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Fuzzy strong implicative hyper BCK-ideals of hyper BCK-algebras
Information Sciences 170 (2005) 351–361 www.elsevier.com/locate/ins
Fuzzy strong implicative hyper BCK-ideals of hyper BCK-algebras Young Bae Jun
*,1,
Wook Hwan Shim
Department of Mathematics Education, Gyeongsang National University, Chinju 660-701, South Korea Received 15 July 2003; accepted 4 March 2004
Abstract The fuzzy setting of strong implicative hyper BCK-ideals in hyper BCK-algebras is considered, some of their properties are investigated. Relations among fuzzy strong hyper BCK-ideals, fuzzy implicative hyper BCK-ideals, and fuzzy strong implicative hyper BCK-ideals are given. A characterization of a fuzzy strong implicative hyper BCK-ideal is provided. The hyper homomorphic preimage of a fuzzy strong implicative hyper BCK-ideal is discussed. 2004 Elsevier Inc. All rights reserved. Keywords: Hyper BCK-algebra; (Fuzzy) Hyper BCK-ideal; (Fuzzy) Strong hyper BCKideal; (Fuzzy) Strong implicative hyper BCK-ideal
1. Introduction BCK-algebras arose from the algebraization of the pure implicative logic BCK [7]. The hyperstructure theory (called also multialgebras) is introduced in 1934 by Marty [6] at the 8th Congress of Scandinavian Mathematicians. Around 1940’s, several authors worked on hypergroups, especially in France and in the US, but also in Italy, Russia and Japan. Over the following decades,
*
Corresponding author. Tel.: +82-55-7515674; fax: +82-55-7516117. E-mail address: [email protected] (Y.B. Jun). 1 The first author is an Executive Research Worker of Educational Research Institute in GSNU.
0020-0255/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2004.03.009
352
Y.B. Jun, W.H. Shim / Information Sciences 170 (2005) 351–361
many important results appeared, but above all since 1970’s the most luxuriant flourishing of hyperstructures has been seen. Hyperstructures have many applications to several sectors of both pure and applied sciences. In Ref. [5], Jun et al. applied the hyperstructures to BCK-algebras, and introduced the concept of a hyper BCK-algebra which is a generalization of a BCK-algebra, and investigated some related properties. They also introduced the notion of a hyper BCK-ideal and a weak hyper BCK-ideal, and gave relations between hyper BCK-ideals and weak hyper BCK-ideals. Jun et al. [4] gave a condition for a hyper BCK-algebra to be a BCK-algebra, and introduced the notion of a strong hyper BCK-ideal, a weak hyper BCK-ideal and a reflexive hyper BCKideal. They showed that every strong hyper BCK-ideal is a hypersubalgebra, a weak hyper BCK-ideal and a hyper BCK-ideal; and every reflexive hyper BCKideal is a strong hyper BCK-ideal. In his pioneering paper [9], Zadeh introduced the notion of a fuzzy set in a set X as a function from X into the closed unit interval ½0; 1. Jun and Shim [1] and Jun and Xin [3] applied this concept to the hyperstructure theory in BCK/BCI-algebras. In this paper, we consider the fuzzy setting of strong implicative hyper BCK-ideals in hyper BCK-algebras, and investigate some of their properties. We give relations among fuzzy strong hyper BCK-ideals, fuzzy implicative hyper BCK-ideals, and fuzzy strong implicative hyper BCK-ideals. We provide a characterization of a fuzzy strong implicative hyper BCK-ideal. We discuss the hyper homomorphic preimage of a fuzzy strong implicative hyper BCK-ideal.
2. Preliminaries Let H be a non-empty set endowed with a hyper operation ‘‘’’, that is, is a function from H H to P ðH Þ ¼ PðH Þ n f;g. For two subsets A and B of H , denote by A B the set [fa b j a 2 A; b 2 Bg: We shall use x y instead of x fyg, fxg y, or fxg fyg. Definition 2.1 [5, Definition 3.1]. By a hyper BCK-algebra we mean a nonempty set H endowed with a hyper operation ‘‘’’ and a constant 0 satisfying the following axioms: (H1) (H2) (H3) (H4)
ðx zÞ ðy zÞ x y, ðx yÞ z ¼ ðx zÞ y, x H fxg, x y and y x imply x ¼ y,
for all x; y; z 2 H , where x y is defined by 0 2 x y and for every A; B H , A B is defined by 8a 2 A, 9b 2 B such that a b.
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353
In a hyper BCK-algebra H , the condition (H3) is equivalent to the condition: (p1) x y fxg for all x; y 2 H . In any hyper BCK-algebra H , the following hold: (p2) x 0 fxg, 0 x f0g and 0 0 f0g for all x; y 2 H , (p3) ðA BÞ C ¼ ðA CÞ B, A B A and 0 A f0g, (p4) 0 0 ¼ f0g, (p5) 0 x, (p6) x x, (p7) A A, (p8) A B implies A B, (p9) 0 x ¼ f0g, (p10) 0 A ¼ f0g, (p11) A f0g implies A ¼ f0g, (p12) A B A, (p13) x 2 x 0, (p14) x 0 fyg implies x y, (p15) y z implies x z x y, (p16) x y ¼ f0g implies ðx zÞ ðy zÞ ¼ f0g and x z y z, (p17) A f0g ¼ f0g implies A ¼ f0g, for all x; y; z 2 H and for all non-empty subsets A, B and C of H . A non-empty subset A of a hyper BCK-algebra H is called a hyper BCK-ideal of H if it satisfies • 0 2 A, • 8x; y 2 H , x y A, y 2 A ) x 2 A. A non-empty subset A of a hyper BCK-algebra H is called a strong hyper BCK-ideal of H if it satisfies • 0 2 A, • 8x; y 2 H , ðx yÞ \ A 6¼ ;, y 2 A ) x 2 A. Note that every strong hyper BCK-ideal of a hyper BCK-algebra is a hyper BCK-ideal, but the converse is not true (see [4]). in a set H is said to satisfy the sup property if for any subset T A fuzzy set A of H there exists x0 2 T such that Aðx0 Þ ¼ supx2T AðxÞ. A fuzzy set A in a hyper BCK-algebra H is called a fuzzy strong hyper BCKideal of H if it satisfies
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(
)
inf AðaÞ P AðxÞ P min AðyÞ; sup AðbÞ ;
a2xx
8x; y 2 H :
b2xy
A fuzzy set A in a hyper BCK-algebra H is called a fuzzy s-weak hyper BCKideal of H if it satisfies • Að0Þ P AðxÞ for all x 2 H , • for every x; y 2 H there exists a 2 x y such that AðxÞ P minfAðaÞ; AðyÞg:
A fuzzy set A in a hyper BCK-algebra H is called a fuzzy weak hyper BCKideal of H if it satisfies Að0Þ P AðxÞ P min AðyÞ; inf AðaÞ ; 8x; y 2 H : a2xy
Note that every fuzzy s-weak hyper BCK-ideal is a fuzzy weak hyper BCKideal, but the converse is not true; and every fuzzy strong hyper BCKideal is both a fuzzy s-weak hyper BCK-ideal and a fuzzy hyper BCK-ideal (see [3]).
