On Fuzzy Ideals of Hyperlattice

Transcription

1 International Journal of Algebra, Vol. 2, 2008, no. 15, On Fuzzy Ideals of Hyperlattice B. B. N. Koguep Department of Mathematics, Faculty of Science University of Yaounde 1, BP 812, Cameroon C. Nkuimi Department of Mathematics, Faculty of Science University of Yaounde 1, BP 812, Cameroon C. Lele Department of Mathematics University of Dschang, BP 67, Cameroon lele Abstract. In this paper we introduce the notion of fuzzy ideal and fuzzy filter of hyperlattice and study some important properties with many examples. Finally, we establish the fuzzy prime ideal theorem for hyperlattice. Mathematics Subject Classification: 06B99, 06D72, 08A72 Keywords: Hyperlattice, ideal, prime ideal, fuzzy hyperlattice, fuzzy ideal, fuzzy prime ideal

2 740 B. B. N. Koguep, C. Nkuimi and C. Lele 1. Backgrounds We recall here some definitions and propositions on hyperlattices [6, 7] and we establish some results which could help to compute some examples of hyperlattices. Definition 1.1. [7] Let L be a non empty set and : L L P(L) be a hyperoperation, where P(L) is a power set of L and P(L) = P(L) { } and : L L L be an operation. Then (L,, ) is a hyperlattice if for all a, b, c L : i) a a a, a a = a; ii) a b = b a, a b = b a; iii) (a b) c = a (b c); (a b) c = a (b c); iv) a [a (a b)] [a (a b)]; v) a a b a b = b. Where for all non empty subsets A and B of L, A B = {a b/a A, b B}, A B = {a b/a A, b B}. Proposition 1.2. Let (L,, ) be a hyperlattice, for each pair (a, b) L L there exist a 1,b 1 a b, such that a a 1 and b b 1. Proof: Since a a (a b), it implies that there exists a 1 (a b) such that a = a a 1 that is a a 1. Likewise, since b b (a b), it implies that there exists b 1 a b such that b = b b 1, that is b b 1. Remark 1.3. Because of axiom v) of definition 1.1, for any a, b L, we have, (i) If a b = L, then a = b. (ii) If a b = {0}, then a = b =0. Definition 1.4. [7] A hyperlattice (L,, ) is said to be distributive if for all a, b, c L : a (b c) =(a b) (a c). Proposition 1.5. Let (L,, ) be a distributive hyperlattice with a least element 0. Then, 0 0={0}. Proof: In fact, since L is a distributive hyperlattice, for any x, y L, 0 (x y) =(0 x) (0 y). On the other hand, 0 (x y) ={0} and (0 x) (0 y) =0 0. Thus, 0 0={0}.

3 On Fuzzy ideals of hyperlattice 741 Example 1.6. Let L = {0,a,b,1} and define and by the following Cayley tables 0 a b a 0 a 0 a b o 0 b b 0 a b 1 0 L {a, 1} {b, 1} {1 } a {a, 1} {a, 1} {1 } {1 } b {b, 1} {1 } {b, 1} {1 } 1 {1 } {1 } {1 } {1 } 1 0 a b 1 (L,,, 0, 1) is a bounded hyperlattice, but is not distributive. Since 0 0= L {0}. Example 1.7. Let L = {0,a,b,1} and define and by the following Cayley tables 0 a b 1 0 a b { 0 } {a } {b } {1 } a 0 a 0 a a {a } L {b} {0, 1 } L {a} b o 0 b b b {b } {0, 1 } L {a} L {b} 1 0 a b 1 1 {1 } L {a} L {b} L (L,,, 0, 1) is a bounded hyperlattice, but is not distributive. Since a (a 1) = {a 0,a b, a 1} = {0,a} and (a a) (a 1) = {0,a,1}. Thus a (a 1) (a a) (a 1). Example 1.8. Let L = {0,a,b,1} and define and by the following Cayley tables 0 a b 1 0 a b { 0 } {a } {b } {1 } a 0 a 0 a a {a } {0, a} {1 } {b, 1 } b o 0 b b b {b } {1 } {0, b } {a, 1 } 1 0 a b 1 1 {1 } {b, 1 } {a, 1 } L (L,,, 0, 1) is a bounded distributive hyperlattice. Proposition 1.9. Let (L,, ) be a distributive hyperlattice. For any a L, there is no element in a a greater than a. Proof: Let a L. Suppose that there exists x a a, such that x>a. Since L is distributive, we have a (a x) =(a a) (a x) =a a. Therefore, there exists y a x, such that x = a y a, this is a contradiction. Definition [7] Let (L,, ) be a hyperlattice. A nonempty subset J of L is called an ideal of L if for all x, y L (i) x, y J implies x y J.

