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Soft BCH-ideals and idealistic soft BCH-algebras Peng Jiayin Key Laboratory of Numerical Simulation of Sichuan Province and School of Mathematics and Information Scienc Neijiang Normal University Neijiang, Sichuan 641100, Chin e-mail: [email protected] AbstractApplying soft sets to BCH-algebras, the concepts of soft BCH-ideals and idealistic soft BCH- algebras are introduced, and several examples are given. The intersection, union and “AND” operations of soft BCH-ideals and idealistic soft BCH-algebras are established. The relations between these soft BCH-ideal, idealis- tic soft BCH-algebra and their homomorphic i- mages are provided.) Keywords-BCH-algebra; Soft set; soft BCH- algebra; soft BCH-ideal; idealistic soft BCH- algebra I. INTRODUCTION To solve a complicated problem in econo- mics, engineering, environment, sociology, me- dical science, busi- ness management, etc., we cannot successfully use classical methods because of various types of uncertainties present in these problems. Uncertainties cannot be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of fuzzy sets [1-2] , theory of intuitionistic fuzzy sets [3] , theory of vague sets [4] , theory of interval mathematics [5] , theory of rough set [6] , etc. and these many be utilized as mathematical tools for dealing with diverse types of uncertainties and imprecision embedded in a system. But all these theories have their inherent difficulties as point- ed out by Molodtsov [7] . Maji et al. [8] and Molodtsov [7] suggested that one reason for these difficulties may be due to the inadequacy of the parameterization tools of the theory. To over- come these difficulties, Molodtsov [7] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties and vagueness that is free from the difficulties that have troubled usual theoretical approaches. Maji et al. [8-9] described the application of soft set theory to a decision making problem. Chen et al. [10] presented a new definition of soft set parame- terization reduction in rough set theory. The algebraic structure of set theories dealing with uncertainties has been studied by some authors. The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh [2] . Jun et al [11] applied the fuzzy set theory to BCK-algebras. Hu et al [12] applied it to BCH-algebras. In this paper we apply the notion of soft sets by Molodtsov to the theory of BCH-algebras. We introduce the notion of soft BCH-algebras and soft subalgebras, and then derive their basic properties II. PRELIMINARES In what follows, a binary multiplication will be denoted by juxtaposition. Dots we use only to avoid repetitions of brackets. For example, the formula (( )( ))( ) 0 xy zy xz will be written as ( ) 0 xy zy xz . A non-empty set X with a constant 0 and a binary operation denoted by juxtaposition is called a BCH-algebra if for all , , x yz X the following axioms hold: (1) 0 xx , (2) xy 0 yx x y , (3) xyz xz y . On any BCH-algebra one can define the natural order d ” by putting (4) 0 x y xy d . A nonempty subset of a BCH-algebra S X is called a BCH-subalgebra of X if xy for all S , x y S . A subset I of a BCH-algebra X is called a BCH-ideal of X if 0 I and , y xy I imply x I . A mapping : f X Y o of BCH-algebras is called a homomorphism if ( ) f xy () () f xf y for all , x y X . For a homomorphism : f X Y o of BCH-algebras, the kernel of f , denot- ed by ker f , is defined to be the set ker { | () 0} f x X fx . Definition 2.1 [7] . A pair ( , is called a soft set over , where ) FA U F is a mapping given by . : F A o ( ) PU Definition 2.2 [8] . For two soft sets ( , and over a common universe U , we say that ( , is a soft subset of ( , , denoted by , if it satisfies: ) FA F ) (G ( , ) GB ) A , ) B ) GB ( , FA (i) , A B (ii) For every e A , and are identical approximations. () Fe () Ge Definition 2.3 [13] . The intersection of two soft sets and over a common universe U is the soft set ( , , where C ( , ) FA ( , ) GB ) HC A B and , e C 2010 International Conference on Artificial Intelligence and Computational Intelligence 978-0-7695-4225-6/10 $26.00 © 2010 IEEE DOI 10.1109/AICI.2010.197 361 2010 International Conference on Artificial Intelligence and Computational Intelligence 978-0-7695-4225-6/10 $26.00 © 2010 IEEE DOI 10.1109/AICI.2010.197 361