3. Fuzzy strong implicative hyper BCK-ideals Definition 3.1 [8]. A non-empty subset A of H is called a strong implicative hyper BCK-ideal of H if it satisfies (I1) 0 2 A, (I2) 8x; y; z 2 H , ððx zÞ ðy xÞÞ \ A 6¼ ;, z 2 A ) x 2 A. Definition 3.2. A fuzzy set A in H is called a fuzzy strong implicative hyper BCK-ideal of H if it satisfies ( ) inf AðaÞ P AðxÞ P min AðzÞ;
a2xx
sup
AðbÞ
b2ðxzÞðyxÞ
for all x; y; z 2 H . Example 3.3. Let H ¼ f0; a; bg. Consider the following Cayley table:
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355
Define a fuzzy set A in H by Að0Þ ¼ AðbÞ ¼ 0:7 and AðaÞ ¼ 0:2. It is routine to verify that A is a fuzzy strong implicative hyper BCK-ideal of H . Theorem 3.4. Every fuzzy strong implicative hyper BCK-ideal is a fuzzy strong hyper BCK-ideal. Proof. Let A be a fuzzy strong implicative hyper BCK-ideal of H . Note that ðx yÞ ð0 xÞ ¼ ðx yÞ 0 ¼ x y for all x; y 2 H . Then ( ) ( ) inf AðaÞ P AðxÞ P min AðyÞ;
a2xx
sup
AðbÞ
¼ min AðyÞ; sup AðbÞ : b2xy
b2ðxyÞð0xÞ
Hence A is a fuzzy strong hyper BCK-ideal of H .
h
Combining Theorem 3.4 and [3, Corollary 3.9], we have the following corollary. Corollary 3.5. Every fuzzy strong implicative hyper BCK-ideal is both a fuzzy s-weak hyper BCK-ideal, and hence a fuzzy weak hyper BCK-ideal, and a fuzzy hyper BCK-ideal, and hence a fuzzy hyper subalgebra. The following example shows that the converse of Theorem 3.4 may not be true. Example 3.6. Consider the hyper BCK-algebra H in Example 3.3. Define a fuzzy set A in H by Að0Þ ¼ AðaÞ ¼ 0:7 and AðbÞ ¼ 0:2. It is routine to verify that A is a fuzzy strong hyper BCK-ideal of H , but not a fuzzy strong implicative hyper BCK-ideal of H , because ( ) 0:2 ¼ AðbÞj min Að0Þ;
sup
AðbÞ
¼ 0:7:
f0;bg2ðb0ÞðabÞ
Proposition 3.7. Let A be a fuzzy strong implicative hyper BCK-ideal of H . Then ii(i) Að0Þ P AðxÞ for all x 2 H , i(ii) x y implies AðyÞ < AðxÞ, (iii) AðxÞ P minfAðaÞ; AðyÞg for all a 2 x y.
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Proof. The proof is by Theorem 3.4 and [3, Proposition 3.7]. h Theorem 3.8. If A is a fuzzy strong implicative hyper BCK-ideal of H , then the set A½t :¼ fx 2 H j AðxÞ P tg is a strong implicative hyper BCK-ideal of H whenever A½t 6¼ ; for t 2 ½0; 1. Proof. Let A be a fuzzy strong implicative hyper BCK-ideal of H and assume that A½t 6¼ ; for t 2 ½0; 1. Since Að0Þ P AðxÞ for all x 2 H , obviously 0 2 A½t. Let x; y; z 2 H be such that ððx zÞ ðy xÞÞ \ A½t 6¼ ; and z 2 A½t. Then there exists w 2 ððx zÞ ðy xÞÞ such that AðwÞ P t. Hence ( AðxÞ P min AðzÞ;
) sup
AðbÞ P minfAðzÞ; AðwÞg P t;
b2ðxzÞðyxÞ
and so x 2 A½t. This completes the proof.