4 742 B. B. N. Koguep, C. Nkuimi and C. Lele (ii) if x J and y x, then y J. Definition [7] Let (L,, ) be a hyperlattice. A nonempty subset F of L is called a filter of L if for all x, y L (i) x, y F implies x y F. (ii) if x F and x y, then y F. Let I(L) and F(L) be respectively the set of all ideals and the set of all filters of the bounded distributive hyperlattice L. It is easy to see that I(L) and F(L) when ordered by inclusion are bounded distributive lattices. Example From the previous examples we have the following sets of all ideals and the sets of all filters of L, For example 1.6 : I(L) ={L} and F(L) ={{1}, {a, 1}, {b, 1},L}. For example 1.7 : I(L) ={{0},L} and F(L) ={{1}, {a, 1}, {b, 1},L}. For example 1.8 : I(L) ={{0}, {0,a}, {0,b},L} and F(L) ={{1}, {a, 1}, {b, 1},L}. Remark (i) We know that in any lattice there always exist a smallest ideal different from the lattice, but in the case of a hyperlattice this is true only when the hyperlattice is distributive and has a bottom element (see example 1.6 and proposition 1.5). (ii) When L is a bounded distributive hyperlattice, for any ideal J of L, 0 J and for any filter F of L, 1 F. Proposition Any intersection of ideals of a bounded distributive hyperlattice is an ideal. Proposition Let (L,, ) be a hyperlattice. For all x, y L, ifx y is an ideal of L, then x = y. Proof: Let x, y L, such that x y is an ideal of L. From proposition 1.2, there exist x 1,y 1 x y, such that x x 1 and y y 1. Since x y is an ideal of L, x, y x y. Thus, x = y because of axiom v) of definition 1.1. Proposition [7] Let (L,, ) be a distributive hyperlattice. If a L then I =(a] ={x L/x a} is an ideal. Proof: Let x, y L, such that x y and y I =(a]. Since x y and y a, we have x I. furthermore, if p, q I, then p a and q a. By the distributivity of L, we have a (p q) =(a p) (a q). Thus for every x p q, there exists y p q such that x = a y a, proving p q I.