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Page 1: [IEEE 2010 International Conference on Artificial Intelligence and Computational Intelligence (AICI) - Sanya, China (2010.10.23-2010.10.24)] 2010 International Conference on Artificial

Soft BCH-ideals and idealistic soft BCH-algebras

Peng Jiayin Key Laboratory of Numerical Simulation of Sichuan Province and

School of Mathematics and Information SciencNeijiang Normal University

Neijiang, Sichuan 641100, Chine-mail: [email protected]

Abstract—Applying soft sets to BCH-algebras, the concepts of soft BCH-ideals and idealistic soft BCH-algebras are introduced, and several examples are given. The intersection, union and “AND” operations of soft BCH-ideals and idealistic soft BCH-algebras are established. The relations between these soft BCH-ideal, idealis- tic soft BCH-algebra and their homomorphic i- mages are provided.)

Keywords-BCH-algebra; Soft set; soft BCH- algebra; soft BCH-ideal; idealistic soft BCH- algebra

I. INTRODUCTION

To solve a complicated problem in econo- mics, engineering, environment, sociology, me- dical science, busi- ness management, etc., we cannot successfully use classical methods because of various types of uncertainties present in these problems. Uncertainties cannot be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of fuzzy sets[1-2], theory of intuitionistic fuzzy sets[3],theory of vague sets[4], theory of interval mathematics[5],theory of rough set[6], etc. and these many be utilized as mathematical tools for dealing with diverse types of uncertainties and imprecision embedded in a system. But all these theories have their inherent difficulties as point- ed out by Molodtsov[7]. Maji et al.[8] and Molodtsov[7] suggested that one reason for these difficulties may be due to the inadequacy of the parameterization tools of the theory. To over- come these difficulties, Molodtsov[7] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties and vagueness that is free from the difficulties that have troubled usual theoretical approaches. Maji et al.[8-9] described the application of soft set theory to a decision making problem. Chen et al.[10] presented a new definition of soft set parame- terization reduction in rough set theory. The algebraic structure of set theories dealing with uncertainties has been studied by some authors. The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh[2]. Jun et al[11]

applied the fuzzy set theory to BCK-algebras. Hu et al[12]

applied it to BCH-algebras. In this paper we apply the notion of soft sets by Molodtsov to the theory of BCH-algebras. We introduce the notion of soft BCH-algebras and soft subalgebras, and then derive their basic properties

II. PRELIMINARES In what follows, a binary multiplication will be denoted by juxtaposition. Dots we use only to avoid repetitions of brackets. For example, the formula (( )( ))( ) 0xy zy xzwill be written as ( ) 0xy zy xz .

A non-empty set X with a constant 0 and a binary operation denoted by juxtaposition is called a BCH-algebra if for all , ,x y z X the following axioms hold:

(1) 0xx ,(2) xy 0yx x y ,(3) xy z xz y .On any BCH-algebra one can define the natural order

“ ” by putting (4) 0x y xy .A nonempty subset of a BCH-algebra S X is called

a BCH-subalgebra of X if xy for all S ,x y S . A subset I of a BCH-algebra X is called a BCH-ideal of

X if 0 I and ,y xy I imply x I . A mapping :f X Y of BCH-algebras is called a homomorphism if ( )f xy ( ) ( )f x f y for all ,x y X .For a homomorphism :f X Y of BCH-algebras, the kernel of f , denot- ed by ker f , is defined to be the set ker { | ( ) 0}f x X f x .

Definition 2.1[7]. A pair ( , is called a soft set

over , where

)F AU F is a mapping given by .:F A ( )P UDefinition 2.2[8]. For two soft sets ( , and

over a common universe U , we say that ( ,is a soft subset of ( , , denoted by ,if it satisfies:

)F AF

) (G( , )G B )A

, )B)G B ( ,F A

(i) ,A B(ii) For every e A , and are identical

approximations. ( )F e ( )G e

Definition 2.3[13]. The intersection of two soft sets and over a common universe U is the

soft set ( , , where C( , )F A ( , )G B

)H C A B and ,e C

2010 International Conference on Artificial Intelligence and Computational Intelligence

978-0-7695-4225-6/10 $26.00 © 2010 IEEE

DOI 10.1109/AICI.2010.197

361

2010 International Conference on Artificial Intelligence and Computational Intelligence

978-0-7695-4225-6/10 $26.00 © 2010 IEEE

DOI 10.1109/AICI.2010.197

361

Page 2: [IEEE 2010 International Conference on Artificial Intelligence and Computational Intelligence (AICI) - Sanya, China (2010.10.23-2010.10.24)] 2010 International Conference on Artificial

( ), if \ ,( ) ( ), if \ ,

( ) ( ), if .