h
Corollary 3.9. If A is a fuzzy strong implicative hyper BCK-ideal of H , then the set A :¼ fx 2 H j AðxÞ ¼ Að0Þg is a strong implicative hyper BCK-ideal of H . Lemma 3.10 [1, Theorem 3.5]. Let A be a fuzzy set in H . Then A is a fuzzy implicative hyper BCK-ideal of H if and only if A½t is an implicative hyper BCKideal of H whenever A½t 6¼ ; for t 2 ½0; 1. Theorem 3.11. Every fuzzy strong implicative hyper BCK-ideal is a fuzzy implicative hyper BCK-ideal. Proof. Let A be a fuzzy strong implicative hyper BCK-ideal of H . Then A½t 6¼ ;, t 2 ½0; 1, is a strong implicative hyper BCK-ideal, and hence an implicative hyper BCK-ideal of H . It follows from Lemma 3.10 that A is a fuzzy implicative hyper BCK-ideal of H . h The following example shows that the converse of Theorem 3.11 need not be true. Example 3.12. Let H ¼ f0; a; bg. Consider the following Cayley table:
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357
Define a fuzzy set A in H by Að0Þ ¼ AðaÞ ¼ 0:7 and AðbÞ ¼ 0:2. It is routine to verify that A is a fuzzy implicative hyper BCK-ideal of H , but not fuzzy strong implicative hyper BCK-ideal of H , because ( ) 0:2 ¼ AðbÞ min AðaÞ;
fAð0Þ; AðaÞ; AðbÞg
sup
¼ 0:7:
f0;a;bg2ðb0ÞðabÞ
Lemma 3.13 [2, Proposition 3.7]. Let A be a subset of H . If I is a hyper BCKideal of H such that A I, then A is contained in I. Theorem 3.14. Let A be a fuzzy set in H satisfying the sup property. If the set A½t :¼ fx 2 H j AðxÞ P tgð6¼ ;Þ is a strong implicative hyper BCK-ideal of H for all t 2 ½0; 1, then A is a fuzzy strong implicative hyper BCK-ideal of H . Proof. Suppose that A½tð6¼ ;Þ is a strong implicative hyper BCK-ideal of H for each t 2 ½0; 1. Combining [8, Theorem 3.4], [4, Theorem 3.8] and [3, Theorem 3.17], A is a fuzzy hyper BCK-ideal of H . If a 2 x x for all x 2 H , then a x and so AðaÞ P AðxÞ. Hence inf a2xx AðaÞ P AðxÞ for all x 2 H . For any x; y; z 2 H , let ( ) t ¼ min AðzÞ;
sup
AðbÞ :
b2ðxzÞðyxÞ
By hypothesis, A½t is a strong implicative hyper BCK-ideal of H . Since A satisfies the sup property, there is w 2 ðx zÞ ðy xÞ such that AðwÞ ¼
sup
AðbÞ:
b2ðxzÞðyxÞ
Thus AðwÞ ¼
( sup b2ðxzÞðyxÞ
AðbÞ P min AðzÞ;
) sup
AðbÞ
¼t
b2ðxzÞðyxÞ
and so w 2 A½t. This shows that ððx zÞ ðy xÞÞ \ A½t 6¼ ;: Since z 2 A½t, it follows from (I2) that x 2 A½t so that
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(
)
AðxÞ P t ¼ min AðzÞ;
sup
AðbÞ :
b2ðxzÞðyxÞ
Therefore, A is a fuzzy strong implicative hyper BCK-ideal of H .
h
Theorem 3.15. Let A be a fuzzy set in H . Then A is a fuzzy strong implicative hyper BCK-ideal of H if and only if A is a fuzzy strong hyper BCK-ideal satisfying AðxÞ P supb2xðyxÞ AðbÞ. Proof. Assume that A is a fuzzy strong implicative hyper BCK-ideal of H . Then, by Theorem 3.4, A is a fuzzy strong hyper BCK-ideal of H . Taking z ¼ 0 in Definition 3.2, we get ( ) AðxÞ P min Að0Þ;
sup
AðbÞ
b2ðx0ÞðyxÞ
(
)
¼ min Að0Þ; sup AðbÞ
since x 0 ¼ x ¼ sup AðbÞ
b2xðyxÞ
b2xðyxÞ
by Proposition 3.7(i). Conversely, let A be a fuzzy strong hyper BCK-ideal of H that satisfies AðxÞ P sup AðbÞ: b2xðyxÞ
Then AðbÞ P minfAðzÞ; supw2bz AðwÞg for all b 2 x ðy xÞ. If min AðzÞ; sup AðwÞ ¼ AðzÞ; w2bz
then AðxÞ P sup AðbÞ P AðzÞ and so b2xðyxÞ
(
)
AðxÞ P min AðzÞ;
sup
AðbÞ :
b2ðxzÞðyxÞ
Assume that minfAðzÞ; sup AðwÞg ¼ sup AðwÞ: Then AðbÞ P sup AðwÞ, and w2bz w2bz w2bz thus AðxÞ P sup AðbÞ P sup sup AðwÞ b2xðyxÞ
¼
sup u2ðxðyxÞÞz
b2xðyxÞ
AðuÞ ¼
w2bz
sup
AðuÞ:
u2ðxzÞðyxÞ
Hence AðxÞ P minfAðzÞ; supu2ðxzÞðyxÞ AðuÞg. This completes the proof.
h
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359
Definition 3.16 [1]. A mapping f : G ! H of hyper BCK-algebras is called a hyper homomorphism if it satisfies • f ð0Þ ¼ 0, • f ðx yÞ ¼ f ðxÞ f ðyÞ for all x; y 2 G.
Proposition 3.17 [1]. Let f : G ! H be a hyper homomorphism of hyper BCKalgebras. If x y in G, then f ðxÞ f ðyÞ in H . Proposition 3.18. Let f : G ! H be a hyper homomorphism of hyper BCKalgebras. If A B in G, then f ðAÞ f ðBÞ in H . Proof. Let A and B be subsets of G such that A B. Let y 2 f ðAÞ. Then there exists a 2 A such that y ¼ f ðaÞ. Since A B, for a 2 A there exists b 2 B such that a b. It follows from Proposition 3.17 that y ¼ f ðaÞ f ðbÞ and f ðbÞ 2 B. Hence f ðAÞ f ðBÞ. h Theorem 3.19. Let f : G ! H be a hyper homomorphism of hyper BCK-algebras. If A is a strong implicative hyper BCK-ideal of H , then f 1 ðAÞ is a strong implicative hyper BCK-ideal of G. Proof. Clearly 0 2 f 1 ðAÞ. Let x; y; z 2 G be such that z 2 f 1 ðAÞ and ððx zÞ ðy xÞÞ \ f 1 ðAÞ 6¼ ;: Then there exists a 2 G such that a 2 ðx zÞ ðy xÞ and a 2 f 1 ðAÞ. Then f ðaÞ f ððx zÞ ðy xÞÞ ¼ ðf ðxÞ f ðzÞÞ ðf ðyÞ f ðxÞÞ and so ððf ðxÞ f ðzÞÞ ðf ðyÞ f ðxÞÞÞ \ A 6¼ ;. Since f ðzÞ 2 A and A is strong implicative hyper BCK-ideal, it follows that f ðxÞ 2 A, i.e., x 2 f 1 ðAÞ. This completes the proof. h Lemma 3.20 [8, Theorem 3.8]. Let I be a non-empty subset of a hyper BCKalgebra H . Then I is a strong implicative hyper BCK-ideal of H if and only if I is a strong hyper BCK-ideal of H and ðx ðy xÞÞ \ I 6¼ ; implies x 2 I for all x; y 2 H . Theorem 3.21. If f : G ! H is a hyper homomorphism of hyper BCK-algebras, then Kerðf Þ :¼ fx 2 G j f ðxÞ ¼ 0g, called the kernel of f , is a strong implicative hyper BCK-ideal of G.
