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FUZZY IMPLICATIVE HYPER BCK-IDEALS
OF HYPER BCK-ALGEBRAS
YOUNG BAE JUN and WOOK HWAN SHIM
Received 8 February 2001
We consider the fuzzification of the notion of implicative hyper BCK-ideals, and then investigate several properties. Using the concept of level subsets, we give a characteri-zation of a fuzzy implicative hyper BCK-ideal. We state a relation between a fuzzy hyper BCK-ideal and a fuzzy implicative hyper BCK-ideal. We establish a condition for a fuzzy hyper BCK-ideal to be a fuzzy implicative hyper BCK-ideal. Finally, we introduce the notion of hyper homomorphisms of hyper BCK-algebras, and discuss related properties.
2000 Mathematics Subject Classification: 06F35, 03G25, 20N20, 03B52.
1. Introduction. The hyper structure theory (called also multialgebras) is intro-duced in 1934 by Marty [5] at the 8th Congress of Scandinavian Mathematicians. Around the 40’s, several authors worked on hypergroups, especially in France and in the United States, but also in Italy, Russia, and Japan. Over the following decades, many important results appeared, but above all since the 70’s onwards the most luxu-riant flourishing of hyper structures has been seen. Hyper structures have many appli-cations to several sectors of both pure and applied sciences. Jun et al. [4] applied the hyper structures 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 and Xin [1] considered the fuzzification of the notion of a (weak, strong, reflexive) hyper BCK-ideal, and investigated the relations among them. In this paper, we deal with the fuzzification of the notion of implicative hyper BCK-ideals. We show that every fuzzy implicative hyper BCK-ideal is a fuzzy hyper BCK-ideal, but not converse. We provide a condition for a fuzzy hyper BCK-ideal to be a fuzzy implicative hyper BCK-ideal. Using the notion of level subsets of a fuzzy set, we give a characterization of a fuzzy implicative hyper BCK-ideal. We introduce the concept of hyper homomorphisms of hyper BCK-algebras, and we show that the hyper homomorphic preimage of a fuzzy implicative hyper BCK-ideal is also a fuzzy implicative hyper BCK-ideal.
2. Preliminaries. We include some elementary aspects of hyper BCK-algebras that are necessary for this paper, and for more details we refer to [2,4].
By ahyper BCK-algebrawe mean a nonempty setHendowed with a hyper operation “◦” and a constant 0 satisfying the following axioms:
(HK1)(x◦z)◦(y◦z)x◦y, (HK2)(x◦y)◦z=(x◦z)◦y, (HK3)x◦H {x},
(HK4)xyandyximplyx=y,
for allx,y,z∈H, wherexyis defined by 0∈x◦yand for everyA,B⊆H,AB is defined by for alla∈A, there existsb∈Bsuch thatab.
In any hyper BCK-algebraH, the following hold (see [2,4]): (P1) A⊆BimpliesAB,
(P2) 0◦x= {0},
(P3) (A◦B)◦C=(A◦C)◦B,A◦BA, and 0◦A {0}, (P4) x◦0= {x}andA◦0=A
for allx∈Hand for all nonempty subsetsA,B, andCofH.
Proposition2.1(see Jun et al. [4, Proposition 3.3]). In a hyper BCK-algebraH, the
condition (HK3) is equivalent to the condition
(P5) x◦y {x}for allx,y∈H.
Definition2.2(see Jun et al. [4, Definition 3.14]). A nonempty subsetIof a hyper BCK-algebraHis called ahyper BCK-ideal ofHif
(HI1) 0∈I,
(HI2)x◦yIandy∈Iimplyx∈I for allx,y∈H.
By afuzzy setµinHwe mean a functionµ:H→[0,1]. For a fuzzy setµinHand t∈[0,1], the setU(µ;t):= {x∈H|µ(x)≥t}is called alevel subset ofµ.
Definition2.3(see Jun et al. [1, Definition 3.1]). A fuzzy setµ in a hyper
BCK-algebraHis called afuzzy hyper BCK-idealofHif (FHI1)xyimpliesµ(y)≤µ(x),
(FHI2)µ(x)≥min{infa∈x◦yµ(a),µ(y)} for allx,y∈H.