5 On Fuzzy ideals of hyperlattice 743 Corollary [7] Let (L,, ) be a distributive hyperlattice and a, b L, then we have a b (c] c a c b Definition Let (L,, ) be a bounded distributive hyperlattice and let A be a non empty subset of L. The ideal of L induced by A is the intersection of all ideals of L containing A and is denoted by <A>. Definition [7] Let J and F be respectively a proper ideal and a proper filter of a hyperlattice L. (i) J is said to be prime if a, b L and a b J implies a J or b J. (ii) F is said to be prime if a, b L and (a b) F implies a F or b F. Lemma [7] Let (L,, ) be a distributive hyperlattice. If P is an ideal of L, and a L then P (a] is an ideal of L. Lemma [7] Let D be a filter in a distributive hyperlattice L, let a, p, q L, such that q a. If(p q) D, then (p a) D. The next theorem is the prime ideal theorem for hyperlattice, which is establish in [7] by using the two previous lemmas. Theorem [7] Let the axiom of choise hold. Let (L,,, 0) be a distributive hyperlattice with a least element 0. IfI and D are an ideal and a filter, respectively, such that I D =, then there exists a prime ideal P of L such that I P and P D =. Through this paper, L stands for the bounded distributive hyperlattice (L,,, 0, 1). Definition A fuzzy subset of L is a function μ : L [0, 1]. Let μ be a fuzzy subset of L. Forα [0, 1], the set μ α = {x L/μ(x) α} is called a α level subset of μ or α-cut set of μ. Definition [1] A fuzzy subset μ of L is proper if it is a non constant function. 2. Fuzzy prime ideal of hyperlattice From the definition of fuzzy ideal and fuzzy filter respectively of a lattice and a BCI-Algebra given in [5] and [8], we introduce the notion of fuzzy ideal and fuzzy filter in hyperlattice as follow.

6 744 B. B. N. Koguep, C. Nkuimi and C. Lele Definition 2.1. Let μ be a fuzzy set of L. Then : (i) μ is a fuzzy ideal of L if, for all x, y L, μ(a) μ(x) μ(y) inf a x y x y μ(x) μ(y) (ii) μ is a fuzzy filter of L if for all x, y L, μ(x y) μ(x) μ(y) x y μ(x) μ(y) Example 2.2. Consider the bounded distributive hyperlattice L of example 1.8, and consider the following fuzzy sets of L μ and η defined by, μ(0) = 1, μ(a) = 1, μ(b) = 1, μ(1) = and η(0) = 0, η(a) =1, η(b) = 0, η(1) = 1. Then μ is a fuzzy ideal of L and η is a fuzzy filter of L. Remark 2.3. (i) If μ is a fuzzy ideal of L, then μ(0) μ(x) μ(1), for all x L. (ii) If μ is a fuzzy filter of L, then μ(0) μ(x) μ(1), for all x L. The following proposition gives the characterization of fuzzy ideal and fuzzy filter of hyperlattice in term of α - cut set. Proposition 2.4. Let μ be a fuzzy subset of L. Then : (i) μ is a fuzzy ideal of L, if and only if, for any α [0; 1], such that μ α, μ α is an ideal of L. (ii) μ is a fuzzy filter of L, if and only if, for any α [0; 1], such that μ α, μ α is a filter of L. Proof: (i) ( ) Let μ be a fuzzy ideal of L and α [0; 1], such that μ α. Let x μ α and y x. We have μ(x) α and μ(y) μ(x). So μ(y) α. Hence y μ α. Let x, y μ α. We have μ(x) α and μ(y) α. So μ(x) μ(y) α. Since μ is a fuzzy ideal of L, inf μ(a) α. a x y We obtain that, a (x y),μ(a) α. Hence, a (x y),a μ α. Then, x y μ α. Thus, μ α is an ideal of L. ( ) Suppose that for any α [0; 1], such that μ α, μ α is an ideal of L. Let x, y L, and α = μ(x) μ(y), we have x μ α and y μ α. Because μ α is an ideal of L, we have x y μ α and x y μ α. i.e., μ(x y) α = μ(x) μ(y) and for all a x y, μ(a) α = μ(x) μ(y). Therefore,