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Proof. Clearly 0 2 Kerðf Þ. Let x; y 2 G be such that ðx yÞ \ Kerðf Þ 6¼ ; and y 2 Kerðf Þ. Then there exists a 2 G such that a 2 x y and a 2 Kerðf Þ. Hence 0 ¼ f ðaÞ 2 f ðx yÞ ¼ f ðxÞ f ðyÞ, and so f ðxÞ f ðyÞ. Since f ðyÞ ¼ 0, it follows from (p11) that f ðxÞ ¼ 0. This shows that Kerðf Þ is a strong hyper BCKideal of G. Let x; y 2 G be such that ðx ðy xÞÞ \ Kerðf Þ 6¼ ;. Then there exists a 2 G such that a 2 ðx ðy xÞÞ \ Kerðf Þ, and so 0 ¼ f ðaÞ 2 f ðx ðy xÞÞ ¼ f ðxÞ ðf ðyÞ f ðxÞÞ:
ð1Þ
Taking y ¼ 0 in (1) and using (p9), we get 0 2 f ðxÞ 0, that is, f ðxÞ 0. Hence f ðxÞ ¼ 0, i.e., x 2 Kerðf Þ: Using Lemma 3.20, we conclude that Kerðf Þ is a strong implicative hyper BCK-ideal of G. h Theorem 3.22. Let f : G ! H be an onto hyper homomorphism of hyper BCKalgebras. If A is a strong implicative hyper BCK-ideal of G containing Kerðf Þ, then f ðAÞ is a strong implicative hyper BCK-ideal of H . Proof. Note that 0 ¼ f ð0Þ 2 f ðAÞ. Let x; y; z 2 H be such that z 2 f ðAÞ and ððx zÞ ðy xÞÞ \ f ðAÞ 6¼ ;: Then there exist a; b; c 2 G such that f ðaÞ ¼ x, f ðbÞ ¼ y; f ðcÞ ¼ z and f ðða cÞ ðb aÞÞ ¼ ðf ðaÞ f ðcÞÞ ðf ðbÞ f ðaÞÞ ¼ ðx zÞ ðy xÞ: Hence f ðða cÞ ðb aÞÞ \ f ðAÞ 6¼ ;. Let v 2 H be such that v 2 f ðða cÞ ðb aÞÞ and v 2 f ðAÞ. Then there exists u 2 G such that f ðuÞ ¼ v and hence u 2 ðða cÞ ðb aÞÞ \ A 6¼ ;. Since A is a strong implicative hyper BCK-ideal and f ðcÞ ¼ z 2 f ðAÞ, i.e., c 2 A, we get a 2 A. Hence x ¼ f ðaÞ 2 f ðAÞ, and therefore, f ðAÞ is strong implicative hyper BCK-ideal. This completes the proof. h Theorem 3.23. Let f : G ! H be an onto hyper homomorphism of hyper BCK is a fuzzy strong implicative hyper BCK-ideal of H , then the hyper algebras. If B under f , i.e., the fuzzy set A in G defined by homomorphic preimage A of B ðxÞÞ for all x 2 G, is a fuzzy strong implicative hyper BCK-ideal of G. AðxÞ ¼ Bðf Proof. For any x 2 G, we have ( ðxÞÞ P min BðaÞ; AðxÞ ¼ Bðf
) sup
BðcÞ
c2ðf ðxÞaÞðbf ðxÞÞ
for all a; b; c 2 H . Let z; y 2 G be arbitrary preimages of a, b under f , respectively. Then c 2 ðf ðxÞ aÞ ðb f ðxÞÞ ¼ ðf ðxÞ f ðzÞÞ ðf ðyÞ f ðxÞÞ ¼ f ððx zÞ ðy xÞÞ which implies c ¼ f ðwÞ for some w 2 ðx zÞ ðy xÞ. Hence
Y.B. Jun, W.H. Shim / Information Sciences 170 (2005) 351–361
(
361
)
ðzÞÞ; AðxÞ P min Bðf
ðwÞÞ Bðf
sup f ðwÞ2f ððxzÞðyxÞÞ
(
)
¼ min AðzÞ;
sup
AðwÞ :
w2ðxzÞðyxÞ
Since a and b are arbitrary elements of H , the above result is true for all z; y 2 G. Let t be an arbitrary preimage of d under f . Then f ðtÞ ¼ d 2 f ðx xÞ which implies d ¼ f ðtÞ for some t 2 x x. Hence inf AðtÞ ¼
t2xx
inf
f ðtÞ2f ðxxÞ
ðtÞÞ ¼ Bðf
inf
d2f ðxÞf ðxÞ
BðdÞ P AðxÞ:
Since d is an arbitrary element of H , the above result is true for all t 2 G. Hence we conclude that A is a fuzzy strong implicative hyper BCK-ideal of G. h
References [1] Y.B. Jun, W.H. Shim, Fuzzy implicative hyper BCK-ideals of hyper BCK-algebras, Int. J. Math. Math. Sci. 29 (2) (2002) 63–70. [2] Y.B. Jun, X.L. Xin, Scalar elements and hyper atoms of hyper BCK-algebras, Sci. Math. 2 (3) (1999) 303–309. [3] Y.B. Jun, X.L. Xin, Fuzzy hyper BCK-ideals of hyper BCK-algebras, Sci. Math. Jpn. 53 (2) (2001) 353–360. [4] Y.B. Jun, X.L. Xin, M.M. Zahedi, E.H. Roh, Strong hyper BCK-ideals of hyper BCK-algebras, Math. Jpn. 51 (3) (2000) 493–498. [5] Y.B. Jun, M.M. Zahedi, X.L. Xin, R.A. Borzoei, On hyper BCK-algebras, Ital. J. Pure Appl. Math. 8 (2000) 127–136. [6] F. Marty, Sur une generalization de la notion de groupe, 8th Congress Math. Scandinaves, Stockholm, 1934, pp. 45–49. [7] A.N. Prior, Formal Logic, second ed., Clarendon Press, Oxford, 1962. [8] E.H. Roh, Q. Zhang, Y.B. Jun, Some results in hyper BCK-algebras, Sci. Math. Jpn. 55 (2) (2002) 297–304. [9] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353.