F e e A BH e G e e B A

F e G e e A BIn this case, we write

( , ) ( , ) ( , )F A G B H C .Remark In [13], Xu et al. pointed out that the

definition for the intersection of two soft sets by Maji et al. may cause a contradiction, and redefined it in [13].

Definition 2.4[8]. The union of two soft sets ( ,and over a common universe U is the soft set (H, C) , where C A and for all ,

)F A( , )G B

B e C( ), if \ ,

( ) ( ), if \ ,( ) ( ), if .

F e e A BH e G e e B A

F e G e e A BIn this case, we write

( , ) ( , ) ( , )F A G B H C .Definition 2.5[8]. If and be two soft sets

over a common universe U , then “ ( , AND ”denoted by ( , is defined by

( , )F A

( , )G B

( , )G B)F A ( , )G B

( , )F A)F A= , where ( , )G B ( ,H A B) H ( , ) ( )F ( )G

for all ( , ) A B .

III. SOFT BCH-IDEALS

In what follows, let X and be a BCH-algebra and Aa nonempty set, respectively, and R will refer to an arbitrary binary relation between an element of and an element of

AX , that is, R is a subset of A X unless

otherwise specified. A set-valued function can be defined as

:F A}

( )P U( )F x {y |X xRy for all x A .

The pair ( , is then a soft set over )F A X .Definition 3.1. Let be a BCH-sub- algebra of S X .

A subset I of X is called an ideal of X related (briefly, -ideal of

SS X ), denoted by I S , if 0 I and

( )( )( )x S y I xy I x I .Note that if is a subalgebra of S X and I is a subset

of X that contains , then S I is an -ideal of S X .Obviously, every BCH- ideal of X is an -ideal of S X ,but the converse is not true in general as seen in the following example.

Example 3.1. Let {0, , , , }X a b c d be a BCH-algebra with the following Cayley table:

Then is a subalgebra of {0, }S

0 a b c d0 0 0 0 0 0a a 0 a 0 0

b b b 0 0 bc c b a 0 bd d a d a 0

Definition 3.2[14]. Let be a soft set over ( , )F A X .Then is called a soft BCH-algebra over ( , )F A X if

is a BCH- subalgebra of (F x) X for all x A .Definition 3.3. Let ( , be a soft BCH-algebra

over )F A

X . A soft set ( , over )G I X is called a soft BCH-ideal of ( , , denoted by ( , , if it satisfies:

)F A ) ( , )G I F A

(i) I A ,(ii) ( )( ( ) ( ))x I G x F x .Let us illustrate this definition using the following

examples. Example 3.2. Let {0, , , , }X a b c d be a BCH-

algebra with the following Cayley table: 0 a b c d

0 0 0 0 0 0a a 0 a a ab b b 0 b bc c c c 0 cd d d d d 0

Let ( , be a soft set over )F A X , where A X and is a set-valued function defined by : (F A P X

( ) {F x y X)

| y yx {0, }a for all } x A . Then (0) ( )F F a X , ,(F b) {0, ,a c, }d ( )F c {0,a ,

, }b d , and ( ) {0, , , }F d a b c are BCH-sub- algebras of X . Therefore ( , is a soft BCH- algebra over )F A X .

(1) Let ( , be a soft set over )G I X , where { , , }I a b c and is a set- valued function

defined by : (P X )G I

( ) { | {0, }}G x y X y yx dfor all x I . Then ( ) {0, , , }G a b c d X ( )F a ,( ) {0, , , } ( )G b a c d F a , and ( )G c {0, , , }a b d

( )F c . Hence ( , .)G I ( , )F A (2) For { , , }I a b c , let be a set-

valued function defined by :H I ( )P X

( ) {0} { | }H x y X x y

b X and {0, , }I b d S , but I is not an ideal of X because

and .0ad I a I

for all x I . Then ( ) {0, } ( )H a a F a ,

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( ) {0, } ( )H b b F b( , ) (H I F

, and .Therefore .

