Scientiae Mathematicae Japonicae Online, Vol. 4(2001), FUZZY HYPERBCK IDEALS OF HYPERBCK ALGEBRAS Young Bae Jun and Xiao LongXin Received

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1 Scientiae Mathematicae Japonicae Online, Vol. 4(2001), FUZZY HYPERBCK IDEALS OF HYPERBCK ALGEBRAS Young Bae Jun and Xiao LongXin Received August 7, 2000 Abstract. The fuzzification of the notion of a (weak, strong, reflexive) hyperbck-ideal is considered, and relations among them and some related properties are given. 1. Introduction The study of BCK-algebras was initiated by Iséki in 1966 as a generalization of the concept of set-theoretic difference and propositional calculus. Since then a great deal of literature has been produced on the theory of BCK-algebras. In particular, emphasis seems to have been put on the ideal theory of BCK-algebras (see [1, 2, 7]). The hyperstructure theory (called also multialgebras) was introduced in 1934 by Marty [6] at the 8th congress of Scandinavian Mathematiciens. In [5], Jun et al. applied the hyperstructures to BCKalgebras, and introduced the concept of a hyperbck-algebra which is a generalization of a BCK-algebra, and investigated some related properties. They also introduced the notion of a(weak, strong, reflexive) hyperbck-ideal, and gave relations among them. In this paper we consider the fuzzification of the notion of a (weak, strong, reflexive) hyperbck-ideal, gave relations among them and investigate some related properties. 2. Preliminaries An algebra (X; ; 0) of type (2; 0) is said to be a BCK-algebra if it satisfies: for all x; y; z 2 X, (I) ((x y) (x z)) (z y) =0; (II) (x (x y)) y =0; (III) x x =0; (IV) 0 x =0; (V) x y =0andyx = 0 imply x = y. Note that an algebra (X; ; 0) of type (2,0) is a BCK-algebra if and only if (i) ((y z) (x z)) (y x) =0; (ii) ((z x) y) ((z y) x) =0; (iii) (x y) x =0; (iv) x y =0andyx = 0 imply that x = y, for all x; y 2 X (see [7]). Note that the identity x (x (x y)) = x y holds in a BCKalgebra. A non-empty subseti of a BCK-algebra X is called an ideal of X if 0 2 I, and x y 2 I and y 2 I imply x 2 I for all x; y 2 X. A fuzzy set μ in a set X is a function μ : X! [0; 1]. A fuzzy set μ in a set X is said to satisfy the (resp. sup) property if for any subset T of X there exists x 0 2 T such that μ(x 0 )= x2t μ(x) (resp. μ(x 0 ) = sup x2t μ(x)) Mathematics Subject Classification. 06F35, 03E72, 20N20. Key words and phrases. Fuzzy (weak, s-weak, strong) hyperbck-ideal.

2 416 Y. B. JUN AND X. L. XIN Let H be a non-empty set endowed with a hyperoperation ffi". For two subsets A S and B of H, denote by A ffi B the set a ffi b. We shall use x ffi y instead of x ffifyg, fxgffiy, or fxgffifyg. a2a;b2b Definition 2.1 (Jun et al. [5]). By a hyperbck-algebra we mean a non-empty set H endowed with a hyperoperation ffi" and a constant 0 satisfing the following axioms: (HK1) (x ffi z) ffi (y ffi z) fi x ffi y, (HK2) (x ffi y) ffi z =(x ffi z) ffi y, (HK3) x ffi H fifxg, (HK4) x fi y and y fi x imply x = y, for all x; y; z 2 H, where x fi y is defined by 02 x ffi y and for every A; B H; A fi B is defined by 8a 2 A, 9b 2 B such that a fi b. In such case, we call fi" thehyperorder in H. Example 2.2 (Jun et al. [5]). (1) Let (H; ; 0) be a BCK-algebra and define a hyperoperation ffi" onh by x ffi y = fx yg for all x; y 2 