NAFIPS 2005 - 2005 Annual Meeting of the North American Fuzzy Information Processing Society

H-ideals and Fuzzy H-ideals of BCH-algebras

B. Q. Hu, J. J. HeSchool ofMathematics and statistics, Wuhan University, Wuhan 430072, Hubei, P R. ofChina

bqhu@public.wh.hb.cn

Abstract - Fuzzy H-ideals were

discussed in BCI-algebras, but inBCH-algebras. In this paper we

introduce notions of fuzzy H-ideals inBCH-algebras and discuss theirproperties in BCH-algebras.

I. INTRODUCTION

Q.P. Hu and X. Li [1] introduced the concept ofBCH-algebras base on BCK/BCI-algebras [2-5], andsubsequently gave examples of proper BCH-algebras[6]. WA. Dudek [7] and B. Ahmad [8] studied some

classifications of BCH-algebras.Fuzzy ideals were discussed in BCK/BCI-algebras

[9-18] and generalized in BCH-algebras by B.Q. Hu, J.J. He [19]. Fuzzy H-ideals were studied inBCKIBCI-algebras but in BCH-algebras. So, thispaper introduces notions of H-ideals and fuzzyH-ideals and discusses their properties inBCH-algebras.

II. PRELIMINARIES

An algebra (X; *, 0) of type (2, 0) is called a

BCH-algebra if it satisfies the following axioms:(BCHI) x*x=0,

(BCH2) x*y=y*x=0=>x=y,(BCH3) (x*y)*z=(x*z)*y,

for all x, y, z E X. In a BCH-algebra X, we define a

partial ordering < by putting x < y if and only ifx*y=O.

A BCH-algebra X is said to be a BCI-algebra if itsatisfies the identity:

0-7803-91 87-X/05/$20.00 ©2005 IEEE.

(BCIl) ((x*y)*(x*z))*(z*y)=Oforall x,y,ze X.

If a BCI-algebraX satisfies the following condition:(BCKl) 0*x=0

for all x E X thenX is called a BCK-algebra.A BCH-algebra X is said to be non-negative if it

satisfies condition (BCK1). A BCH-algebra X is saidto be proper if it doesn't satisfies condition (BCI1). ABCI-algebra X is said to be proper if it doesn'tsatisfies condition (BCKI).

In any BCH-algebra X, the following hold:(1) x*(x*y)< y,

(2) 0*(x*y)=(O*x)*(O*y),(3) x*O=x,(4) x<O implies x=O,for all x, y eX .

A nonempty subset I of a BCH-algebra X is calledan ideal ofXif(I1) Oe I,(12) x *ye I and y E I imply xE I .

Let (X; *, 0) be a BCH-algebra. A fuzzy ,u inXis said to be a fuzzy ideal ofXif it satisfies(FIl) u(O) > p(x), for all x E X,(F12) ,u(x) . min{u(x*y),,u(y)},for all x, y eX .

III. H-IDEAL OF BCH-ALGEBRAS

Definition 3.1 A nonempty subset H of aBCH-algebra Xis called an H-ideal ofXif(HII) OEH,(HI2) x*(y*z)e H, yeH imply x*ze H.

Obviously it follows every H-ideal is an ideal in theBCH-algebras. For showing that H-ideals exist in theBCH-algebras we firstly propose a theorem ofconstructing proper BCH-algebras.515

Definition 3.2 Let < X; *,0 > be a BCH-algebra and

xo E X. Then x0 is called a sub-element of X ifthere exist a x (. x0o) E X such that x0 * x' . x0.

It is obvious to 0 is not sub-element in thenon-negative BCH-algebras (esp. BCK-algebras).

Theorem 3.3 Let < X;*,0 > be a BCK-algebra witha sub-element x0 and IX.>3. In XU{a}(a X)we define *' as follows

x*y, x,ye X,

{x, xe X,y=a,O, x=y=a,or,x=a,y=xo,

Then (i) < X U {a}; *', 0 > is a proper BCH-algebra.(ii) If I is an ideal of X then I is also an ideal ofX U {a} iff sub-element x0 o I .

