RATIO MATHEMATICA 24 (2013), 3–10 ISSN:1592-7415 Fuzzy hyperalgebras and direct product R. Ameri, T. Nozari School of mathematics, statistics and computer sciences, College of sciences, University of Tehran, Tehran, Iran Departement of mathematics, Faculty of science, Golestan university, Gorgan, Iran rameri@ut.ac.ir,t.nozari@gu.ac.ir Abstract We introduce and study the direct product of a family of fuzzy hyperalgebras of the same type and present some properties of it. Key words: Fuzzy hyperalgebras, Term function, Direct product. MSC2010: 97U99. 1 Introduction In this section we present some definitions and simple properties of hy- peralgebras which will be used in the next section. 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 [1]). Recall that 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 hyperalgebra H = 〈H, (βi, | i ∈ I)〉 (which is called hyperalgebraic system or a multialgebra ) is the set H with together a collection (βi, | i ∈ I) of hyperoperations on H. R. Ameri, T. Nozari 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. Two hyperalgebras of the same type are called similar hyperalgrbras. 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)} It is easy to see that 〈P∗(H), (βi : i ∈ I)〉 is an algebra of the same type of H. Definition 1.1 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 )). Definition 1.2 Let H be a nonempty set. A fuzzy subset µ of H is a function µ : H → [0, 1]. Definition 1.3 A fuzzy n-ary hyperoperation fn on S is a map fn : S × ···× S −→ F∗(S), which associated a nonzero fuzzy subset fn(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 1.4 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 hyper- algebra, where (ni : i ∈ I) is type of this fuzzy hyperalgebra. Definition 1.5 If µ1, . . . ,µni be ni nonzero fuzzy subsets of a fuzzy huperal- gebra 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, for nonempty subsets A1, . . . ,Ank of H, set A = A1 × . . . × Ani . Then for all t ∈ H βk(A1, . . . ,Ank )(t) = ∨(a1,...,ank )∈A(βk(a1, . . . ,ank )(t)). 4 Fuzzy hyperalgebras and direct product For nonempty subset A of H, χA denote the characteristic function of A . Note that, if f : H1 −→ H2 is a map and a ∈ H1, then f(χa) = χf(a). Definition 1.6 Let H = 〈H, (βi : i ∈ I)〉 and H′ = 〈H′, (β′i : i ∈ I)〉 be two fuzzy hyperalgebras with the same type, and f : H −→ H′ be a map. We say that f is a homomorphism of fuzzy hyperalgebras if for every i ∈ I and every a1, . . . ,ani ∈ H we have f(βi(a1, . . . ,ani )) ≤ β′i(f(a1), . . . ,f(ani )). 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 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). In [3] 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)〉. 2 On the Direct Product of Fuzzy Hyperal- gebras Proposition 2.1 Let H=〈H, (βi : i ∈ I)〉 and B=〈B, (βi : i ∈ I)〉 are fuzzy hyperalgebras of the same type, h : H → B a fuzzy homomorphism and p ∈ P (n)(F∗(H)). Then for all a1, . . . ,an ∈ H we have h(p(χa1, . . . ,χan )) ⊆ p(h(χa1 ), . . . ,h(χan )). 