Theorem 2.4(see Jun et al. [1, Theorem 3.17]). Letµ be a fuzzy set in a hyper
BCK-algebraH. Thenµis a fuzzy hyper BCK-ideal ofHif and only ifU(µ;t)is a hyper BCK-ideal ofHwheneverU(µ;t)= ∅fort∈[0,1].
3. Fuzzy implicative hyper BCK-ideals. In what follows letHdenote a hyper BCK-algebra unless otherwise specified.
Definition3.1(see Jun et al. [3]). A nonempty subsetIofHis called animplicative
hyper BCK-idealofHif it satisfies (HI1) and (HI3)(x◦z)◦(y◦x)Iandz∈Iimplyx∈I for allx,y,z∈H.
Definition3.3. A fuzzy setµinHis called afuzzy implicative hyper BCK-ideal ofHif it satisfies (FHI1) and
(FHI3)µ(x)≥min{infw∈(x◦z)◦(y◦x)µ(w),µ(z)}for allx,y,z∈H.
Example3.4. Consider a hyper BCK-algebraH= {0,a,b}with the following Cayley table:
◦ 0 a b
0 {0} {0} {0} a {a} {0} {0} b {b} {a,b} {0,a,b}
Define a fuzzy setµinHbyµ(0)=µ(a)=0.7 andµ(b)=0.2. It is routine to verify thatµis a fuzzy implicative hyper BCK-ideal ofH.
Theorem3.5. Letµbe a fuzzy set inH. Thenµis a fuzzy implicative hyper
BCK-ideal of H if and only if U(µ;t) is an implicative hyper BCK-ideal ofH whenever
U(µ;t)= ∅fort∈[0,1].
Proof. Letµbe a fuzzy implicative hyper BCK-ideal ofHand assume thatU(µ;t)= ∅fort∈[0,1]. Letx,y,z∈Hbe such that(x◦z)◦(y◦x)U(µ;t)andz∈U(µ;t). For anyw∈(x◦z)◦(y◦x), there existsa∈U(µ;t)such thatwa. It follows from (FHI1) thatµ(w)≥µ(a)≥t. Hence
µ(x)≥min
inf
w∈(x◦z)◦(y◦x)µ(w),µ(z)
≥mint,µ(z)≥t, (3.1)
and sox∈U(µ;t). ThusU(µ;t)is an implicative hyper BCK-ideal ofH. Conversely, suppose that for eacht∈[0,1],U(µ;t) (= ∅)is an implicative hyper BCK-ideal ofH. For anyx,y,z∈H, let
t=min
inf
w∈(x◦z)◦(y◦x)µ(w),µ(z)
. (3.2)
Thenz∈U(µ;t)and for everya∈(x◦z)◦(y◦x), we have
µ(a)≥ inf
w∈(x◦z)◦(y◦x)µ(w)≥min
inf
w∈(x◦z)◦(y◦x)µ(w),µ(z)
=t. (3.3)
Thusa∈U(µ;t), and so(x◦z)◦(y◦x)⊆U(µ;t). Hence(x◦z)◦(y◦x)U(µ;t)by (P1). Combiningz∈U(µ;t)andU(µ;t)being an implicative hyper BCK-ideal ofH, we getx∈U(µ;t)and thus
µ(x)≥t=min
inf
w∈(x◦z)◦(y◦x)µ(w),µ(z)
. (3.4)
Thereforeµis a fuzzy implicative hyper BCK-ideal ofH.
Theorem3.6. For any subsetIofH, letµIbe a fuzzy set inHdefined by
µI(x):= t
1 ifx∈I, t2 otherwise,
(3.5)
for allx∈H, wheret1,t2∈[0,1]witht1> t2. ThenIis an implicative hyper BCK-ideal
Proof. Assume thatIis an implicative hyper BCK-ideal ofH. Note that
U µI;t=
∅ ift1< t≤1, I ift2< t≤t1, H if 0≤t≤t2,
(3.6)
which is an implicative hyper BCK-ideal ofHfor allt∈[0,1]. It follows fromTheorem 3.5thatµI is a fuzzy implicative hyper BCK-ideal ofH. Conversely, ifµIis a fuzzy implicative hyper BCK-ideal ofH, thenU(µI;t1)=Iis an implicative hyper BCK-ideal ofH.
