Q{FUZZY

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SUBALGEBRAS OF BCK/BCI{ALGEBRAS

Young Bae Jun Received July 11, 2000 Q

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Given a set , we introduce the notion of -fuzzy subalgebras of BCK/BCIalgebras, and provide some appropriate examples. Using fuzzy subalgebras, we describe fuzzy subalgebras. Conversely, we construct fuzzy subalgebras by using -fuzzy subalgebras. How the homomorphic images and inverse images of -fuzzy subalgebras become -fuzzy subalgebras is stated. Abstract.

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1. Introduction

The notion of BCK-algebras was proposed by Iami and Iseki in 1966. In the same year, Iseki [1] introduced the notion of a BCI-algebra which is a generalization of a BCK-algebra. Since then numerous mathematical papers have been written investigating the algebraic properties of the BCK/BCI-algebras and their relationship with other universial structures including lattices and Boolean algebras. Fuzzy sets were initiated by Zadeh [5]. In this paper, given a set Q, we introduce the notion of Q-fuzzy subalgebras of BCK/BCI-algebras, and provide some appropriate examples. Using fuzzy subalgebras, we describe Q-fuzzy subalgebras. Conversely, we construct fuzzy subalgebras by using Q-fuzzy subalgebras. How the homomorphic images and inverse images of Q-fuzzy subalgebras become Q-fuzzy subalgebras is stated. 2. Preliminaries

In this section we include some elementary aspects that are necessary for this paper. Recall that a BCI-algebra is an algebra (X; ; 0) of type (2, 0) satisfying the following axioms: (I) ((x y) (x z )) (z y) = 0; (II) (x (x y)) y = 0; (III) x x = 0; and (IV) x y = 0 and y x = 0 imply x = y for every x; y; z 2 X . A BCI-algebra X satisfying the condition (V) 0 x = 0 for all x 2 X is called a BCK-algebra. A non-empty subset S of a BCK/BCI-algebra X is called a subalgebra of X if x y 2 S whenever x; y 2 S . A mapping f : X ! Y of BCK/BCIalgebras is called a homomorphism if f (x y) = f (x) f (y) for all x; y 2 X . For further information on BCK/BCI-algebras, the reader refer to the textbook BCK-algebras (Meng and Jun [4]) and [1, 2, 3]. 3.

Q-fuzzy

subalgebras

2000 Mathematics Subject Classi cation. 06F35; 03B52. Key words and phrases. Q-fuzzy set; Q-fuzzy subalgebra .

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Let X be a BCK/BCI-algebra. A fuzzy set A in X , i.e., a mapping A : X ! [0; 1], is called a fuzzy subalgebra of X if A(x y) minfA(x); A(y)g for all x; y 2 X . Note that if A is a fuzzy subalgebra of a BCK/BCI-algebra X , then A(0) A(x) for all x 2 X .

Proposition 3.1. Let A be a fuzzy subalgebra of a BCK/BCI-algebra X . De ne a fuzzy (x) for all x 2 X . Then B is a fuzzy subalgebra of X and B (0) = 1. set B in X by B (x) = A A(0) Proof.