H. Then H is a hyperbck-algebra. (2) Define a hyperoperation ffi" onh := [0; 1) by 8 >< [0;x] if x» y x ffi y := (0;y] if x>y6= 0 >: fxg if y =0 for all x; y 2 H. Then H is a hyperbck-algebra. (3) Let H = f0; 1; 2g. Consider the following table: Then H is a hyperbck-algebra. ffi f0g f0g f0g 1 f1g f0; 1g f0; 1g 2 f2g f1; 2g f0; 1; 2g Proposition 2.3 (Jun et al. [5]). In a hyperbck-algebra H, the condition (HK3) is equivalent to the condition: (2-1) x ffi y fifxg for all x; y 2 H. In any hyperbck-algebra H, the following hold (see Jun et al. [5]): (2-2) x ffi 0 fifxg, 0ffi x fif0g and 0 ffi 0 fif0g for all x; y 2 H, (2-3) (A ffi B) ffi C =(A ffi C) ffi B, A ffi B fi A and 0 ffi A fif0g; (2-4) 0 ffi 0=f0g; (2-5) 0 fi x, (2-6) x fi x, (2-7) A fi A; (2-8) A B implies A fi B, (2-9) 0 ffi x = f0g, (2-10) 0 ffi A = f0g, (2-11) A fi f0g implies A = f0g, (2-12) A ffi B fi A, (2-13) x 2 x ffi 0, (2-14) x ffi 0 fifyg implies x fi y, (2-15) y fi z implies x ffi z fi x ffi y, (2-16) x ffi y = f0g implies (x ffi z) ffi (y ffi z) =f0g and x ffi z fi y ffi z, (2-17) A ffif0g = f0g implies A = f0g, for all x; y; z 2 H and for all non-empty subsetes A, B and C of H.

3 FUZZY HYPERBCK IDEALS OF HYPERBCK ALGEBRAS 417 Definition 2.4 (Jun et al. [5]). Let I be a non-empty subset of a hyperbck-algebra H. Then I is said to be a hyperbck-ideal of H if (HI1) 0 2 I, (HI2) x ffi y fi I and y 2 I imply x 2 I for all x; y 2 H. Definition 2.5 (Jun et al. [4]). AhyperBCK-ideal I of H is said to be reflexive if xffix I for all x 2 H. Definition 2.6 (Jun et al. [4]). Let I be a non-empty subset of H. Then I is called a strong hyperbck-ideal of a hyperbck-algebra H if it satisfies (HI1) and (SHI) (x ffi y) I 6= ; and y 2 I imply x 2 I for all x; y 2 H. Note that every strong hyperbck-ideal of a hyperbck-algebra is a hyperbck-ideal (see [4, Theorem 3.8]). Definition 2.7 (Jun et al. [5]). Let I be a non-empty subset of a hyperbck-algebra H. Then I is called a weak hyperbck-ideal of H if it satisfies (HI1) and (WHI) x ffi y I and y 2 I imply x 2 I for all x; y 2 H. 3. Fuzzy HyperBCK-ideals In what follows, H shall mean a hyperbck-algebra unless specified otherwise. Definition 3.1. A fuzzy set μ in H is called a fuzzy hyperbck-ideal of H if (i) x fi y implies μ(y)» μ(x), (ii) μ(x) m μ(a);μ(y)g; for all x; y 2 H. Example 3.2. Let H be the hyperbck-algebra in Example 2.2(3). Define a fuzzy set μ in H by μ(0) = 1; μ(1) = 0:3 and μ(2) = 0: It is easily verified that μ is a fuzzy hyperbck-ideal of H. Definition 3.3. A fuzzy set μ in H is called a fuzzy strong hyperbck-ideal of H if for all x; y 2 H. μ(a) μ(x) m sup μ(b);μ(y)g Definition 3.4. A fuzzy set μ in H is called a fuzzy s-weak hyperbck-ideal of H if (i) μ(0) μ(x) for all x 2 H, (ii) for every x; y 2 H there exists a 2 x ffi y such that μ(x) mμ(a);μ(y)g: Definition 3.5. A fuzzy set μ in H is called a fuzzy weak hyperbck-ideal of H if for all x; y 2 H. μ(0) μ(x) m μ(a);μ(y)g Let μ be a fuzzy s-weak hyperbck-ideal of H and let x; y 2 H. Then there exists a 2 x ffi y such that μ(x) mμ(a);μ(y)g: Since μ(a) μ(b); it follows that μ(x) m μ(b);μ(y)g: Hence every fuzzy s-weak hyperbck-ideal is a fuzzy weak hyperbck-ideal. It is not easy to find an example of a fuzzy weak hyperbck-ideal which is not a fuzzy s-weak hyperbck-ideal. But we have the following proposition.