Theorem3.7. Every fuzzy implicative hyper BCK-ideal is a fuzzy hyper BCK-ideal.
Proof. Letµbe a fuzzy implicative hyper BCK-ideal ofH. Then
µ(x)≥min
inf
w∈(x◦y)◦(0◦x)µ(w),µ(y)
=min
inf
w∈(x◦y)◦0µ(w),µ(y)
=min
inf
w∈x◦yµ(w),µ(y)
(3.7)
for allx,y∈H. Henceµis a fuzzy hyper BCK-ideal ofH.
The converse ofTheorem 3.7may not be true as seen in the following example.
Example3.8. Consider a hyper BCK-algebraH= {0,a,b}with the following Cayley table:
◦ 0 a b
0 {0} {0} {0} a {a} {0} {0} b {b} {a} {0,a}
Define a fuzzy setµ inH byµ(0)=t1and µ(a)=µ(b)=t2for everyt1> t2in [0,1]. It can be easily checked thatµis a fuzzy hyper BCK-ideal ofH, butµis not a fuzzy implicative hyper BCK-ideal ofHsince
µ(a)=t2< t1=min
inf
w∈(a◦0)◦(b◦a)µ(w),µ(0)
. (3.8)
We give a condition for a fuzzy hyper BCK-ideal to be a fuzzy implicative hyper BCK-ideal.
Theorem3.9. Letµbe a fuzzy hyper BCK-ideal ofH. Thenµis a fuzzy implicative
hyper BCK-ideal ofHif and only ifµ(x)≥infw∈x◦(y◦x)µ(w)for allx,y∈H.
Proof. Assume thatµis a fuzzy implicative hyper BCK-ideal ofH. Then
µ(x)≥min inf
w∈(x◦0)◦(y◦x)µ(w),µ(0)
= inf
w∈x◦(y◦x)µ(w) (3.9)
the inequalityµ(x)≥infw∈x◦(y◦x)µ(w)for allx,y∈H. It is sufficient to show that U(µ;t) (= ∅)is an implicative hyper BCK-ideal of H. Letx,y,z∈H be such that (x◦z)◦(y◦x)U(µ;t)andz∈U(µ;t). Thenµ(z)≥tand(x◦(y◦x))◦zU(µ;t), and sow◦zU(µ;t)for allw∈x◦(y◦x). SinceU(µ;t)is a hyper BCK-ideal ofH (seeTheorem 2.4), it follows thatw∈U(µ;t), that is,x◦(y◦x)⊆U(µ;t). Hence
µ(x)≥ inf
w∈x◦(y◦x)µ(w)≥w∈U(µinf;t)µ(w)≥t, (3.10)
which implies thatx∈U(µ;t). Therefore,U(µ;t)is an implicative hyper BCK-ideal of H, and soµis a fuzzy implicative hyper BCK-ideal ofH.
Theorem3.10. Ifµis a fuzzy implicative hyper BCK-ideal ofH, then the set
I:=x∈H|µ(x)=µ(0) (3.11)
is an implicative hyper BCK-ideal ofH.
Proof. Clearly 0∈I. Letx,y∈Hbe such thatx◦yIandy∈I. Thenµ(y)= µ(0), and for eacha∈x◦y there existsz∈Isuch thataz. Thusµ(a)≥µ(z)= µ(0)by (FHI1), and soµ(a)=µ(0). It follows from (FHI2) that
µ(x)≥min
inf
a∈x◦yµ(a),µ(y)
=µ(0), (3.12)
so thatµ(x)=µ(0). Hencex∈I, which shows thatIis a hyper BCK-ideal ofH. Now let x,y∈Hbe such thatx◦(y◦x)I. Then for everyb∈x◦(y◦x)there existsw∈I such thatbw. Thusµ(b)≥µ(w)=µ(0)and soµ(b)=µ(0). UsingTheorem 3.9, we have
µ(x)≥ inf
b∈x◦(y◦x)µ(b)=µ(0), (3.13)
and thusµ(x)=µ(0), that is,x∈I. Therefore, by means ofLemma 3.2,I is an im-plicative hyper BCK-ideal ofH.