For any x; y 2 X , we have B (x

1 A(x y) 1 minfA(x); A(y)g y) = A(0) A(0) x) A(y ) ; g = minfB (x); B (y)g: = minf AA((0) A(0)

Hence B is a fuzzy subalgebra of X , and clearly B (0) = 1. According to Proposition 3.1, we may suppose that a fuzzy subalgebra A of a BCK/BCIalgebra X satis es A(0) = 1. In what follows, let Q and X denote a set and a BCK/BCI-algebra, respectively, unless otherwise speci ed. A mapping H : X Q ! [0; 1] is called a Q-fuzzy set in X . De nition 3.2. A Q-fuzzy set H : X Q ! [0; 1] is called a fuzzy subalgebra of X over Q (brie y, Q-fuzzy subalgebra of X ) if H (x y; q ) minfH (x; q ); H (y; q )g for all x; y 2 X and q 2 Q. Example 3.3. Let X = f0; a; b; cg be a BCK-algebra with the following Cayley table: 0 a b c 0 0 0 0 0 a 0 0 a a b b a 0 b c c c c 0 De ne a Q-fuzzy set H in X as follows: for every q 2 Q, H (0; q) = H (b; q) = 0:6 and H (a; q ) = H (c; q ) = 0:2. It is easy to verify that H is a Q-fuzzy subalgebra of X . Example 3.4. Consider a BCI-algebra X = f0; xg with Cayley table as follows (Iseki [2]): 0 x 0 0 x x x 0 Let Q = f1; 2g and let H be a Q-fuzzy set in X de ned by H (0; 1) = H (0; 2) = 1; H (x; 1) = 0:8 and H (x; 2) = 0:5. It is easy to verify that H is a Q-fuzzy subalgebra of X . Example 3.5. Let X be a BCK/BCI-algebra and let Q = fA j A is a fuzzy subalgebra of X g: Let H be a mapping from X Q into [0; 1] de ned by H (x; A) = A(x) for all x 2 X and A 2 Q. Then H is a Q-fuzzy subalgebra of X . Note that for q 2 Q, if H is a q-fuzzy subalgebra of X , then H (0; q ) = H (x x; q ) minfH (x; q ); H (x; q )g = H (x; q ) for all x 2 X .

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Proposition 3.6. Let H be a Q-fuzzy subalgebra of X . De ne a Q-fuzzy set G in X by

G(x; q ) Proof.

= ((0 )) for all x 2 X and q 2 Q. Then G is a Q-fuzzy subalgebra of X . Let x; y 2 X and q 2 Q. Then H (x y; q ) 1 minfH (x; q); H (y; q)g G(x y; q ) = H (0; q ) H (0; q ) H (x; q ) H (y; q ) = minf H (0; q) ; H (0; q) g = minfG(x; q); G(y; q)g: H x;q H

;q

Hence G is a Q-fuzzy subalgebra of X . Let X denote the collection of all functions from Q into X , and de ne a binary operation ~ on X by (u ~ v)(q) = u(q) v(q) for all u; v 2 X and q 2 Q. Then (X ; ~; ) is a BCK/BCI-algebra, where is the zero map in X , i.e., (q) = 0 for all q 2 Q. Q

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Proposition 3.7. Let A be a fuzzy subalgebra of X and let H be a mapping from X into [0; 1] de ned by H (u; q ) subalgebra of X Q . Proof.

= A(u(q)) for all u 2 X

Q

and q

2 Q.

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Then H is a Q-fuzzy

For any u; v 2 X , we have H (u ~ v; q ) = A((u ~ v )(q )) = A(u(q ) v (q )) minfA(u(q)); A(v(q))g = minfH (u; q); H (v; q)g: Q

Hence H is a Q-fuzzy subalgebra of X . Q

Proposition 3.8. Let H be a Q-fuzzy subalgebra of X . For any q 2 Q, de ne H : X !

[0; 1] by H (x) = H (x; q) for all x 2 X . Then H is a fuzzy subalgebra of X . Proof. Let x; y 2 X and q 2 Q. Then H (x y ) = H (x y; q ) minfH (x; q ); H (y; q )g = minfH (x); H (y )g: Hence H is a fuzzy subalgebra of X . We now consider the converse of Proposition 3.8. q

q

q

q

q

q

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Proposition 3.9. Let H X de ned by H (x; q ) X.