Fuzzy strong implicative hyper BCK-ideals of hyper BCK-algebras Young Bae Jun
*,1,
Wook Hwan Shim
Department of Mathematics Education, Gyeongsang National University, Chinju 660-701, South Korea Received 15 July 2003; accepted 4 March 2004
Abstract The fuzzy setting of strong implicative hyper BCK-ideals in hyper BCK-algebras is considered, some of their properties are investigated. Relations among fuzzy strong hyper BCK-ideals, fuzzy implicative hyper BCK-ideals, and fuzzy strong implicative hyper BCK-ideals are given. A characterization of a fuzzy strong implicative hyper BCK-ideal is provided. The hyper homomorphic preimage of a fuzzy strong implicative hyper BCK-ideal is discussed. 2004 Elsevier Inc. All rights reserved. Keywords: Hyper BCK-algebra; (Fuzzy) Hyper BCK-ideal; (Fuzzy) Strong hyper BCKideal; (Fuzzy) Strong implicative hyper BCK-ideal
1. Introduction BCK-algebras arose from the algebraization of the pure implicative logic BCK [7]. The hyperstructure theory (called also multialgebras) is introduced in 1934 by Marty [6] at the 8th Congress of Scandinavian Mathematicians. Around 1940’s, several authors worked on hypergroups, especially in France and in the US, but also in Italy, Russia and Japan. Over the following decades,
*
Corresponding author. Tel.: +82-55-7515674; fax: +82-55-7516117. E-mail address: [email protected] (Y.B. Jun). 1 The first author is an Executive Research Worker of Educational Research Institute in GSNU.
0020-0255/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2004.03.009
352
Y.B. Jun, W.H. Shim / Information Sciences 170 (2005) 351–361
many important results appeared, but above all since 1970’s the most luxuriant flourishing of hyperstructures has been seen. Hyperstructures have many applications to several sectors of both pure and applied sciences. In Ref. [5], Jun et al. applied the hyperstructures to BCK-algebras, and introduced the concept of a hyper BCK-algebra which is a generalization of a BCK-algebra, and investigated some related properties. They also introduced the notion of a hyper BCK-ideal and a weak hyper BCK-ideal, and gave relations between hyper BCK-ideals and weak hyper BCK-ideals. Jun et al. [4] gave a condition for a hyper BCK-algebra to be a BCK-algebra, and introduced the notion of a strong hyper BCK-ideal, a weak hyper BCK-ideal and a reflexive hyper BCKideal. They showed that every strong hyper BCK-ideal is a hypersubalgebra, a weak hyper BCK-ideal and a hyper BCK-ideal; and every reflexive hyper BCKideal is a strong hyper BCK-ideal. In his pioneering paper [9], Zadeh introduced the notion of a fuzzy set in a set X as a function from X into the closed unit interval ½0; 1. Jun and Shim [1] and Jun and Xin [3] applied this concept to the hyperstructure theory in BCK/BCI-algebras. In this paper, we consider the fuzzy setting of strong implicative hyper BCK-ideals in hyper BCK-algebras, and investigate some of their properties. We give relations among fuzzy strong hyper BCK-ideals, fuzzy implicative hyper BCK-ideals, and fuzzy strong implicative hyper BCK-ideals. We provide a characterization of a fuzzy strong implicative hyper BCK-ideal. We discuss the hyper homomorphic preimage of a fuzzy strong implicative hyper BCK-ideal.
2. Preliminaries Let H be a non-empty set endowed with a hyper operation ‘‘’’, that is, is a function from H H to P ðH Þ ¼ PðH Þ n f;g. For two subsets A and B of H , denote by A B the set [fa b j a 2 A; b 2 Bg: We shall use x y instead of x fyg, fxg y, or fxg fyg. Definition 2.1 [5, Definition 3.1]. By a hyper BCK-algebra we mean a nonempty set H endowed with a hyper operation ‘‘’’ and a constant 0 satisfying the following axioms: (H1) (H2) (H3) (H4)
ðx zÞ ðy zÞ x y, ðx yÞ z ¼ ðx zÞ y, x H fxg, x y and y x imply x ¼ y,
for all x; y; z 2 H , where x y is defined by 0 2 x y and for every A; B H , A B is defined by 8a 2 A, 9b 2 B such that a b.
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In a hyper BCK-algebra H , the condition (H3) is equivalent to the condition: (p1) x y fxg for all x; y 2 H . In any hyper BCK-algebra H , the following hold: (p2) x 0 fxg, 0 x f0g and 0 0 f0g for all x; y 2 H , (p3) ðA BÞ C ¼ ðA CÞ B, A B A and 0 A f0g, (p4) 0 0 ¼ f0g, (p5) 0 x, (p6) x x, (p7) A A, (p8) A B implies A B, (p9) 0 x ¼ f0g, (p10) 0 A ¼ f0g, (p11) A f0g implies A ¼ f0g, (p12) A B A, (p13) x 2 x 0, (p14) x 0 fyg implies x y, (p15) y z implies x z x y, (p16) x y ¼ f0g implies ðx zÞ ðy zÞ ¼ f0g and x z y z, (p17) A f0g ¼ f0g implies A ¼ f0g, for all x; y; z 2 H and for all non-empty subsets A, B and C of H . A non-empty subset A of a hyper BCK-algebra H is called a hyper BCK-ideal of H if it satisfies • 0 2 A, • 8x; y 2 H , x y A, y 2 A ) x 2 A. A non-empty subset A of a hyper BCK-algebra H is called a strong hyper BCK-ideal of H if it satisfies • 0 2 A, • 8x; y 2 H , ðx yÞ \ A 6¼ ;, y 2 A ) x 2 A. Note that every strong hyper BCK-ideal of a hyper BCK-algebra is a hyper BCK-ideal, but the converse is not true (see [4]). in a set H is said to satisfy the sup property if for any subset T A fuzzy set A of H there exists x0 2 T such that Aðx0 Þ ¼ supx2T AðxÞ. A fuzzy set A in a hyper BCK-algebra H is called a fuzzy strong hyper BCKideal of H if it satisfies
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(
)
inf AðaÞ P AðxÞ P min AðyÞ; sup AðbÞ ;
a2xx
8x; y 2 H :
b2xy
A fuzzy set A in a hyper BCK-algebra H is called a fuzzy s-weak hyper BCKideal of H if it satisfies • Að0Þ P AðxÞ for all x 2 H , • for every x; y 2 H there exists a 2 x y such that AðxÞ P minfAðaÞ; AðyÞg:
A fuzzy set A in a hyper BCK-algebra H is called a fuzzy weak hyper BCKideal of H if it satisfies Að0Þ P AðxÞ P min AðyÞ; inf AðaÞ ; 8x; y 2 H : a2xy
Note that every fuzzy s-weak hyper BCK-ideal is a fuzzy weak hyper BCKideal, but the converse is not true; and every fuzzy strong hyper BCKideal is both a fuzzy s-weak hyper BCK-ideal and a fuzzy hyper BCK-ideal (see [3]).