5 R. Ameri, T. Nozari Proof. The prove is by induction over the steps of construction of a term.2 Remark 2.1 If h : H → B be fuzzy good homomorphism then h(p(χa1, . . . ,χan )) = p(h(χa1 ), . . . ,h(χan )). Remark 2.2 We can easily construct the category of the fuzzy hyperalgebras of the same type, where the morphisms are considered to be the fuzzy ho- momorphisms and the composition of two morphisms is the usual mapping composition and we will denote it by FHA Definition 2.1 Let q,p ∈ P (n)(F∗(H)). The n−ary (strong) identity p = q is said to be satisfied on a fuzzy hyperalgebra H if p(χa1, . . . ,χan ) = q(χa1, . . . ,χan ) for all a1, . . . ,an ∈ H. We can also consider that a weak identity p∩ q 6= ∅ is said to be satisfied on a fuzzy hyperalgebra H if p(χa1, . . . ,χan ) ∧ q(χa1, . . . ,χan ) > 0 for all a1, . . . ,an ∈ H. Definition 2.2 Let ((Hk, (β k i : i ∈ I)),k ∈ K) be an indexed family of fuzzy hyperalgebras with the same type. The direct product ∏ k∈K Hk is a fuzzy hy- peralgebra with univers Πk∈KHk and for every i ∈ I and (a1k)k∈K, . . . , (a ni k )k∈K ∈ Πk∈KHk : β Q i ((a 1 k)k∈K, . . . , (a ni k )k∈K)(tk)k∈K = ∧ k∈K βki (a 1 k, . . . ,a ni k )(tk) Theorem 2.1 The fuzzy hyperalgebra ∏ k∈K Hk constructed this way, to- gether with the canonical projections, is the product of the fuzzy hyperalgebras (Hk,k ∈ K) in the category FHA. Proof. For any fuzzy hyperalgebra (B, (βBi : i ∈ I)) and for any family of fuzzy hyperalgebra homomorphisms (αk : B → Hk|k ∈ K) there is only one homomorphism α : B → Πk∈KHk such that αk = πKk ◦α for any k ∈ K. Indeed, there exists only one mapping α such that the diagram is commuta- tive. - 6 B Hk � αk Πk∈KHk πK k α 6 Fuzzy hyperalgebras and direct product This mapping is defined by α(b) = (αk(b))k∈K. Now we have to do is to verify that α is fuzzy hyperalgebra homomorphism. If we consider i ∈ I and b1, . . . ,bni ∈ B, (tk)k∈K ∈ Πk∈KHk then if r ∈ α−1((tk)k∈K) we have α(r) = (tk)k∈K and α(r) = (αk(r))k∈K, hence ∀k ∈ K; tk = αk(r), it means that ∀k ∈ K; r ∈ α−1k (tk), therefore ∀k ∈ K; α −1((tk)k∈K) ⊆ α−1k (tk). We have α(βBi (b1, . . . ,bni ))(tk)k∈K = ∨ r∈α−1((tk)k∈K ) (βBi (b1, . . . ,bni ))(r) ≤ ∨ s∈α−1 k (tk)) βBi (b1, . . . ,bni ))(s) = αk(β B i (b1, . . . ,bni ))(tk) then α(βBi (b1, . . . ,bni ))(tk)k∈K ≤ ∧ k∈K αk(β B i (b1, . . . ,bni ))(tk) ≤ ∧ k∈K βki (αk(b1), . . . ,αk(bni ))(tk) = β Q i (α(b1), . . . ,α(bni ))(tk)k∈K. Which finishes the proof.2 Proposition 2.2 For every n ∈ N, p ∈ P (n)(F∗(H)) and (a1k)k∈K, . . . , (a n k)k∈K, we have p(χ(a1 k )k∈K , . . . ,χ(an k )k∈K )(tk)k∈K = ∧ k∈K p(χa1 k , . . . ,χan k )(tk) Proof. We will use the steps of construction of a term. i. If p = ejn(j = 1, 2, . . . ,n) then p(χ(a1 k ) k∈K , . . . ,χ(an k ) k∈K )(tk)k∈K = e j n(χ(a1k)k∈K , . . . ,χ(an k ) k∈K )(tk)k∈K = χ (a j k ) k∈K (tk)k∈K = ∧ k∈K ejn(χa1k, . . . ,χa n k )(tk) = ∧ k∈K p(χa1 k , . . . ,χan k )(tk) ii. Suppose that the statement has been proved for p1, . . . ,pni and that p = βi(p1, . . . ,pni ). Then we have p(χ(a1 k )k∈K , . . . ,χ(an k )k∈K )(tk)k∈K = βi(p1, . . . ,pni )(χ(a1k)k∈K , . . . ,χ(an k )k∈K )(tk)k∈K = βi(p1(χ(a1 k )k∈K , . . . ,χ(an k )k∈K ), . . . ,pni (χ(a1k)k∈K , . . . ,χ(an k )k∈K ))(tk)k∈K = ∨ (s1 k )k∈K,...,(s ni k )k∈K [p1(χ(a1 k )k∈K , . . . ,χ(an k )k∈K )(s1k)k∈K∧. . .