q

;q

2 Q, be a fuzzy subalgebra of X: Let H be a Q-fuzzy set in

= H (x) for all x 2 X and q 2 Q. Then H is a Q-fuzzy subalgebra of q

For every x; y 2 X and q 2 Q, we have H (x y; q ) = H (x y ) minfH (x); H (y )g = minfH (x; q ); H (y; q )g: Thus H is a Q-fuzzy subalgebra of X . Proposition 3.10. Let be a subalgebra of X . Then for any q 2 Q, the set := fu(q) j u 2 g is a subalgebra of X . Proof. For any q 2 Q, let u(q ); v (q ) 2 . Then u(q ) v (q ) = (u ~ v )(q ) 2 since u ~ v 2 : Hence , q 2 Q, is a subalgebra of X . Proof.

q

q

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q

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q

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Theorem 3.11. Let A be a fuzzy subalgebra of X . De ne a mapping Q

H :X

Q ! [0; 1]

by H (x; q ) := supfA(u) j u 2 X Q ; u(q ) = xg

for all x 2 X and q 2 Q. Then H is a Q-fuzzy subalgebra of X . Proof.

Let x; y 2 X and q 2 Q. Then

H (x

y; q) = supfA(u) j u 2 X ; u(q) = x yg supfA(u ~ v) j u; v 2 X ; u(q) = x; v(q) = yg supfminfA(u); A(v)g j u; v 2 X ; u(q) = x; v(q) = yg = minfsupfA(u) j u 2 X ; u(q) = xg; supfA(v) j v 2 X = minfH (x; q); H (y; q)g: Q

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; v (q )

= ygg

Hence H is a Q-fuzzy subalgebra of X . Example 3.12. Let X = f0; xg be a BCI-algebra in Example 3.4 and let Q = f1; 2g. Then X := f; u; v; wg, where (1) = (2) = 0; u(1) = u(2) = x; v (1) = 0; v (2) = x; w(1) = x and w(2) = 0; is a BCI-algebra under the following Cayley table: Q

~

u v w

u v w

u v w

u w v

v w u

w v u

Let A be a fuzzy subalgebra of X de ned by A() = 0:8; A(u) = A(v) = 0:3 and A(w) = 0:7. Then we can obtain a Q-fuzzy subalgebra of X as follows: Q

f j 2 X ; u(1) = 0g = supfA(); A(v)g = supf0:8; 0:3g = 0:8;

H (0; 1) = sup A(u) u

Q

and similarly we have H (0; 2) = 0:8; H (x; 1) = 0:7 and H (x; 2) = 0:3. Theorem 3.13. Let

H be a Q-fuzzy subalgebra of X and let A be a fuzzy set in X Q de ned by A(u) := inf H (u(q ); q ) q Q for all u X Q . Then A is a fuzzy subalgebra of X Q. Proof.

f

j 2 g

2

Let u; v 2 X . Then Q

A(u

~ v) = inf fH ((u ~ v)(q); q) j q 2 Qg = inf fH (u(q) v(q); q) j q 2 Qg inf fminfH (u(q); q); H (v(q); q)g j q 2 Qg = minfinf fH (u(q); q) j q 2 Qg; inf fH (v(q); q) j q 2 Qgg = minfA(u); A(v)g:

Therefore A is a fuzzy subalgebra of X . Q

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Example 3.14. Let H be a Q-fuzzy subalgerba of X in Example 3.4. Then we can induce

a fuzzy subalgebra A of X as follows: Q

A()

= inf fH ((q); q) j q 2 Qg = inf fH ((1); 1); H ((2); 2)g = 1

and similarly we obtain A(u) = A(v) = 0:5 and A(w) = 0:8; where X is a BCI-algebra in Example 3.12. De nition 3.15. Let f : X ! Y be a homomorphism of BCK/BCI-algebras and let H be a Q-fuzzy set in Y . Then the inverse image of H , denoted by f 1 [H ], is the Q-fuzzy set in X given by f 1 [H ](x; q) = H (f (x); q) for all x 2 X and q 2 Q. Conversely, let G be a Q-fuzzy set in X . The image of G, written as f [G], is a Q-fuzzy set in Y de ned by sup 1 G(z; q) if f 1 (y) 6= ;, 2 () f [G](y; q ) = 0 otherwise; Q

z

f

y

for all y 2 Y and q 2 Q, where f 1 (y) = fx j f (x) = yg: Theorem 3.16. Let f : X ! Y be a homomorphism of BCK/BCI-algebras. If H is a Q-fuzzy subalgebra of Y; then the inverse image f 1 [H ] of H is a Q-fuzzy subalgebra of X . Proof. Let x; y 2 X and q 2 Q. Then f