3. Fuzzy strong implicative hyper BCK-ideals Definition 3.1 [8]. A non-empty subset A of H is called a strong implicative hyper BCK-ideal of H if it satisfies (I1) 0 2 A, (I2) 8x; y; z 2 H , ððx zÞ ðy xÞÞ \ A 6¼ ;, z 2 A ) x 2 A. Definition 3.2. A fuzzy set A in H is called a fuzzy strong implicative hyper BCK-ideal of H if it satisfies ( ) inf AðaÞ P AðxÞ P min AðzÞ;
a2xx
sup
AðbÞ
b2ðxzÞðyxÞ
for all x; y; z 2 H . Example 3.3. Let H ¼ f0; a; bg. Consider the following Cayley table:
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Define a fuzzy set A in H by Að0Þ ¼ AðbÞ ¼ 0:7 and AðaÞ ¼ 0:2. It is routine to verify that A is a fuzzy strong implicative hyper BCK-ideal of H . Theorem 3.4. Every fuzzy strong implicative hyper BCK-ideal is a fuzzy strong hyper BCK-ideal. Proof. Let A be a fuzzy strong implicative hyper BCK-ideal of H . Note that ðx yÞ ð0 xÞ ¼ ðx yÞ 0 ¼ x y for all x; y 2 H . Then ( ) ( ) inf AðaÞ P AðxÞ P min AðyÞ;
a2xx
sup
AðbÞ
¼ min AðyÞ; sup AðbÞ : b2xy
b2ðxyÞð0xÞ
Hence A is a fuzzy strong hyper BCK-ideal of H .
h
Combining Theorem 3.4 and [3, Corollary 3.9], we have the following corollary. Corollary 3.5. Every fuzzy strong implicative hyper BCK-ideal is both a fuzzy s-weak hyper BCK-ideal, and hence a fuzzy weak hyper BCK-ideal, and a fuzzy hyper BCK-ideal, and hence a fuzzy hyper subalgebra. The following example shows that the converse of Theorem 3.4 may not be true. Example 3.6. Consider the hyper BCK-algebra H in Example 3.3. Define a fuzzy set A in H by Að0Þ ¼ AðaÞ ¼ 0:7 and AðbÞ ¼ 0:2. It is routine to verify that A is a fuzzy strong hyper BCK-ideal of H , but not a fuzzy strong implicative hyper BCK-ideal of H , because ( ) 0:2 ¼ AðbÞj min Að0Þ;
sup
AðbÞ
¼ 0:7:
f0;bg2ðb0ÞðabÞ
Proposition 3.7. Let A be a fuzzy strong implicative hyper BCK-ideal of H . Then ii(i) Að0Þ P AðxÞ for all x 2 H , i(ii) x y implies AðyÞ < AðxÞ, (iii) AðxÞ P minfAðaÞ; AðyÞg for all a 2 x y.
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Proof. The proof is by Theorem 3.4 and [3, Proposition 3.7]. h Theorem 3.8. If A is a fuzzy strong implicative hyper BCK-ideal of H , then the set A½t :¼ fx 2 H j AðxÞ P tg is a strong implicative hyper BCK-ideal of H whenever A½t 6¼ ; for t 2 ½0; 1. Proof. Let A be a fuzzy strong implicative hyper BCK-ideal of H and assume that A½t 6¼ ; for t 2 ½0; 1. Since Að0Þ P AðxÞ for all x 2 H , obviously 0 2 A½t. Let x; y; z 2 H be such that ððx zÞ ðy xÞÞ \ A½t 6¼ ; and z 2 A½t. Then there exists w 2 ððx zÞ ðy xÞÞ such that AðwÞ P t. Hence ( AðxÞ P min AðzÞ;
) sup
AðbÞ P minfAðzÞ; AðwÞg P t;
b2ðxzÞðyxÞ
and so x 2 A½t. This completes the proof.