( ) {0, }H c c ( )F c, )A

Theorem 3.1. Let ( , be a soft BCH-algebras

over

)F AX . For any soft sets and over 1( ,G I1) 2 2( , )G I X ,

we have

1 1( , ) ( , )G I F A , 2 2( , ) ( , )G I F A

1 1 2 2( , ) ( , ) ( , )G I G I F A .Proof. Using Definition 2.3, we can write

, where C1 1 2 2( , ) ( , ) ( , )G I G I H C 1I 2I and

or or for all 1( ) ( )H x G x 2 ( )G x 1( )G x 2G x( )x C :H C

( , )H C. Note that is a mapping, and

hence is a soft set over (P X )

X . Obviously, C . If A

1 2x I I , then since 1( )H x G ( )x ( )F x 1( ,G I1)( , )F A . If x 2 1I I , then

since . If2( ) ( ) ( )H x G x F x

1 22 2 , )F A( ,G I ) ( x I I ( )H x, then

1( )G x 2 ( )G x ( )F x since 1( ,G 1)I ( , )F A and

. Thus for all 2 2( , )G I ( , )F A ( )H x ( )F x x C ,

and so ( , .)H C ( , )F ACorollary 3.2. Let be a soft BCH-algebras

over

( , )F AX . For any soft sets and over 1( ,G I ) 2( , )G I X ,

we have

1( , ) ( , )G I F A , 2( , ) ( , )G I F A .1 2( , ) ( , ) ( , )G I G I F A

Proof. Straightforward. Theorem 3.3. Let ( , be a soft BCH-algebras

over

)F AX . For any soft sets and over1( ,G I1) 2 2( , )G I X ,

where 1I 2I , we have

1( , ) ( , )G I F A , 2( , ) ( , )G I F A

1 2( , ) ( , ) ( , )G I G I F A .Proof. Using Definition 2.4, we can write

where 1 1 2 2( , ) ( , ) ( , )G I G I H C 1C I 2I and

for every x C ,

1 1 2

2 2 1

1 2 1

( ), if \ ,( ) ( ), if \ ,

( ) ( ), if .

G x x I IH x G x x I I

G x G x x I I2

Since 1 2I I , it follows that either 1 \ 2x I I or

2 \ 1x I I for all x C . If x 1 \ 2I I( , )F A

,then

since . If 1( )H x G ( )x F ( )x 1( , )G I

2 \ 1x I I , then since

. Hence H x for all

( )H x( )

2 ( ) ( )G x F x( )F x2( , )G I ( , )F A x C ,

and so ( , .) (F , )AH C

1If I and 2I are not disjoint in Theorem 3.3, then Theorem 3.3 is not true in gengral as seen in the following example.

Example 3.3. Let {0, , , , }X a b c d

c

be a BCH-algebra with the following Cayley table:

0 a b d0 0 0 0 0 0a a 0 0 0 0

b b b 0 b 0

c c c c 0 0

d d d c b 0

Let ( , be a soft set over )F A X , where A X and: (F A P X ) is a set-valued function defined by

( ) { | {0, }}F x y X y yx bfor all x A . Then ,(0)F X ( )F a

( )F b {0, , , }b c d , and ( )F c ( )F d {0 which are BCH-subalgebras of

, }bX . Therefore ( , is a soft

BCH- algebra over

)F AX . Let ( , be a soft sets over 1G I1) X ,

where 1I and is a set-

valued function defined by G x{ , ,b c d} 1 1:G I P

( )(X

{ |y X)

yx 0}1

for all 1x I . Then G b ={0 ,1( ) , , } ( )a b F b 1( )G c{0, , }a c

1 1( , )G I( )F c

( ,F A, and , and so

. Let be a soft sets over 1( )G d X

2 2( , )G I( )F b

) X ,

where 2 { }I b and is a set-valued

function defined by G x for all 2 2: (G I P X

( ) {y X y)

yx2 | 0}

2x I . Then G2 ( )b {0, } ( )c F b , and so 1 1( , )G I( , )H C( , )F A . Obviously, , but1 2I I

is not a soft BCH-ideal of ( ,since

1 1( , )G IH b

2 2( , )G I

1( ) ( )G b)F A

2 ( )G b {0, is not an

-ideal of

,a ,b c}( )F b X because ,( )d F b 0 ( )F b c( )H b , ( )dc b H b and ( )d d H bLet :f X Y