4 418 Y. B. JUN AND X. L. XIN Proposition 3.6. Let μ be a fuzzy weak hyperbck-ideal of H. property, thenμ is a fuzzy s-weak hyperbck-ideal of H. Proof. Since μ satisfies the property, there exists a 0 2 xffiy such that μ(a 0 )= It follows that ending the proof. μ(x) m μ(a);μ(y)g = mμ(a 0);μ(y)g; If μ satisfies the μ(a): Note that, in a finite hyperbck-algebra, every fuzzy set satisfies (also sup) property. Hence the concept of fuzzy weak hyperbck-ideals and fuzzy s-weak hyperbck-ideals coincide in a finite hyperbck-algebra. Proposition 3.7. Let μ be a fuzzy strong hyperbck-ideal of H and let x; y 2 H. Then (i) μ(0) μ(x), (ii) x fi y implies μ(y)» μ(x); (iii) μ(x) mμ(a);μ(y)g for all a 2 x ffi y: Proof. (i) Since 0 2 x ffi x for all x 2 H, wehave μ(0) μ(a) μ(x); which proves (i). (ii) Let x; y 2 H be such thatxfiy. Then 0 2 x ffi y and so sup μ(b) μ(0): If follows from (i) that μ(x) m sup μ(b);μ(y)g mμ(0);μ(y)g = μ(y): (iii) Let x; y 2 H. Since μ(x) m sup μ(b);μ(y)g mμ(a);μ(y)g for all a 2 x ffi y, we conclude that (iii) is true. Corollary 3.8. If μ is a fuzzy strong hyperbck-ideal of H, then for all x; y 2 H. Proof. Since μ(a) μ(x) m μ(a);μ(y)g μ(b) for all a 2 x ffi y, the result is by Proposition 3.7(iii). Corollary 3.9. Every fuzzy strong hyperbck-ideal is both a fuzzy s-weak hyperbckideal (and hence a fuzzy weak hyperbck-ideal) and a fuzzy hyperbck-ideal. Proof. Straightforward. Proposition Let μ be a fuzzy hyperbck-ideal of H and let x; y 2 H. Then (i) μ(0) μ(x), (ii) if μ satisfies the property, thenμ(x) mμ(a);μ(y)g for some a 2 x ffi y. Proof. (i) Since 0 fi x for each x 2 H; we have μ(x)» μ(0) by Definition 3.1(i) and hence (i) holds. (ii) Since μ satisfies the property, there is a 0 2 x ffi y such that μ(a 0 )= μ(a): Hence which implies that (ii) is true. μ(x) m μ(a);μ(y)g = mμ(a 0);μ(y)g;

5 FUZZY HYPERBCK IDEALS OF HYPERBCK ALGEBRAS 419 Corollary (i) Every fuzzy hyperbck-ideal of H is a fuzzy weak hyperbck-ideal of H. (ii) If μ is a fuzzy hyperbck-ideal of H satisfying property, then μ is a fuzzy s-weak hyperbck-ideal of H. Proof. Straightforward. The following example shows that the converse of Corollary 3.9 and Corollary 3.11(i) may not be true. Example (1) Consider the hyperbck-algebra H in Example 2.2(3). Define a fuzzy set μ in H by 8 >< 1 if x =0, 1 μ(x) = if x =1; 2 >: 0 if x =2: Then we can see that μ is a fuzzy hyperbck-ideal of H and hence it is also a fuzzy weak hyperbck-ideal of H. But μ is not a fuzzy strong hyperbck-ideal of H since m sup μ(a);μ(1)g = mμ(1);μ(1)g = μ(1) = 1 2 > 0=μ(2). a22ffi1 (2) Consider the hyperbck-algebra H in Example 2.2(3). Define a fuzzy set μ in H by 8 >< 1 if x =0; 1 μ(x) = if x =2; 2 >: 0 if x =1: Then μ is a fuzzy weak hyperbck-ideal of H but it is not a fuzzy hyperbck-ideal of H since 1 fi 2 but μ(1) 6 μ(2). Theorem If μ is a fuzzy strong hyperbck-ideal of H, then the set