Definition3.11. A mappingf:G→H of hyper BCK-algebras is called ahyper
homomorphismif (HH1)f (0)=0,
(HH2)f (x◦y)=f (x)◦f (y)for allx,y∈G.
Proposition3.12. Letf:G→Hbe a hyper homomorphism of hyper BCK-algebras.
IfxyinG, thenf (x)f (y)inH.
Proof. Let x,y ∈G be such that x y. Then 0∈x◦y, and so 0=f (0)∈ f (x◦y)=f (x)◦f (y). Therefore,f (x)f (y)inH.
Theorem3.13. Letf:G→Hbe a hyper homomorphism of hyper BCK-algebras. If Iis an implicative hyper BCK-ideal ofH, thenf−1(I)is an implicative hyper BCK-ideal
ofG.
zw, that is, 0∈z◦w. It follows that
0=f (0)∈f (z◦w)=f (z)◦f (w)⊆f (x◦y)◦I= f (x)◦f (y)◦I (3.14)
so thatf (x)◦f (y)I. SinceIis a hyper BCK-ideal ofH, we havef (x)∈I, that is, x∈f−1(I)by (HI2). Hencef−1(I)is a hyper BCK-ideal ofG. Now letx,y∈Gbe such thatx◦(y◦x)f−1(I). Then for eacha∈x◦(y◦x), there existsb∈f−1(I)such thatab, that is, 0∈a◦b. Thus
0=f (0)∈f (a◦b)=f (a)◦f (b)⊆f x◦(y◦x)◦I
= f (x)◦ f (y)◦f (x)◦I, (3.15)
and sof (x)◦(f (y)◦f (x))I. SinceIis implicative, it follows fromLemma 3.2that f (x)∈Iso thatx∈f−1(I). This shows thatf−1(I)is an implicative hyper BCK-ideal ofG.
Lemma3.14(see Jun et al. [2, Proposition 3.7]). LetAbe a subset ofH. IfIis a hyper
BCK-ideal ofHsuch thatAI, thenAis contained inI.
Theorem3.15. Iff:G→His a hyper homomorphism of hyper BCK-algebras, then ker(f ):= {x∈G|f (x)=0}, called the kernel off, is an implicative hyper BCK-ideal ofG.
Proof. Clearly 0∈ker(f ). Let x,y ∈G be such that x◦y ker(f ) and y∈ ker(f ). Thenf (y) =0, and for each a∈x◦y there existsb∈ ker(f ) such that ab. It follows from (HH2) and (P4) thata∈ker(f )so that
0=f (a)∈f (x◦y)=f (x)◦f (y)=f (x)◦0=f (x), (3.16)
that is,x∈ker(f ). Hence ker(f )is a hyper BCK-ideal ofG. Letx,y∈Gbe such that x◦(y◦x)ker(f ). Thenx◦(y◦x)⊆ker(f )byLemma 3.14. Takingy=0 and using (P2) and (P4), we have{x} ⊆ker(f ), that is, x∈ker(f ). It follows fromLemma 3.2 that ker(f )is an implicative hyper BCK-ideal ofG.
Using Theorems3.6and3.15, we know that iff:G→His a hyper homomorphism of hyper BCK-algebras, thenµker(f )is a fuzzy implicative hyper BCK-ideal ofG.
Theorem 3.16. Letf :G→H be an onto hyper homomorphism of hyper
BCK-algebras. IfIis an implicative hyper BCK-ideal ofGcontainingker(f ), thenf (I)is an implicative hyper BCK-ideal ofH.