7 On Fuzzy ideals of hyperlattice 745 inf μ(a) μ(x) μ(y). a x y Let x, y L such that x y. Let α = μ(y), then y μ α. Since μ α is an ideal of L, we have x μ α.soμ(x) α = μ(y). So μ is a fuzzy ideal of L. (ii) Similar to (i) and we omitted. Corollary 2.5. A nonempty subset I of L is an ideal of L if and only if the characteristic function of I is a fuzzy ideal of L. Let FI(L) and FF(L) be respectively the set of all fuzzy ideals of L and the set of all fuzzy filters of L. Remark 2.6. (FI(L), ) and (FF(L), ) are completely bounded distributive lattices, with define by : for any μ, ν FI(L) (or in FF(L)), μ ν μ(x) ν(x), for all x L and for any family {μ α /α Λ} of fuzzy ideals (or fuzzy filters) of L, ( μ α )(x) = supμ α (x) and ( μ α )(x) = inf μ α(x). α Λ α Λ In the case of lattices, M. Attallah [1] gives the definition of fuzzy prime ideal and fuzzy prime filter, from that we give the following definition in the case of hyperlattice. Definition 2.7. (fuzzy prime ideal and fuzzy prime filter) (i) A proper fuzzy ideal μ of L is called fuzzy prime ideal, if μ(x y) μ(x) μ(y), for all x, y L. (ii) A proper fuzzy filter μ of L is called fuzzy prime filter, if μ(a) μ(x) μ(y), for all x, y L. inf a x y Example 2.8. Consider the bounded distributive hyperlattice L of example 1.8, and consider the following fuzzy sets of Lμand η defined by, μ(0) = 1,μ(a) = 1,μ(b) = 1,μ(1) = 1, and η(0) = 0, η(a) =1,η(b) =0,η(1) = Then μ is a fuzzy prime ideal of L and η is a fuzzy prime filter of L. The following theorem characterize a fuzzy prime ideal by his α cut sets. Theorem 2.9. Let μ FI(L). Then, μ is a fuzzy prime ideal of L if and only if for any α [0; 1], such that μ α is a proper ideal of L, μ α is a prime ideal of L. α Λ α Λ

8 746 B. B. N. Koguep, C. Nkuimi and C. Lele Proof: ( ) Suppose that μ is a fuzzy prime ideal of L. Let α [0; 1], such that μ α is a proper ideal of L. Let a, b, L, a b μ α μ(a b) α μ(a) μ(b) α (since μ is fuzzy prime ideal) μ(a) αorμ(b) α (since any α [0, 1] is irreducible) a μ α or b μ α Using the fact that μ α is an ideal of L (proposition 2.4), we conclude that μ α is a prime ideal of L. ( ) Conversely suppose that for any α [0; 1] such that μ α is a proper ideal of L, μ α is a prime ideal of L. Let x, y L and α = μ(x y). Then we have x y μ α. Hence x μ α or y μ α (since μ α is a prime ideal).it follows that μ(x) α = μ(x y) orμ(y) α = μ(x y), then μ(x y) μ(x) μ(y). Therefore, μ is a fuzzy prime ideal of L. Corollary A proper subset I of L is a prime ideal of L, if and only if the characteristic function of I is a fuzzy prime ideal of L. Now we want to establish how to construct the fuzzy ideal of a bounded distributive hyperlattice induced by a fuzzy set. Definition Let μ be a fuzzy subset of L. The least fuzzy ideal of L containing μ is called a fuzzy ideal of L induced by μ and denoted by <μ>. Lemma Let μ be a fuzzy subset of L. Then μ(x) = sup{α [0; 1]/x μ α } for all x L. Proof: Let x L. Let β = sup{α [0; 1]/x μ α }. For any ɛ>0, there is α 0 [0; 1], such that β ɛ <α 0 and x μ α0. Thus for any ɛ>0, β ɛ <μ(x). i.e., β μ(x). Since μ(x) {α [0; 1]/x μ α }, we have μ(x) β. Then, μ(x) =β = sup{α [0; 1]/x μ α }. Let Γ be a subset of [0, 1]. Theorem Let {I α /α Γ} be a collection of ideals of L such that, (i) L = I α ; α Γ (ii) α>βif and only if I α I β for all α, β Γ. Define a fuzzy subset ν of L by ν(x) = sup{α Γ/x I α } for all x L. Then ν is a fuzzy ideal of L.