μ t := fx 2 H j μ(x) tg is a strong hyperbck-ideal of H when μ t 6= ; for t 2 [0; 1]: proof. Let μ be a fuzzy strong hyperbck-ideal of H and μ t 6= ; for t 2 [0; 1]: Then there is a 2 μ t and so μ(a) t: By Proposition 3.7(i), μ(0) μ(a) t and so 0 2 μ t : Let x; y 2 H be such that (xffiy) μ t 6= ; and y 2 μ t : Then there exists a 0 2 (xffiy) μ t and hence μ(a 0 ) t: By Definition 3.3, we have μ(x) m sup μ(a);μ(y)g mμ(a 0 );μ(y)g mt; tg = t and so x 2 μ t : It follows that μ t is a strong hyperbck-ideal of H: Lemma 3.14 ([3, Proposition 3.7]). Let A be a subset of a hyperbck-algebra H. If I is ahyperbck-ideal of H such that A fi I, then A is contained in I. Theorem Let μ be a fuzzy set in H satisfying the sup property. If the set μ t := fx 2 H j μ(x) tg(6= ;) is a strong hyperbck-ideal of H for all t 2 [0; 1], then μ is a fuzzy strong hyperbck-ideal of H: Proof. Assume that μ t (6= ;) is a strong hyperbck-ideal of H for all t 2 [0; 1]: Then there exists x 2 μ t and hence x ffi x fi x 2 μ t. Using Lemma 3.14, we have x ffi x μ t. Thus for each a 2 x ffi x; we have a 2 μ t and hence μ(a) t. It follows that μ(a) t = μ(x): Moreover let x; y 2 H and put k = m sup μ(a);μ(y)g: By hypothesis, μ k is a strong hyperbck-ideal of H. Since μ satisfies the sup property, there is a 0 2 x ffi y such that μ(a 0 )= sup μ(a): Thus μ(a 0 )= sup μ(a) m sup μ(a);μ(y)g = k and so a 0 2 μ k : This shows that a 0 2 xffiy μ k and hence xffiy μ k 6= ;: Combining y 2 μ k and noticing that μ k is a strong hyperbck-ideal of H we get x 2 μ k : Hence μ(x) k =m sup Therefore μ is a fuzzy strong hyperbck-ideal of H. μ(a);μ(y)g:

6 420 Y. B. JUN AND X. L. XIN Example (1) Let H = f0; 1; 2g: Consider the following table: ffi f0g f0g f0g 1 f1g f0g f1g 2 f2g f2g f0; 2g Then H is a hyperbck-algebra. Moreover we can see that I 1 := f0; 1g and I 2 := f0; 2g are strong hyperbck-ideals of H. Define μ by μ(0) = 1;μ(1) = 0:3 andμ(2) = 0:2. Then we can see that 8 >< H if 0» t» 0:2; μ t = f0; 1g if 0:2 <t» 0:3; >: f0g if 0:3 <t» 1: Since f0g; f0; 1g and H are strong hyperbck-ideals of H, It follows from Theorem 3.15 that μ is a fuzzy strong hyperbck-ideal of H: (2) Consider the hyperbck-algebra H as in Example 2.2(2). We can see that there exist only two strong hyperbck-ideals f0g and H itself (see [4, Example 3.6(2)]). Define a fuzzy set μ by ρ 1 if x 2 [0; 1]; μ(x) = 0 if x 2 (1; 1): Then μ 1 = [0; 1] is not a strong hyperbck-ideal of H and so μ is not a fuzzy strong hyperbck-ideal of H by Theorem Theorem Let μ be a fuzzy set in H. Then μ is a fuzzy hyper-bck-ideal of H if and only if μ t := fx 2 H j μ(x) tg is a hyperbck-ideal of H whenever μ t 6= ; for t 2 [0; 1]: Proof. Let μ be a fuzzy hyperbck-ideal of H and assume μ t 6= ; where t 2 [0; 1]. Then there exists a 2 μ t and hence μ(a) t: By Proposition 3.10(i), μ(0) μ(a) t and so 0 2 μ t : Let x; y 2 H be such thatx