ(x) ∧ (ν ⊕ η)(y) ∧ (x � y)(r)] =∨ x,y∈R [µ(x) ∧ ( ∨ s,t∈R ν(s) ∧ η(t) ∧ (s ⊕ t)(y)) ∧ (x � y)(r)] =∨ x,y∈R [ ∨ s,t∈R (µ(x) ∧ ν(s) ∧ η(t) ∧ (s ⊕ t)(y) ∧ (x � y)(r))] =∨ x,s,t∈R [µ(x) ∧ ν(s) ∧ η(t) ∧ ( ∨ y∈R (x � y)(r) ∧ (s ⊕ t)(y))] =∨ x,s,t∈R [µ(x) ∧ ν(s) ∧ η(t) ∧ ( ∨ p,q∈R (x � s)(p) ∧ (x � t)(q) ∧ (p ⊕ q)(r))] = Ratio Mathematica, 20, 2010 60 ∨ x,s,t∈R [ ∨ p,q∈R (µ(x) ∧ η(t) ∧ (x � t)(q) ∧ µ(x) ∧ ν(s) ∧ (x � s)(p) ∧ (p ⊕ q)(r))] =∨ p,q∈R [( ∨ x,t∈R µ(x) ∧ η(t) ∧ (x � t)(q)) ∧ ( ∨ x,s∈R µ(x) ∧ ν(s) ∧ (x � s)(p)) ∧ (p ⊕ q)(r)] =∨ p,q∈R [(µ � η)(q) ∧ (µ � ν)(p) ∧ (p ⊕ q)(r)] = ((µ � ν) ⊕ (µ � η))(r). And analogously, (µ ⊕ ν) � η = (µ � η) ⊕ (ν � η). Now we can construct the universal algebra (with two binary operations) of the polynomial functions of F∗(R) for any n ∈ N. The images of the elements of this algebra are the sums of products of nonzero fuzzy subsets of R. Thus we can define α on R by; aαb ⇐⇒ ∃xij ∈ R, i ∈ {1, ..., kj}, j ∈ {1, ..., l}, kj, l ∈ N: (⊕lj=1(� kj i=1χxij ))(a) > 0 and (⊕ l j=1(� kj i=1χxij ))(b) > 0. Consider the quotient set R/α∗ withe two following hyperoperations : α∗(a) ⊕ α∗(b) = {α∗(c) | (a′ ⊕ b′)(c) > 0, a′α∗a, b′α∗b}, and α∗(a) � α∗(b) = {α∗(c) | (a′ � b′)(c) > 0, a′α∗a, b′α∗b} Actually ⊕ and � are operations, because α∗ is the fundamental relation on R. By con- sidering the previous example, obviously 〈R/α∗, ⊕〉 is a commutative group. We verify the distributivity of � with respect to ⊕ for the universal algebra R/α∗=〈R/α∗, ⊕, �〉. We have α∗(a) � (α∗(b) ⊕ α∗(c)) = α∗(a) � α∗(d) = α∗(e), where (b ⊕ c)(d) > 0 and (a � d)(e) > 0 0 < (a � (b ⊕ c))(e) = ∨ p∈R (a � p)(e) ∧ (b ⊕ c)(p). Thus 0 < ((a � b) ⊕ (a � c))(e) = ∨ x,y∈R (a � b)(x) ∧ (a � c)(y) ∧ (x ⊕ y)(e). Therefore, there exist x, y ∈ R such that α∗(e) = α∗(x) + α∗(y) = (α∗(a) + α∗(b))⊕(α∗(a)�α∗(c)), and hence it was proved that α∗(a) � (α∗(b) ⊕ α∗(c)) = (α∗(a) + α∗(b)) ⊕ (α∗(a) � α∗(c)). Analogously, we can prove that (α∗(b) ⊕ α∗(c)) � α∗(a)) = (α∗(b) � α∗(a)) ⊕ (α∗(c) � α∗(a)). Thus it concluded that R/α∗=〈R/α∗, ⊕, �〉 is a ring, as desired.� Conclusion Ratio Mathematica, 20, 2010 61 We introduced and studied term functions over fuzzy hyperalgebras, as the largest class of fuzzy algebraic systems. We use the idea that the set of nonzero fuzzy subsets of a fuzzy hyperalgebra can be organized naturally as a universal algebra, and constructed the term functions over this algebra. We gave the form of generated subfuzzy hyperalgebra of a given fuzzy hyperalgebra as a generalization of universal algebras and multialgebras. Finally, we characterized the form of the fundamental relation of a fuzzy hyperalgebra, to construct the fundamental universal algebra corresponding to a given fuzzy hyperalgebra, and this result guarantee that that fundamental relation on any fuzzy algebraic hyper- structures, such as fuzzy hypergroups, fuzzy hyperrings, fuzzy hypermodules,... exists. Acknowledgement This research is partially supported by the “Fuzzy Systems and Its Applications Center of Excellence, Shahid Bahonar University of Kerman, Iran” and “Research Center in Algebraic Hyperstructures and Fuzzy Mathematics, University of Mazandaran, Babolsar, Iran”. References [1] R. Ameri, On categories of