∧pni (χ(a1k)k∈K, . . . ,χ(ank )k∈K) (snik )k∈K ∧βi((s 1 k)k∈K, . . . , (s ni k )k∈K)(tk)k∈K] 7 R. Ameri, T. Nozari = ∨ (s1 k )k∈K,...,(s ni k )k∈K [ ∧ k∈K p1(χa1 k , . . . ,χan k )(s1k)∧ . . .∧ ∧ k∈K pni (χa1k, . . . ,χa n k )(snik )∧∧ k∈K βi(s 1 k, . . . ,s ni k )(tk)] = ∧ k∈K [ ∨ (s1 k )k∈K,...,(s ni k )k∈K p1(χa1 k , . . . ,χan k )(s1k)∧. . .∧pni (χa1k, . . . ,χank )(s ni k )∧βi(s 1 k, . . . ,s ni k )(tk)] = ∧ k∈K βi(p1(χa1 k , . . . ,χan k ), . . . ,pni (χa1k, . . . ,χa n k ))(tk) = ∧ k∈K βi(p1, . . . ,pni )(χa1k, . . . ,χa n k )(tk) = ∧ k∈K p(χa1 k , . . . ,χan k )(tk). which finishes the proof of the proposition.2 Theorem 2.2 If ((Hk, (β k i : i ∈ I)),k ∈ K) be an indexed family of fuzzy hy- peralgebras with the same type I such that p∩q 6= ∅ is satisfied on each fuzzy hyperalgebra Hk, then is also satisfied on the fuzzy hyperalgebra ∏ k∈K Hk. Proof. Let p,q ∈ P (n)(F∗(H)) and suppose that p∩ q 6= ∅ is satisfied on each fuzzy hyperalgebra Hk. This means that for all k ∈ K and for any a1k, . . . ,a n k ∈ Hk we have p(χa1k, . . . ,χank )∧q(χa1k, . . . ,χank ) > 0. By proposition 3.7 , we conclude that p(χ(a1 k )k∈K , . . . ,χ(an k )k∈K ) ∧ r(χ(a1 k )k∈K , . . . ,χ(an k )k∈K ) = = ∧ k∈K p(χa1 k , . . . ,χan k ) ∧ ∧ k∈K q(χa1 k , . . . ,χan k ) = ∧ k∈K (p(χa1 k , . . . ,χan k ) ∧ q(χa1 k , . . . ,χan k )) > 0 and the proof is finished.2 Theorem 2.3 If ((Hk, (β k i : i ∈ I)),k ∈ K) be an indexed family of fuzzy hy- peralgebras with the same type I such that p = q is satisfied on each fuzzy hy- peralgebra Hk, then p = q is also satisfied on the fuzzy hyperalgebra ∏ k∈K Hk. Proof. Let p,q ∈ P (n)(F∗(H)) and suppose that p = q is satisfied on each fuzzy hyperalgebra Hk. This means that for all k ∈ K and for any a1k, . . . ,a n k ∈ Hk we have p(χa1k, . . . ,χank ) = q(χa1k, . . . ,χank ). By proposition 3.7 , we conclude that p(χ(a1 k )k∈K , . . . ,χ(an k )k∈K ) = ∧ k∈K p(χa1 k , . . . ,χan k ) = ∧ k∈K q(χa1 k , . . . ,χan k ) 8 Fuzzy hyperalgebras and direct product = r(χ(a1 k )k∈K , . . . ,χ(an k )k∈K ) and the proof is finished.2 3 Acknowledgement The first author partially has been supported by the ”Research Center in Al- gebraic Hyperstructures and Fuzzy Mathematics, University of Mazandaran, Babolsar, Iran” and ”Algebraic Hyperstructure Excellence, Tarbiat Modares University, Tehran, Iran”. References [1] P. Corsini, Prolegomena of Hypergroup Theory, Second edition, Aviani editor (1993). [2] P. Corsini and I. Tofan, On Fuzzy Hypergroups, PU.M.A, 8 (1997), 29- 37. [3] G. Gratzer, Universal algebra, 2nd edition, Springer Verlage, 1970. [4] J. N. Mordeson,M.S. Malik, Fuzzy Commutative Algebra, Word Publ. 1998. [5] C. Pelea, On the direct product of multialgebras, Studia uni, Babes-bolya, Mathematica, vol. XLVIII (2003), 93-98. [6] C. Pelea, Multialgebras and term functions over the algebra of their non- void subsets, Mathematica (Cluj), 43 (2001), 143-149. [7] M.K. Sen, R.Ameri, G. Chowdhury, Fuzzy hypersemigroups, Soft Com- puting, Soft Comput, 12 (2008), 891-900. [8] T. Vougiuklis, Hyperstructures and their representations, Hardonic Press (1994). 9 10