ffi y fi μ t and y 2 μ t. Thus for any a 2 x ffi y; there exists a 0 2 μ t such that a fi a 0 and so μ(a 0 )» μ(a). Hence μ(a) t for all a 2 x ffi y: It follows that μ(a) t so that μ(x) m μ(a);μ(y)g mt; μ(y)g t; which shows that x 2 μ t and μ t isahyperbck-ideal of H. Conversely assume that for each t 2 [0; 1], μ t (6= ;) isahyperbck-ideal of H. Let x fi y and t = μ(y). Then y 2 μ t ; and thus x fi μ t. It follows from Lemma 3.14 that x 2 μ t and hence μ(x) t = μ(y): Moreover let x; y 2 H and put t =m y 2 μ t ; and for each a 2 x ffi y we have μ(a) μ(b) m μ(a);μ(y)g: Then μ(b);μ(y)g = t and hence a 2 μ t, which follows that x ffi y μ t. Thus x ffi y fi μ t. Combining y 2 μ t and μ t being a hyperbck-ideal of H, wegetx2μ t and so μ(x) t =m μ(a);μ(y)g; which shows that μ is a fuzzy hyperbck-ideal of H. Theorem Let μ be a fuzzy set in H. Then μ is a fuzzy weak hyperbck-ideal of H if and only if the set μ t := fx 2 H j μ(x) tg is a weak hyperbck-ideal of H whenever μ t 6= ; for t 2 [0; 1]. Proof. The proof is similat to the proof of Theorem For any subset I ρ H we define a fuzzy set μ I in H by ρ 1 if x 2 I; μ I (x) := 0 otherwise.

7 FUZZY HYPERBCK IDEALS OF HYPERBCK ALGEBRAS 421 Theorem Let I be a subset of H. Then (i) I is a strong hyperbck-ideal of H if and only if μ I is a fuzzy strong hyperbck-ideal of H: (ii) I is a hyperbck-ideal of H if and only if μ I is a fuzzy hyper-bck-ideal of H. (iii) I is a weak hyperbck-ideal of H if and only if μ I is a fuzzy weak hyperbck-ideal of H. Proof. Let I be a subset of H. Then clearly μ I is a fuzzy set in H satisfying and sup property. (i) Let I be a strong hyperbck-ideal of H. Note that μ I satisfies ρ I if 0 <t» 1; (μ I ) t = H if t =0: Then for each t 2 [0; 1], (μ I ) t is a strong hyperbck-ideal of H. By Theorem 3.15, μ I is a fuzzy strong hyperbck-ideal of H. Conversely let μ I be a fuzzy strong hyperbck-ideal of H. Then (μ I ) 1 = I is a strong hyperbck-ideal of H by Theorem Hence (i) is true. (ii) and (iii) are similar to (i). Using Theorems 3.13, 3.15, 3.17, 3.18, 3.19 and Corollaries 3.9 and 3.11, we have the following corollary. Corollary (i) Every strong hyperbck-ideal of H is a hyper-bck-ideal of H; (ii) Every hyperbck-ideal of H is a weak hyperbck-ideal of H: Definition Let μ be a fuzzy set in H. Then μ is called a fuzzy reflexive hyperbckideal of H if it satisfies (i) μ(a) μ(y), (ii) μ(x) m sup for all x; y 2 H: μ(a);μ(y)g; Theorem Every fuzzy reflexive hyperbck-ideal of H is a fuzzy strong hyperbckideal of H. Proof. Straightforward. Theorem If μ is a fuzzy reflexive hyperbck-ideal of H, then μ t is a reflexive hyperbck-ideal of H whenever μ t 6= ; for t 2 [0; 1]. Proof. Let μ be a fuzzy reflexive hyperbck-ideal of H and μ t 6= ; for t 2 [0; 1]. Then there exists a 2 H such thata 2 μ t and so μ(a) t. By Theorem 3.22, μ is a fuzzy strong hyperbck-ideal and moreover a