(iii) IfH is a H-ideal ofX then H is also an ideal ofXU{a} iff h*y=xo ( heH,yeX ) impliesy = x .

Proof. (i) Obviously, X U {a} satisfies H-I and H-2.We only show that X U {a} satisfies H-3.

If x = a,y = x0,ze X-{x0}, then(a *'xo) *'z = O*z = 0,(a*'z)*'x0 =a* x0 =0.

If x=y=a,zeX,then(a *'a) *'z = 0 * z = 0

{0*'a=O, z=x0,(a *'z) *'a a*a = 0, zeX-{x0}.

If xe X,y=a,ze X,then(x *'a) *'z = x * z

(x * z) *'a = x * z.

In a word, for all x,y,ze XU{a} , we have(x*'y)*'z=(x*'z)*'y. Moreover if x'(.x0)eXsuch that x0 * x' # xo, then

((a*x')*'(a* x)) (x *' x0)=a*'(x0*x')=a,

i.e., <XU{a}; *',0> doesn't satisfy (BCII). Thus< X U {a}; *', 0 > is a proper BCH-algebra.(ii) Let I be an ideal of X. If I is also an ideal ofXU{a} and xoEI,then a*'xo=OeI. So ae I.This is contradiction with I c X. Therefore x0 o I.Whereas if x0 o I, then for all y e I a *'y = a t I.Thus I is also an ideal of X U {a} .

(iii) Let Hbe a H-ideal ofX. IfH is also an ideal of

XU{a}and h*y=xo (heH,yeX)thena*'(h*'y) = a*'xo = 0eH .

So a *'ye H by (HI2). Therefore y = x0. Whereas,

h*y=xo ( he H,yeX ) implies y=x0 . Thenwhen a*'(h*'a)e H and he H , we give thata*'a=OeH. When x*'(h*'a)eH (xeX) andhe H, we give that x*he H since h*'a = h .Thereby x*'a=xe H since H is an ideal of X.When a*'(h*'y)eiH (yE X) and he H, wehave that h *'y = xo by the definition of *' andy=x0 byhypothesis. So a*'y=a*''xo =Oe H. Byabove discussion we have that H satisfies (HI2) inXU {a} . ThereforeH is an H-ideal of X U {a} . El

Example 3.4 Let X = {O, a, b, c, d} in which * isdefined in Table I. Then < X;*,0 > is aBCK-algebra and A = {O, a, c} and B = {O, b, d} areideals of X. It is easy to see that A = {0, a, c} andB = {0, b, d} are H-ideal of X by Definition 3.1. ByDefinition 3.2 we see c is sub-element. Let e o Xand X U {e} in which *' is defined in Table II.Then < X U {e}; *', 0 > is a proper BCH-algebra.And more, we have that B = {0, b, d} is a H-ideal ofX U {e} (certainly ideal of X U {e} ), butA = {0, a, c} not an ideal of X U{e} .

TABLE I* 0 a b c do 0 0 0 0 0a a 0 a 0 ab b b 0 b 0c c a c 0 cd d d b d 0

TABLE II*' 0 a b c d eo o 0 0 0 0 0a a 0 a 0 a ab b b 0 b 0 bc c a c 0 c cd d d b d 0 de e e e 0 e 0

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IV. FUZZY H-IDEAL OF BCH-ALGEBRAS

We define fuzzy H-ideals in BCH-algebras as

follows.Definition 4.1 A fuzzy set u : X -> [0,1] is called a

fuzzy H-ideal, if for all x, y, z in X, (FI1) and(FI3) ,u(x* z) 2,u(x* (y * z)) Ay8(y) .