hypergroups and hypermodules , Italian journal of pure and applid mathematics, Vol. 6 (2003) 121-132. [2] R. Ameri and I. G. Rosenberg, Congruences of multialgebras, Multivalued Logic and Soft Computing (to appaear). [3] R. Ameri and M.M. Zahedi, Hyperalgebraic system, Italian journal of pure and applid mathematics, Vol. 6 (1999) 21-32. Ratio Mathematica, 20, 2010 62 [4] R. Ameri and M.M. Zahedi, Fuzzy subhypermodules over fuzzy hyperrings, Sixth International on AHA, Democritus University, 1996, 1-14,(1997). [5] S. Burris, H. P. Sankappanavar, A course in universal algebra, Springer Verlage 1981. [6] P. Corsini, Prolegomena of hypergroup theory, Supplement to Riv. Mat.Pura Appl., Aviani Editor, 1993. [7] P. Corsini, V. Leoreanu, Applications of hyperstructure theory, Kluwer, Dordrecht 2003. [8] P. Corsini, I. Tofan, On fuzzy hypergroups, PU.M.A., 8 (1997) 29-37. [9] B. Davvaz, Fuzzy Hv-groups, Fuzzy sets and systems, 101 (1999) 191-195. [10] B. Davvaz, Fuzzy Hv-Submodules, Fuzzy sets and systems, 117 (2001) 477-484. [11] B. Davvaz, P. Corsini, Generalized fuzzy sub-hyperquasigroups of hyperquasigroups, Soft Computing, 10 (11) (2006), 1109-1114. [12] M. Mehdi Ebrahimi, A. Karimi and M. MahmoudiOn Quotient and Isomorphism Theorems of Universal Hyperalgebras, Italian Journal of Pure and Applied Mathe- matics, 18 (2005), 9-22. [13] G. Gratzer, Universal algebra, 2nd edition, Springer Verlage, 1970. [14] V. Leoreanu-Fotea, B. Davvaz, Fuzzy hyperrings, Fuzzy sets and systems, 2008, DOI 10.1016/j.fss.2008.11.007. [15] V. Leoreanu-Fotea, Fuzzy Hypermodules, Computes and Mathematics with Applica- tions, vol. 57 (2009) 466-475. Ratio Mathematica, 20, 2010 63 [16] F. Marty, Sur une generalization de la nation de groupe, 8th congress des Mathe- maticiens Scandinaves, Stockholm (1934) 45-49. [17] J.N. Mordeson, M.S. Malik, Fuzzy commutative algebra, Word Publ., 1998. [18] C. Pelea, On the direct product of multialgebras, Studia uni. Babes-bolya, Mathemat- ica, vol. XLVIII (2003) 93-98. [19] C. Pelea, Multialgebras and termfunctions over the algebra of their nonvoid subsets, Mathematica (Cluj), vol. 43 (2001) 143-149. [20] C. Pelea, On the fundamental relation of a multialgebra, Italian Journal of Pure and Applid Mathematics, Vol. 10 (2001) 141-146. [21] H. E. Pickett, Homomorphism and subalgebras of multialgebras, Pacific J. Math, vol. 10 (2001) 141-146. [22] M.K. Sen, R. Ameri, G. Chowdhury, Fuzzy hypersemigroups, Soft Computing, 2007, DOI 10.1007/s00500-007-025709. [23] A. Rosenfeld , Fuzzy groups, J. Math. Anal. Appl. 35. (1971) 512-517. [24] D. Schweigert, Congruence Relations of Multialgebras , Discrete Mathematics 53 (1985) 249-253. [25] S. Spartalis, T. Vougiouklis, The Fundamental Relations on Hv−rings, Math. Pura Appl., 13 (1994) 7-20. [26] T. Vougiouklis, The fundamental Relations in Hyperrings, The general hyperfield Proc. 4th International Congress in Algebraic Hyperstructures and Its Applications (AHA 1990) World Scientific, (1990) 203-211. Ratio Mathematica, 20, 2010 64 [27] T. Vougiouklis, Hyperstructures and their representations, Hardonic, press Inc., 1994. [28] L. A. Zadeh, Fuzzy Sets, Inform. and Control, vol. 8 (1965) 338-353. Ratio Mathematica, 20, 2010 65