h
Corollary 3.9. If A is a fuzzy strong implicative hyper BCK-ideal of H , then the set A :¼ fx 2 H j AðxÞ ¼ Að0Þg is a strong implicative hyper BCK-ideal of H . Lemma 3.10 [1, Theorem 3.5]. Let A be a fuzzy set in H . Then A is a fuzzy implicative hyper BCK-ideal of H if and only if A½t is an implicative hyper BCKideal of H whenever A½t 6¼ ; for t 2 ½0; 1. Theorem 3.11. Every fuzzy strong implicative hyper BCK-ideal is a fuzzy implicative hyper BCK-ideal. Proof. Let A be a fuzzy strong implicative hyper BCK-ideal of H . Then A½t 6¼ ;, t 2 ½0; 1, is a strong implicative hyper BCK-ideal, and hence an implicative hyper BCK-ideal of H . It follows from Lemma 3.10 that A is a fuzzy implicative hyper BCK-ideal of H . h The following example shows that the converse of Theorem 3.11 need not be true. Example 3.12. Let H ¼ f0; a; bg. Consider the following Cayley table:
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Define a fuzzy set A in H by Að0Þ ¼ AðaÞ ¼ 0:7 and AðbÞ ¼ 0:2. It is routine to verify that A is a fuzzy implicative hyper BCK-ideal of H , but not fuzzy strong implicative hyper BCK-ideal of H , because ( ) 0:2 ¼ AðbÞ min AðaÞ;
fAð0Þ; AðaÞ; AðbÞg
sup
¼ 0:7:
f0;a;bg2ðb0ÞðabÞ
Lemma 3.13 [2, Proposition 3.7]. Let A be a subset of H . If I is a hyper BCKideal of H such that A I, then A is contained in I. Theorem 3.14. Let A be a fuzzy set in H satisfying the sup property. If the set A½t :¼ fx 2 H j AðxÞ P tgð6¼ ;Þ is a strong implicative hyper BCK-ideal of H for all t 2 ½0; 1, then A is a fuzzy strong implicative hyper BCK-ideal of H . Proof. Suppose that A½tð6¼ ;Þ is a strong implicative hyper BCK-ideal of H for each t 2 ½0; 1. Combining [8, Theorem 3.4], [4, Theorem 3.8] and [3, Theorem 3.17], A is a fuzzy hyper BCK-ideal of H . If a 2 x x for all x 2 H , then a x and so AðaÞ P AðxÞ. Hence inf a2xx AðaÞ P AðxÞ for all x 2 H . For any x; y; z 2 H , let ( ) t ¼ min AðzÞ;
sup
AðbÞ :
b2ðxzÞðyxÞ
By hypothesis, A½t is a strong implicative hyper BCK-ideal of H . Since A satisfies the sup property, there is w 2 ðx zÞ ðy xÞ such that AðwÞ ¼
sup
AðbÞ:
b2ðxzÞðyxÞ
Thus AðwÞ ¼
( sup b2ðxzÞðyxÞ
AðbÞ P min AðzÞ;
) sup
AðbÞ
¼t
b2ðxzÞðyxÞ
and so w 2 A½t. This shows that ððx zÞ ðy xÞÞ \ A½t 6¼ ;: Since z 2 A½t, it follows from (I2) that x 2 A½t so that
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(
)
AðxÞ P t ¼ min AðzÞ;
sup
AðbÞ :
b2ðxzÞðyxÞ
Therefore, A is a fuzzy strong implicative hyper BCK-ideal of H .
h
Theorem 3.15. Let A be a fuzzy set in H . Then A is a fuzzy strong implicative hyper BCK-ideal of H if and only if A is a fuzzy strong hyper BCK-ideal satisfying AðxÞ P supb2xðyxÞ AðbÞ. Proof. Assume that A is a fuzzy strong implicative hyper BCK-ideal of H . Then, by Theorem 3.4, A is a fuzzy strong hyper BCK-ideal of H . Taking z ¼ 0 in Definition 3.2, we get ( ) AðxÞ P min Að0Þ;
sup
AðbÞ
b2ðx0ÞðyxÞ
(
)
¼ min Að0Þ; sup AðbÞ
since x 0 ¼ x ¼ sup AðbÞ
b2xðyxÞ
b2xðyxÞ
by Proposition 3.7(i). Conversely, let A be a fuzzy strong hyper BCK-ideal of H that satisfies AðxÞ P sup AðbÞ: b2xðyxÞ
Then AðbÞ P minfAðzÞ; supw2bz AðwÞg for all b 2 x ðy xÞ. If min AðzÞ; sup AðwÞ ¼ AðzÞ; w2bz
then AðxÞ P sup AðbÞ P AðzÞ and so b2xðyxÞ
(
)
AðxÞ P min AðzÞ;
sup
AðbÞ :
b2ðxzÞðyxÞ
Assume that minfAðzÞ; sup AðwÞg ¼ sup AðwÞ: Then AðbÞ P sup AðwÞ, and w2bz w2bz w2bz thus AðxÞ P sup AðbÞ P sup sup AðwÞ b2xðyxÞ
¼
sup u2ðxðyxÞÞz
b2xðyxÞ
AðuÞ ¼
w2bz
sup
AðuÞ:
u2ðxzÞðyxÞ
Hence AðxÞ P minfAðzÞ; supu2ðxzÞðyxÞ AðuÞg. This completes the proof.
h
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Definition 3.16 [1]. A mapping f : G ! H of hyper BCK-algebras is called a hyper homomorphism if it satisfies • f ð0Þ ¼ 0, • f ðx yÞ ¼ f ðxÞ f ðyÞ for all x; y 2 G.
Proposition 3.17 [1]. Let f : G ! H be a hyper homomorphism of hyper BCKalgebras. If x y in G, then f ðxÞ f ðyÞ in H . Proposition 3.18. Let f : G ! H be a hyper homomorphism of hyper BCKalgebras. If A B in G, then f ðAÞ f ðBÞ in H . Proof. Let A and B be subsets of G such that A B. Let y 2 f ðAÞ. Then there exists a 2 A such that y ¼ f ðaÞ. Since A B, for a 2 A there exists b 2 B such that a b. It follows from Proposition 3.17 that y ¼ f ðaÞ f ðbÞ and f ðbÞ 2 B. Hence f ðAÞ f ðBÞ. h Theorem 3.19. Let f : G ! H be a hyper homomorphism of hyper BCK-algebras. If A is a strong implicative hyper BCK-ideal of H , then f 1 ðAÞ is a strong implicative hyper BCK-ideal of G. Proof. Clearly 0 2 f 1 ðAÞ. Let x; y; z 2 G be such that z 2 f 1 ðAÞ and ððx zÞ ðy xÞÞ \ f 1 ðAÞ 6¼ ;: Then there exists a 2 G such that a 2 ðx zÞ ðy xÞ and a 2 f 1 ðAÞ. Then f ðaÞ f ððx zÞ ðy xÞÞ ¼ ðf ðxÞ f ðzÞÞ ðf ðyÞ f ðxÞÞ and so ððf ðxÞ f ðzÞÞ ðf ðyÞ f ðxÞÞÞ \ A 6¼ ;. Since f ðzÞ 2 A and A is strong implicative hyper BCK-ideal, it follows that f ðxÞ 2 A, i.e., x 2 f 1 ðAÞ. This completes the proof. h Lemma 3.20 [8, Theorem 3.8]. Let I be a non-empty subset of a hyper BCKalgebra H . Then I is a strong implicative hyper BCK-ideal of H if and only if I is a strong hyper BCK-ideal of H and ðx ðy xÞÞ \ I 6¼ ; implies x 2 I for all x; y 2 H . Theorem 3.21. If f : G ! H is a hyper homomorphism of hyper BCK-algebras, then Kerðf Þ :¼ fx 2 G j f ðxÞ ¼ 0g, called the kernel of f , is a strong implicative hyper BCK-ideal of G.
