960 JOURNAL OF COMPUTERS, VOL. 8, NO. 4, APRIL 2013

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1 960 JORNAL OF COMPTERS, VOL 8, NO 4, APRIL 03 Study on Soft Groups Xia Yin School of Science, Jiangnan niversity, Wui, Jiangsu 4, PR China Zuhua Liao School of Science, Jiangnan niversity, Wui, Jiangsu 4, PR China Abstract Soft set theory, proposed by Molodtsov, has been regarded as an effective mathematical tool to deal with uncertainties In this paper, we further study the properties of soft sets and their duality Moreover, we introduce the new concepts of soft subgroups and normal soft subgroups of a group Some algebraic properties about soft subgroups and normal soft subgroups are investigated Inde Terms Duality, Soft Subgroups, Normal Soft Subgroups I INTRODCTION In order to solve complicated problems in economics, engineering, environmental science, medical science and social science, methods in classical mathematics are not always successfully used because various uncertainties are typical for these problems Therefore, there has been a great deal of alternative research and applications in the literature concerning some special tools such as probability theory, fuzzy set theory [,], rough set theory [3-6], vague set theory [7] and interval mathematics [8] Although they are all useful approaches to describe uncertainty, each of these theories has its inherent difficulties, as mentioned by Molodtsov [9] Consequently, Molodtsov [9] proposed a completely new approach, called soft set theory, for modeling vagueness and uncertainty Soft set theory has potential applications in many fields, including the smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability theory and measurement theory Most of these applications have already been demonstrated in Molodtsov's paper [9] Currently, work on soft set theory is progressing rapidly Maji et al [0] investigated the applications of soft set theory to a decision making problem Roy and Maji [] proposed the concept of a fuzzy soft set and provided its properties and an application in decision making under an imprecise environment Chen et al [] presented a definition for soft set parameterization reduction and showed an application in another decision making problem Kong et al [3] further studied the problem of the reduction of soft sets and fuzzy soft sets by introducing a definition for normal parameter Corresponding author: Liao Zuhua reduction Maji et al [4] defined and studied several operations on soft sets, and Ali et al [5] gave some new notions such as restricted intersection, restricted union, restricted difference, and etended intersection of soft sets Jun [6] applied Molodtsov's notion of soft sets to the theory of BCK/BCI-algebras and introduced the notion of soft BCK/BCI-algebras and soft subalgebras and then investigated their basic properties Jun and Park [7] dealt with the algebraic structure of BCK/BCIalgebras by applying soft set theory They introduced the notion of soft ideals and idealistic soft BCK/BCI-algebras and gave several eamples Jun et al [8] introduced the notion of soft p-ideals and p-idealistic soft BCI-algebras and investigated their basic properties Aktas et al [9] gave a definition of soft groups and derived their basic properties Sezgin et al [0] corrected some of the problematic cases in a previous paper by Aktas and Cagman, and introduced the concept of normalistic soft groups and the homomorphism of normalistic soft groups, studied their several related properties, and investigated some structures that are preserved under normalistic soft group homomorphisms Atagün et al [] introduced soft subrings and soft ideals of a ring, soft subfields of a field and soft submodule of a left R-module, and studied some related properties Sezgin and Atagün [] proved that certain De Morgan s laws hold in soft set theory with respect to different operations and etended the theoretical aspect of operations on soft sets Jin et al [3] studied the equivalence soft set relations and obtained soft analogues of many results concerning ordinary equivalence relations and partitions The algebraic structure of set theories dealing with uncertainties has also been studied by some authors Rosenfeld [4] proposed the concept of fuzzy groups in order to establish the algebraic structures of fuzzy sets Yuan [] gave the definition of a fuzzy subgroup with thresholds which is a generalization of Rosenfeld's fuzzy subgroup and Bhakat and Das's fuzzy subgroup Liao [5] studied generalized fuzzy k-ideals and (, q( λ, )) - fuzzy k-ideals in semigroups Rough groups were defined by Biswas et al [6], and some other authors have studied the algebraic properties of rough sets as well Wen [7], Yuan Xuehai's graduate student, presented the new definitions of soft subgroups and normal soft subgroups and obtained some primary results 03 ACADEMY PBLISHER doi:04304/jcp