hyperbck-ideal of H: It follows from Theorem 3.17 that μ t is a hyperbck-idael of H. Moreover for each x 2 H and c 2 x ffi x, μ(c) μ(b) b2xffix μ(a) t by Definition 3.21(i). Hence c 2 μ t for each c 2 x ffi x, which shows that x ffi x μ t : Therefore μ t is a reflexive hyperbck-ideal of H. Theorem Let μ is a fuzzy set in H satisfying the sup property. If μ t (6= ;) is a reflexive hyperbck-ideal of H for all t 2 [0; 1], thenμ is a fuzzy reflexive hyperbck-ideal of H. Proof. Let μ t (6= ;) be a reflexive hyperbck-ideal of H for all t 2 [0; 1]. By [4, Theorem 3.6], μ t (6= ;) isa strong hyperbck-ideal of H for all t 2 [0; 1]: Using Theorem 3.15, we have thatμ is a fuzzy strong hyperbck-ideal of H and thus it satisfies the condition (ii) of

8 422 Y. B. JUN AND X. L. XIN the Definition Nowweshow that μ satisfies (i) in the Definition Let x; y 2 H and μ(y) =t. Since μ t (6= ;) is a reflexivehyperbck-ideal of H, itfollows that xffix μ t. Thus for each c 2 x ffi x, wehave c 2 μ t ; and so μ(c) t. This shows that μ(c) t = μ(y). Therefore μ is a fuzzy reflexive hyperbck-ideal of H: c2xffix Theorem Let μ be a fuzzy strong hyperbck-ideal of H satisfying the sup property. Then μ is a fuzzy reflexive hyperbck-ideal of H if and only if μ(a) μ(0) for all x 2 H: Proof. Let μ be a fuzzy reflexive hyperbck-ideal of H. Then, by Definition 3.21(i), we have μ(a) μ(0). Conversely, assume μ(a) μ(0) for all x 2 H. Since μ is a fuzzy hyperbck-ideal of H, wegetμ(0) μ(y) forally 2 H and hence μ(a) μ(y) μ(a); μ(y)g where x; y 2 H. Using Theorem for all x; y 2 H: Moreover put t =m sup 3.13 and 3.22, the set μ t is a strong hyperbck-ideal of H. Since μ satisfies the sup property, there exists a 0 2 x ffi y such that μ(a 0 )= sup μ(a) andsoμ(a 0 ) t; i.e., a 0 2 μ t : This shows that x ffi y μ t 6= ;: It follows from y 2 μ t that x 2 μ t since μ t is a strong hyperbck-ideal of H. Therefore μ(x) t = m sup reflexive hyperbck-ideal of H. References μ(a);μ(y)g, and hence μ is a fuzzy [1] K. Iséki and S. Tanaka, Ideal theory of BCK-algebras, Math. Japonica 21 (1976), [2] K. Iséki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japonica 23(1) (1978), [3] Y. B. Jun and X. L. Xin, Scalar elements and hyperatoms of hyperbck-algebras, Scientiae Mathematicae 2(3) (1999), [4]Y.B.Jun,X.L.Xin,E.H.RohandM.M.Zahedi,Strong hyperbck-ideals of hyperbck-algebras, Math. Japonica 51(3) (2000), [5] Y. B. Jun, M. M. Zahedi, X. L. Xin and R. A. Borzoei, On hyperbck-algebras, Italian J. Pure and Appl. Math. (to appear). [6] F. Marty, Sur une generalization de la notion de groupe, 8th Congress Math. Scandinaves, Stockholm (1934), [7] J. Meng and Y. B. Jun, BCK-algebras, Kyungmoonsa, Seoul, Korea, Y. B. Jun Department of Mathematics Education Gyeongsang National University Chinju , Korea ybjun@nongae.gsnu.ac.kr X. L. Xin Department of Mathematics Northwest University Xian , P. R. China