( , )F A be a mapping of BCH- algebras. For a

soft set over X , then ( ( ), )f F A is a soft set over where Y ( ) :f F is defined by ( )YA P

( )( )f F x ( ( ))f F x for all x A .

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Theorem 3.4. Let :f X Y be a homomorphism of BCH-algebras. For any soft BCH-algebra ( , and soft set ( , over

)G B)F A X , we have

( , ) ( , ) ( ( ), ) ( ( ), )F A G B f F A f G BProof. From ( , and Definition 3.3, we

have that A and for all) ( , )F A G B

B ( )F x G x( ) x A . By Hypothesis and Theorem 3.6[14], ( ( ), )f G B is a BCH- algebra over Y . Since f is a homomorphism,

( )( ) ( ( ))f F x f F x ( ( ))f G x ( )( )f G xfor all x A , and so ( ( ), )f F A ( ( ), )f G B ..

IV. IALISTIC SOFT-ALGEBRAS

Definition 4.1. A soft set ( , over )F A X is called an idealistic soft BCH-algebra over X if is an ideal of

(F x)X for all x A .

Example 4.1. The soft set ( , in Example 3.2 is an idealistic soft BCH-algebra over

)F AX .

Theorem 4.1. Let ( , and be two soft sets over

)F A ( , )F BX where B A . If ( , is an idealistic soft

BCH-algebra over )F A

X , then so is ( , .)F BProof. Straightforward. Corollary 4.2. Let ( , be an idealistic soft BCH-

algebra over

)F AX . If B is a set of , then ( | is an

idealistic soft BCH-algebra over

A , )BF BX .

Theorem 4.3. Let ( , and be two idealistic soft BCH-algebras over

)F A ( , )G BX . Then the intersection

is an idealistic soft BCH-algebra over ( , ) ( , )F A G BX .

Proof. Using Definition 2.3, we can write , where C A( , ) ( , ) ( , )F A G B H C B and for

all x C ,( ), if \ ,

( ) ( ), if \ ,( ) ( ), if .

F x x A BH x G x x B A

F x G x x A B)Note that is a mapping, and hence

is a soft set over : (H C P X

( , )H C X . Since ( , and ( ,are idealistic soft BCH-algebras over

)F A )G BX , it follows that

for ( )H x ( )F x x A B is an ideal of X ,

for ( )H x ( )G x x B A is an of X , and for all ( )H x ( )F x ( )G x x A B is an ideal of

X . Therefore ( , is an idealistic soft BCH- algebra over

) ( ,G B)F AX .

Theorem 4.4. Let ( , and be two idealistic soft BCH-algebras over

)F A ( , )G BX . If A B ,

then is an idealistic soft BCH-algebra over

( , ) ( , )F A G BX .

Proof. Using Definition 2.4, we can write , where C A( , ) ( , ) ( , )F A G B H C B and for

every x C ,( ), if \ ,

( ) ( ), if \ ,( ) ( ), if .

F x x A BH x G x x B A

F x G x x A BSince A B , it follows that either \x A B or

\x B A for all x C . If x \A B , then ( )H x ( )F x is an ideal of X since is an

idealistic soft BCH-algebras over ( , )F A

X , and if \x B A ,then ( )H x is an ideal of (G x) X since is an idealistic soft BCH-algebras over

( , )G BX . Hence ( , is an

idealistic soft BCH-algebra over )H C

X .Theorem 4.5. Let ( , and be two

idealistic soft BCH-algebras over )F A ( , )G BX .Then

( , ) ( , )F A G B is an idealistic soft BCH-algebra over X .