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Proof. Clearly 0 2 Kerðf Þ. Let x; y 2 G be such that ðx yÞ \ Kerðf Þ 6¼ ; and y 2 Kerðf Þ. Then there exists a 2 G such that a 2 x y and a 2 Kerðf Þ. Hence 0 ¼ f ðaÞ 2 f ðx yÞ ¼ f ðxÞ f ðyÞ, and so f ðxÞ f ðyÞ. Since f ðyÞ ¼ 0, it follows from (p11) that f ðxÞ ¼ 0. This shows that Kerðf Þ is a strong hyper BCKideal of G. Let x; y 2 G be such that ðx ðy xÞÞ \ Kerðf Þ 6¼ ;. Then there exists a 2 G such that a 2 ðx ðy xÞÞ \ Kerðf Þ, and so 0 ¼ f ðaÞ 2 f ðx ðy xÞÞ ¼ f ðxÞ ðf ðyÞ f ðxÞÞ:
ð1Þ
Taking y ¼ 0 in (1) and using (p9), we get 0 2 f ðxÞ 0, that is, f ðxÞ 0. Hence f ðxÞ ¼ 0, i.e., x 2 Kerðf Þ: Using Lemma 3.20, we conclude that Kerðf Þ is a strong implicative hyper BCK-ideal of G. h Theorem 3.22. Let f : G ! H be an onto hyper homomorphism of hyper BCKalgebras. If A is a strong implicative hyper BCK-ideal of G containing Kerðf Þ, then f ðAÞ is a strong implicative hyper BCK-ideal of H . Proof. Note that 0 ¼ f ð0Þ 2 f ðAÞ. Let x; y; z 2 H be such that z 2 f ðAÞ and ððx zÞ ðy xÞÞ \ f ðAÞ 6¼ ;: Then there exist a; b; c 2 G such that f ðaÞ ¼ x, f ðbÞ ¼ y; f ðcÞ ¼ z and f ðða cÞ ðb aÞÞ ¼ ðf ðaÞ f ðcÞÞ ðf ðbÞ f ðaÞÞ ¼ ðx zÞ ðy xÞ: Hence f ðða cÞ ðb aÞÞ \ f ðAÞ 6¼ ;. Let v 2 H be such that v 2 f ðða cÞ ðb aÞÞ and v 2 f ðAÞ. Then there exists u 2 G such that f ðuÞ ¼ v and hence u 2 ðða cÞ ðb aÞÞ \ A 6¼ ;. Since A is a strong implicative hyper BCK-ideal and f ðcÞ ¼ z 2 f ðAÞ, i.e., c 2 A, we get a 2 A. Hence x ¼ f ðaÞ 2 f ðAÞ, and therefore, f ðAÞ is strong implicative hyper BCK-ideal. This completes the proof. h Theorem 3.23. Let f : G ! H be an onto hyper homomorphism of hyper BCK is a fuzzy strong implicative hyper BCK-ideal of H , then the hyper algebras. If B under f , i.e., the fuzzy set A in G defined by homomorphic preimage A of B ðxÞÞ for all x 2 G, is a fuzzy strong implicative hyper BCK-ideal of G. AðxÞ ¼ Bðf Proof. For any x 2 G, we have ( ðxÞÞ P min BðaÞ; AðxÞ ¼ Bðf
) sup
BðcÞ
c2ðf ðxÞaÞðbf ðxÞÞ
for all a; b; c 2 H . Let z; y 2 G be arbitrary preimages of a, b under f , respectively. Then c 2 ðf ðxÞ aÞ ðb f ðxÞÞ ¼ ðf ðxÞ f ðzÞÞ ðf ðyÞ f ðxÞÞ ¼ f ððx zÞ ðy xÞÞ which implies c ¼ f ðwÞ for some w 2 ðx zÞ ðy xÞ. Hence
Y.B. Jun, W.H. Shim / Information Sciences 170 (2005) 351–361
(
361
)
ðzÞÞ; AðxÞ P min Bðf
ðwÞÞ Bðf
sup f ðwÞ2f ððxzÞðyxÞÞ
(
)
¼ min AðzÞ;
sup
AðwÞ :
w2ðxzÞðyxÞ
Since a and b are arbitrary elements of H , the above result is true for all z; y 2 G. Let t be an arbitrary preimage of d under f . Then f ðtÞ ¼ d 2 f ðx xÞ which implies d ¼ f ðtÞ for some t 2 x x. Hence inf AðtÞ ¼
t2xx
inf
f ðtÞ2f ðxxÞ
ðtÞÞ ¼ Bðf
inf
d2f ðxÞf ðxÞ
BðdÞ P AðxÞ:
Since d is an arbitrary element of H , the above result is true for all t 2 G. Hence we conclude that A is a fuzzy strong implicative hyper BCK-ideal of G. h
References [1] Y.B. Jun, W.H. Shim, Fuzzy implicative hyper BCK-ideals of hyper BCK-algebras, Int. J. Math. Math. Sci. 29 (2) (2002) 63–70. [2] Y.B. Jun, X.L. Xin, Scalar elements and hyper atoms of hyper BCK-algebras, Sci. Math. 2 (3) (1999) 303–309. [3] Y.B. Jun, X.L. Xin, Fuzzy hyper BCK-ideals of hyper BCK-algebras, Sci. Math. Jpn. 53 (2) (2001) 353–360. [4] Y.B. Jun, X.L. Xin, M.M. Zahedi, E.H. Roh, Strong hyper BCK-ideals of hyper BCK-algebras, Math. Jpn. 51 (3) (2000) 493–498. [5] Y.B. Jun, M.M. Zahedi, X.L. Xin, R.A. Borzoei, On hyper BCK-algebras, Ital. J. Pure Appl. Math. 8 (2000) 127–136. [6] F. Marty, Sur une generalization de la notion de groupe, 8th Congress Math. Scandinaves, Stockholm, 1934, pp. 45–49. [7] A.N. Prior, Formal Logic, second ed., Clarendon Press, Oxford, 1962. [8] E.H. Roh, Q. Zhang, Y.B. Jun, Some results in hyper BCK-algebras, Sci. Math. Jpn. 55 (2) (2002) 297–304. [9] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353.