Clearly, z = 0 gives (FI3) is (FI2). Thus,Proposition 4.2 Every fuzzy H-ideal is a fuzzy idealin BCH-algebras.In (FI3) letting z = y we have the followingproposition.Proposition 4.3 Fuzzy H-ideal ,U satisfies

(7) ,u(x * y) > ,u(X) A A(y)The following example shows that fuzzy H-ideals

exist:Example 4.4 Consider Example 3.4; In BCK-algebraX ={0, a, b, c, d} in which * is defined in Table 1,we define ,u: X [0,1]by ,(0) = t1, ,u(a) =u(c) = t2, A(b) =A(d) = t3,where t, >t2 .t3 and t,,t2,t3 E [0,1] By routinecalculations give that ,u is a fuzzy H-ideal ofX.

Theorem 4.5 A fuzzy subset ,u of a BCH-algebra Xis a fuzzy H-ideal if and only if for any Ae [0,1],AA ={X XEX,/u(x) >2} is an H-ideal of X, where

Proof. Suppose ,u is a fuzzy H-ideal of X and

AAu 0, for Ae [0,1] .

Let xEE ,, then ,u(x) .2 . By Definition 4.1 wehave f,(0) 2,u(x) >2 So, OE HA -

Moreover, Suppose x * (y * z) E ftA, Yy ,UA, then,u(x * (y * z)) 2A and ,u(y) > A. By Definition 4.1we have

,u(x * z) > min{p(x * (y * z)),,u(y)} 2A.So, x*zEfuAA.Hence, tAA isanidealofX.

Conversely, Suppose that for each As [0,1], uHAis either empty or an ideal of X. For any xE X,setting ft(x)=2, then xe-pA. Since u,,(.0) isan ideal of X, we have 0c uAA and hence#u(0)A2=,u(x). Thus ,u(O)2,u(x) for all xe X.Now we prove (FI3) of Definition 4.1. If (FI3) isn'ttrue, then there exist x0, y0, z0 E X, such that

ft(xo * z0 ) < min { (x0 * (y0 *z0 )),yg(y0 )}.

Putting4u(x0 * z0) <A <min{(x, * (y z*zO))sH(yo)

then xo*(yo*zo),yoeu and .,0. But U,is an ideal of X, so xo * zo E #4 by (HI2). Namely

#(xo * zo) 2 2, it is contradictory. Therefore, (FI3) istrue completing the proof. D

Theorem 4.6 A fuzzy set ,u of a BCH-algebraXis afuzzy H-ideal if and only if for any xoe XAxo = {x E X I u(x) 2 Au(xo)} is an H-ideal ofX.Proof. Let a fuzzy set u of a BCH-algebra X be afuzzy H-ideal. Then for any xo E X,

Ax. ={xe XlI(x)>,u(xo)}

is an H-ideal of X with putting A =,u(xo) byTheorem 4.5.

Conversely, let u be a fuzzy set of BCH-algebraXand for any xo E X,

A, ={xeXI1U(x).y(x0)}is an H-ideal ofX. It is clear that ,(0) ,u(x) for allxE X by (HI1), i.e. (Fl1) holds. For all x,ye X,putting

min{u(x * (y * z)),,u(y)} =#(xo),we have ,u(x * (y * z)) 2,u(xo) and ,u(y) 2,u(xo) ,i.e. x*(y*z)e A, , ye Axo . Thus x*ze Ax by(HI2) since Ax is an H-ideal. Namely

,u(x * z) . ,u(x0) = min{p(x * (y * z)), A(y)}and (FI3) holds. Therefore ,u is a fuzzy H-ideal ofXcompleting the proof. DProposition 4.7 If ,u is a fuzzy H-ideal of a

BCH-algebra X, thenA = {xe X I u(x) =u(0)}

is an H-ideal ofX.Proof. Since p(x) <.u(O) for all xe X by (FIl),

A = {xe X I u(x) =,u(0)}={xE X ,u(x) 2 ,(0)} = AO.

Therefore, A is an ideal of X by Theorem 4.6completing the proof. D2

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