9 On Fuzzy ideals of hyperlattice 747 Proof: It is sufficient to show that ν α is an ideal of L, for every α [0; 1] with ν α. Let α [0; 1]. We have two cases : (1) α = sup{β Γ/β < α} (2) α sup{β Γ/β < α}. Case (1) implies that x ν α x I β, for all β < α, β Γ x Hence ν α = I β β <α β Γ β <α β Γ I β, which is an ideal of L. For the case (2), there exists ɛ>0 such that [α ɛ, α] Γ =. Ifx β <α β Γ then x I β for some β α. It follows that ν(x) β α so that x ν α. That is I β ν α. Conversely, if x/ I β, then x/ I β for all β α. β <α β Γ β <α β Γ Which implies that x/ I β for all β>α ɛ, that is, if x I β then β α ɛ. Thus ν(x) α ɛ and so x / ν α. Consequently ν α = I β, which is an ideal of L. β <α β Γ Remark If Γ={0, 1}, then ν is the characteristic function of I 1. Theorem Let μ be a fuzzy subset of L. Then the fuzzy subset μ of L defined by μ (x) = sup{α [0; 1]/x < μ α >} for all x L is the fuzzy ideal of L induced by μ. Proof: We first prove that for any λ Im(μ ), μ λ is an ideal of L. For any λ Im(μ ), let λ n = λ 1 for any n n N, and let x μ λ. Then μ (x) λ, which implies that μ (x) >λ n, for any n N. Hence there exist β {α [0, 1]/x < μ α >} such that β>λ n. Thus μ β μ λn and so x <μ β > <μ λn > for all n N. Therefore, x <μ λn > n N Conversely, if x <μ λn >, λ n {α [0, 1]/x < μ α >}, for any n N n N. Therefore, λ n = λ 1 sup{α [0, 1]/x < μ n α >} = μ (x), for all n N. Hence μ (x) λ, i.e., x μ λ. Then we have μ λ = <μ λn > n N which is an ideal of L. For any x L, let β {α [0, 1]/x μ α }. Then x μ β, and so x < μ β >. Thus β {α [0, 1]/x < μ α >}. Which implies that I β,

10 748 B. B. N. Koguep, C. Nkuimi and C. Lele {α [0, 1]/x μ α } {α [0, 1]/x < μ α >}. Then μ(x) μ (x) (by lemma2.12). Therefore, μ μ. Finally Let ν be a fuzzy ideal of L containing μ. Let x L, ifμ (x) = 0, then μ (x) ν(x). Assume that μ (x) = λ 0. Then x μ λ = < μ λn >. i.e.,x μ λn, for all n N ; it follows that n N ν(x) μ(x) λ n = λ 1, for all n n N ; Then ν(x) λ = μ (x). Thus μ ν. Hence, μ is the least fuzzy ideal of L containing μ. Example Consider the bounded distributive hyperlattice L of example 1.8. Let μ be the fuzzy subset of L defined by μ(0) = 1, μ(a) = 1, μ(b) = 4 5 1, μ(1) = 1. Then, for any α [0; 1] : 3 6 α [0; 1] μ 6 α = L =< μ α >. α ] 1; 1] μ 6 5 α = L {1}, and < μ α >= L. α ] 1; 1] μ 5 4 α = {0,b} =< μ α >. α ] 1; 1] μ 4 3 α = {b}, and < μ α >= {0,b}. α ] 1;1] μ 3 α =, and < μ α >= {0}. Therefore, μ (0) = sup{α [0; 1]/0 < μ α >} = 1, μ (a) = sup{α [0; 1]/a < μ α >} = 1, 5 μ (b) = sup{α [0; 1]/b < μ α >} = 1 and 3 μ (1) = sup{α [0; 1]/x < μ α >} = 1. 5 In [2] the Stone s prime ideal theorem in distributive lattices is given. The following theorem is analogous to it and it was introduce by U.M. Swany in [8]. We prove it here in the case of bounded distributive hyperlattice. Theorem (Fuzzy prime ideal theorem) Letα [0; 1], μ be a fuzzy ideal of L and ν be a fuzzy filter of L such that μ ν α. Then there exists a fuzzy prime ideal γ of L such that μ γ and γ ν α. Proof: Put I = {x L/μ(x) α} and J = {x L/ν(x) α}. Then I is an ideal of L and J is a filter of L ( I = μ α and J = ν α ) such that I J =. Therefore, by Stone s prime ideal theorem (see theorem 1.22)there exists a prime ideal P of L such that I P and P J =. Consider γ a fuzzy subset of L define by : for all x L { γ(x) = 1,if x P α,if x / P Then we have to prove that γ is a fuzzy ideal of L and it is fuzzy prime. We have α 1, then γ is a proper fuzzy subset of L.