Proof. From Definiton 2.5, we can write ( , ) ( , )F A G B = ( , )H A B , where

( , )H x y ( ) ( )F x G y for all ( , )x y A B . Since and are ideals of

( )F x ( )G yX , ( )F x ( )G y is also an ideal of X . Therefore

is an ideal of ( ,H x )y X for all ( , )x y A B . Hence ( ,F A) ( ,G B) = ( , is an idealistic soft BCH-algebra over

)BH AX .

Definition 4.2. An idealistic soft BCH-algebra over ( , )F A X is said to trivial (resp., whole) if

( )F x {0} (resp., ( )F x X ) for all x A .Example 4.2. Let {0, , , , }X a b c d be a BCH-

algebra with the following Cayley table: 0 a b c d

0 0 0 0 0 0a a 0 0 0 0

b b b 0 0 0

c c c c 0 0

d d c c a 0

For A { , , }b c d X : ( )P X, let F D be a set-valued function defined by

( ) { | {0, , }}F x y X yx a c

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for all x A . Then for all ( )F x X x , and so is a whole idealistic soft BCH- algebra over

A( ,F A) X .

( )( )f F x ( ( )) ( )f F x f X Yfor all x A . It follows from Theorem 4.6 and Definition 4.2 that ( ( ), )f F A is the whole idealistic soft BCH-algebra over Y .

Theorem 4.6. Let :f X Y( ,F A

be an onto homomorphism of BCH-algebras. If is an idealistic soft BCH-algebra over

)X , then ( ( ), )f F A is an idealistic

soft BCH-algebra over Y . REFERENCESProof. For every x A , we have

[1] Zadeh L A. From circuit theory to system theory. Proc. Inst. Radio Eng.,50(1962)856-865. ( )( )f F x ( ( ))f F x[2] Zadeh L A. Fuzzy sets. Inform. and Control, 8 (1965) 338-353. is an ideal of Y since is an ideal of ( )F x X and its onto

homomorphic image is a also an ideal of . Hence Y( ( ), )f F A is an idealistic soft BCH-algebra over Y .

[3] Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1986) 87-96. [4] Gau W L, Buehrer D J. Vague sets. IEEE Trans. System Man Cybernet,1993,23 (2):610- 614. [5] Gorzalzany M B. A method of inference in approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets and Systems, 21(1987) 1-17.

Theorem 4.7. Let :f X Y be an onto homomorphism of BCH-algebras and ( , an idealistic soft BCH-algebra over

)F AX .

[6] Pawlak Z. Rough sets. International Journal of Information and Computer Sciences,11 (1982) 341-356.

(i) If for all ( ) kerF x f x A , then ( ( ), )f F Ais the trivial idealistic soft BCH- algebra over Y .

[7] Molodtsov D. Soft set theory-First results. Computers Math. Applic. 1999, 37 (4/5):19-31. [8] Maji P K, Biswas R, Roy A R. Soft set theory. Comput. Math. Appl. 45 (2003) 555–562. (ii) If ( , is whole, then ( ()F A ), )f F A is the whole

idealistic soft BCH-algebra over Y .[9] Maji P K, Roy A R, Biswas R. An application of soft sets in a decision making problem. Comput. Math. Appl. 44 (2002) 1077–1083. [10] Chen D, Tsang E C C, Yeung D S, Wang X. The parameterization reduction of soft sets and its applications. Comput. Math. Appl. 49 (2005) 757–763.

Proof. (i) Since ,( ) kerF x f( )( )f F x = ( ( )) {0 }Yf F x [11] Jun Y B, Xin X L. Fuzzy prime ideals and invertible fuzzy ideals in

BCK-algebras. Fuzzy Sets and Systems 117 (2001) 471–476. for all x A), )

. From Defini- tion 4.2 and Theorem 4.6, ( (f F A is the trivial idealistic soft BCH-algebra over

.Y

[12] Dudek W A, Jun Y B, Stojakovic Z. On fuzzy ideals in BCH-algebras. Fuzzy Sets and Systems 123 (2001) 251–258. [13] Xu W, Ma J, Wang S Y, H, Hao G. Vague soft sets and their properties. Comput. Math. Appl. 59 (2010) 787–794.

(ii) Since f is onto and ( , is whole, then for all

)F A( )F x X x A , and so

[14] Peng J Y. Soft BCH-algebras. To appear..

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