1

[H ](x y; q) = H (f (x y); q) = H (f (x) f (y); q) minfH (f (x); q); H (f (y); q)g = minff 1[H ](x; q); f 1 [H ](y; q)g:

Hence f 1[H ] is a Q-fuzzy subalgebra of X . Theorem 3.17. Let f : X ! Y be a homomorphism between BCK/BCI-algebras X and Y . If G is a Q-fuzzy subalgebra of X; then the image f [G] of G is a Q-fuzzy subalgebra of Y.

Proof.

We rst prove that

(1)

1

f

(y1 ) f 1 (y2 ) f 1 (y1 y2 )

for all y1 ; y2 2 Y . For, if x 2 f 1(y1 ) f 1(y2 ); then x = x1 x2 for some x1 2 f 1 (y1 ) and x2 2 f 1 (y2 ). Since f is a homomorphism, it follows that f (x) = f (x1 x2 ) = f (x1 )f (x2 ) = y1 y2 so that x 2 f 1 (y1 y2 ). Hence (1) holds. Now let y1 ; y2 2 Y and q 2 Q. Assume that y1 y2 2= Im(f ). Then f [G](y1 y2; q) = 0. But if y1 y2 2= Im(f ); i.e., f 1(y1 y2) = ;, then f 1 (y1 ) = ; or f 1 (y2 ) = ; by (1). Thus f [G](y1 ; q) = 0 or f [G](y2 ; q) = 0; and so f [G](y1

y ; q) = 0 = minff [G](y ; q); f [G](y ; q)g: 2

1

2

Suppose that f 1(y1 y2 ) 6= ;: Then we should consider two cases as follows: (i) f 1 (y1 ) = ; or f 1 (y2 ) = ;; (ii) f 1 (y1 ) 6= ; and f 1(y2 ) 6= ;:

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For the case (i), we have f [G](y1 ; q) = 0 or f [G](y2 ; q) = 0; and so f [G](y1

y ; q) 0 = minff [G](y ; q); f [G](y ; q)g: 2

1

2

Case (ii) implies from (1) that f [G](y1

y ; q) = 2

=

z

sup 1

2

(y1

f

y2 )

G(z; q )

sup

x1

2

1 (y1 ); x2

2

f

1 (y2 )

x1

2

1 (y1 ); x2

2

f

1 (y2 )

f

f

sup

= minf sup1 x1

2

f

(y1 )

z

2

f

G(x1

sup

1 (y1 )

f

x ; q)

1 (y2 )

G(z; q )

2

minfG(x1 ; q); G(x2 ; q)g

G(x1 ; q );

x2

sup1

2

f

= minff [G](y1 ; q); f [G](y2 ; q)g:

(y2 )

G(y2 ; q )

g

Hence f [G](y1 y2 ; q) minff [G](y1; q); f [G](y2 ; q)g for all y1 ; y2 2 Y and q 2 Q. This completes the proof.

References [1] [2] [3] [4] [5]

K. Iseki, An algebra related with a propositional calculus, Proc. Japan Acad. 42 (1966), 351-366. K. Iseki, On BCI-algebras, Math. Seminar Notes 8 (1980), 125-130. K. Iseki, Some examples of BCI-algebras, Math. Seminar Notes 8 (1980), 237-240. J. Meng and Y. B. Jun, BCK-algebras, Kyung Moon Sa, Korea, 1994. L. A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965), 338-353.

Department of Mathematics Education Gyeongsang National University Jinju 660-701, Korea. E-mail address: [email protected]