Proof. Note that 0=f (0)∈f (I). Let x,y∈H be such thatx◦y f (I) and y∈f (I). Sincefis onto, there exista,b∈Gsuch thatf (a)=xandf (b)=y. Thus
f (a◦b)=f (a)◦f (b)=x◦yf (I). (3.17)
from (HI2) thata∈I so thatx=f (a)∈f (I). Hencef (I)is a hyper BCK-ideal ofH. Now letx,y∈Hbe such thatx◦(y◦x)f (I). Then
f a◦(b◦a)=f (a)◦ f (b)◦f (a)=x◦(y◦x)f (I) (3.18)
for somea,b∈G. Letu∈a◦(b◦a). Then there existsc∈I such thatf (u)f (c), that is, 0∈f (u)◦f (c)=f (u◦c), which implies thatu◦c⊆ker(f )⊆I. It follows from (P1) thatu◦cIso from (HI2) thatu∈I. Thusa◦(b◦a)⊆I and soa◦(b◦a)I. SinceIis implicative, we geta∈IbyLemma 3.2. Hencex=f (a)∈f (I), and therefore f (I)is implicative. This completes the proof.
Theorem 3.17. Letf :G→H be an onto hyper homomorphism of hyper
BCK-algebras. Ifνis a fuzzy implicative hyper BCK-ideal ofH, then the hyper homomorphic preimageµ ofν underf, that is, the fuzzy setµinGdefined byµ(x)=ν(f (x))for allx∈G, is a fuzzy implicative hyper BCK-ideal ofG.
Proof. Letx,y∈Gbe such thatxy. Thenf (x)f (y)(seeProposition 3.12). Thus
µ(x)=ν f (x)≥ν f (y)=µ(y), (3.19) which proves (FHI1). For anyx∈G, we have
µ(x)=ν f (x)≥min
inf
w∈f (x)◦aν(w),ν(a)
(3.20)
for alla∈H. Letybe an arbitrary preimage ofaunderf. Then
w∈f (x)◦a=f (x)◦f (y)=f (x◦y), (3.21)
which impliesw=f (b)for someb∈x◦y. Hence
µ(x)≥min
inf
f (b)∈f (x◦y)ν f (b)
,ν(f (y))
=min
inf
b∈x◦yµ(b),µ(y)
.
(3.22)
Sinceais an arbitrary element ofH, (3.22) is true for anyy∈G. Thereforeµ is a fuzzy hyper BCK-ideal ofG. Sinceν is a fuzzy implicative hyper BCK-ideal ofH, it follows fromTheorem 3.9that
µ(x)=ν f (x)≥ inf
w∈f (x)◦(a◦f (x))ν(w) (3.23)
for allx∈Ganda∈H. Letybe an arbitrary element ofGsuch thatf (y)=a. Then
w∈f (x)◦ a◦f (x)=f (x)◦ f (y)◦f (x)=f x◦(y◦x), (3.24)
which implies that there existsb∈x◦(y◦x)such thatf (b)=w. Hence
µ(x)≥ inf
f (b)∈f (x◦(y◦x))ν f (b)
= inf
b∈x◦(y◦x)µ(b). (3.25)
Acknowledgement. This work was supported by Korea Research Foundation
Grant (KRF-99-005-D00003).
References
[1] Y. B. Jun and X. L. Xin,Fuzzy hyper BCK-ideals of hyper BCK-algebras, to appear in Math. Japon.
[2] ,Scalar elements and hyper atoms of hyper BCK-algebras, Math. Japon.2(1999), no. 3, 303–309.
[3] ,Implicative hyper BCK-ideals of hyper BCK-algebras, Math. Japon.52(2000), no. 3, 435–443.
[4] Y. B. Jun, M. M. Zahedi, X. L. Xin, and R. A. Borrowed,On hyper BCK-algebras, Ital. J. Pure Appl. Math. (2000), no. 8, 127–136.
[5] F. Marty,Sur une generalization de la notion de groupe, 8th Congress Math. Scandinaves (Stockholm), 1934, pp. 45–49 (French).
Young Bae Jun: Department of Mathematics Education, Gyeongsang National
University, Chinju660-701, Korea
E-mail address:[email protected]
Wook Hwan Shim: Department of Mathematics Education, Gyeongsang National