11 On Fuzzy ideals of hyperlattice 749 i) Let x, y L such that x y. Then, x / P implies y / P (because P is an ideal of L); so x / P implies γ(x) γ(y). Elsewhere, x P implies γ(x) =1 γ(y). Therefore, for any x, y L, x y implies γ(x) γ(y). ii) Let x, y L and a x y. If x, y P, then x y P.Thusa P, therefore γ(a) γ(x) γ(y). If x/ P or y/ P, then γ(x) γ(y) =α γ(a). Thus inf a x y (γ(a) γ(x) γ(y). Hence γ is a fuzzy ideal of L. iii) Now let us prove that γ is fuzzy prime. Let x, y L. If x y P, then x P or y P (since P is a prime ideal of L), hence γ(x) = 1 or γ(y) = 1, i.e., γ(x) γ(y) = 1. So γ(x y) γ(x) γ(y). If x y/ P, then γ(x y) =α γ(x) γ(y). Therefore γ is a fuzzy prime ideal of L. iv) Let x L. If x P, then γ(x) = 1. Hence μ(x) γ(x). If x/ P, then x/ I (since I P ), so μ(x) α = γ(x). Therefore, μ γ. v) Let x L. If x P, then x/ J (due to P J = ), so ν(x) α and (ν γ)(x) α. If x/ P, then γ(x) =α, so(ν γ)(x) γ(x) =α. Therefore, ν γ α. References [1] M. Attallah, Completely Fuzzy Prime Ideals of distributive Lattices, The journal of Fuzzy Mathematics vol. 8, No. 1, pp , 2000, Los Angeles. [2] B.A. Davey and H.A. Priestley, Introduction to Lattices and order, Second edition Cambrige (2002). [3] T.K. Dutta and B.K. Biswas, On completely Fuzzy Semiprime ideals of a Semiring, The journal of Fuzzy Mathematics, Vol 8, No.3, pp , [4] B.B.N. KOGUEP, C. NKUIMI and C. LELE, On Fuzzy Prime Ideal of Lattice, (sumitted). [5] Y.B. Jun, W.H. Shim and C. Lele, Fuzzy filters/ideals in BCI-Algebras, The journal of fuzzy mathematics 10(2), (2002)

12 750 B. B. N. Koguep, C. Nkuimi and C. Lele [6] A. Rahnamai-Barghi, The prime ideal theorem and semiprime ideals in meethyperlattices, Ital. Journal of Pure and Applied Math., vol. 5,pp.53-60, [7] A. Rahnamai-Barghi, The prime ideal theorem for distributive hyperlattices, Ital. Journal of Pure and Applied Math., vol. 10, pp.75-78, [8] U.M. Swany, D. Viswanadha Raju, Fuzzy ideals and congruences of Lattices, Fuzzy Sets and Systemes 95 (1998) [9] J. Zhan, On properties of fuzzy hyperideals in hypernear-ring with t-norms, J. Appl. Math. & Computing, vol. 20, No. 1-2, pp , Received: January 10, 2008