2 JORNAL OF COMPTERS, VOL 8, NO 4, APRIL The purpose of present paper is a further attempt to broaden the theoretical aspects of soft groups The rest of this paper is organized as follows In section, we review the basic concepts of soft sets and list some related properties of soft sets and their duality In section 3, we correct the concepts of soft subgroups and normal soft subgroups defined in [7] which are different from the concepts defined in [0] Also, more algebraic properties of soft subgroups and normal soft subgroups are investigated II BASIC DEFINITIONS AND PRELIMINARY RESLTS OF SOFT SETS In this section, we present the concepts of soft sets and their duality Then we define and study several operations on soft sets Let X be an initial universe set and E be a set of parameters Let PX ( ) be the power set of X, and A E Definition [9] A pair (, A) is called a soft set over X, where is a mapping given by : A PX ( ) In other words, a soft set over X is a parameterized family of subsets of the universe X For ε A, ( ε ) may be considered as the set of ε -elements of the soft set (, A) or as the set of ε -approimate elements of the soft set Clearly, a soft set is not a set For illustration, Molodtsov considered several eamples in [9] Definition [7] Let (, E) be a soft set over X, a pair ( A, X ) is called the duality of (, E), where A is a mapping given by: A : X P( E), a A = { ε E ε ( )} for all X Remark [7] If a pair (, E) is a soft set over X, then the duality ( A, X ) of the soft set (, E) is also a soft set over E In the remains, we consider the set of parameters E be a group and denote by G For convenience sake, we denote the soft set (, G) by The set of all soft sets over X is called the soft power set of G and is denoted by SP ( G) Definition 3 Let, SP ( G), if for all G, then is said to be contained in (or contains ), and we write by Definition 4 Let, SP ( G), the union and intersection of and, denoted by and respectively, is defined as follows: G, ( ) = ( ) = Definition 5 Let GH, be groups and f a function from G into H, and let SP ( G) and SP ( H ) Define the soft sets f ( ) SP ( H ) and f ( ) SP (G) by: for H, if f ( y) f = y f( )( y) = G if f ( y) = and for G, f ( ) = ( f) Then f ( ) is called the image of under f and f ( ) is called the preimage (or inverse image) of under f Definition 6 We define the binary operation o and the unary operation on SP ( G) as following:, SP ( G) and G, ( ) = ( ( y) ( z)), = yz yz, G = ( ) We call o the product of and, and the inverse of It is easy to verify that the binary operation o in Definition 6 is associative The net proposition can be proved easily by using the previously defined notions and thus we omit its proof Proposition Let, SP ( G), then the following assertions hold: () ( ) = ( ( y) ( y )) G ( ( ) ( )) G = y y G; () ( ) = ; (3) ( ) = ; (4) ( ) = ; (5) ( ) = Proposition Let, SP ( G), then the following assertions hold: () A = A A; () A = A A; (3) A = A ; (4) A ( ) = f( A ); f (5) A = f ( A ) ( ) f Proof () For all X, we have g AI ( I)( g) = ( g) I( g) ( )( g)and ( )( g) g Aand g A g A I A Hence A I = A I A () For all X, we have 03 ACADEMY PBLISHER

3 96 JORNAL OF COMPTERS, VOL 8, NO 4, APRIL 03 g G g A o ( o)( g) = ( ( g ) I( g g)) I g = g( g g) A A o g G such that ( g ) ( g g) g G such that g A, g g A Hence A A A On the other hand, g A A g A, g A and g = gg g, g G, such that ( g), ( g)and g = gg ( g) I ( g)and g = gg ( ( g ) I( g )) = ( o)( g) g= gg g A o Hence A o A A (3) For all X, we have g A ( g) = ( g ) g A g ( A ) Hence A = A (4) For all X, we have g A f( )( g) f ( ) ( g ) f ( g ) = g g G g G such that ( g ), f( g ) = g g A and f( g) = g g f( A ) Hence Af ( ) = f( A) (5) For all X, we have g A ( ) f ( )( g ) ( f ( g = )) f ( ) f( g) A g A and f( g) = g g f ( g)and g A g f ( A ) Hence A = f ( A ) f ( ) Proposition 3 Let SP ( G), then the following assertions are equivalent: () ( y) = ( y) G; () ( y) = ( y) y G ; (3) ( y) ( y) y G ; (4) ( y) ( y) y G ; (5) = SP ( G) Proof () () : Let, G, then ( y) = (( y) ) = ( ( y)) = ( y) () (3) : Immediate (3) (4) : Let, G, then ( y) (( ) ( y) ) = ( y) (4) () : Let, G, then ( y) = ( y yy) ( y) = ( y) ( y) Hence ( y) = ( y) () (5) : By Proposition, we have ( ) = ( ( y ) ( y)) G ( ( y) ( y )) = G = ( ) G Thus = SP ( G) (5) () : Suppose, G Define a soft set as follows: X if = y = if y Then we have = by the hypothesis of the theorem It follows that G ( y) = ( y) ( y ) = z z = z G ( ( ) ( )) ( )( ) = = z z ( )( ) ( ( ) ( )) = ( y) z G III SOFT SBGROPS AND NORMAL SOFT SBGROPS Throughout this section, we denote the identity element of a group G by e Definition 3[7] Let SP ( G) Then is called a soft subgroup of G, if () ( y) I ( y) G and () ( ) G Lemma 3[7] () Let SP ( G) Then is a soft subgroup of G if and only if for all X, A is a subgroup of G () Let SP ( G) Then is a subgroup of G for all X if and only if A is a soft subgroup of G We mention that if A is a null set for some X, then the assertions in Lemma 3 are incorrect These difficulties may be due to the definitions of the related notions in [7] Net, we shall endeavor to make the definition of soft subgroup more rational 03 ACADEMY PBLISHER

4 JORNAL OF COMPTERS, VOL 8, NO 4, APRIL Definition 3 Let SP ( G) and () e = X, then is called a soft subgroup of G, denoted by % G, if () ( y) I ( y) G and () ( ) G We denote the set of all soft subgroups of G by S ( G) Remark 3 The condition () of Definition 3 can be replaced by ( ) ( )= G Also satisfies conditions () and () of Definition 3 if and only if ( y ) I ( y) G Theorem 3 Let SP ( G), then is a soft subgroup of G if and only if =, = Proof Suppose be a soft subgroup of G By lemma 3, we know that A is a subgroup of G for all X (i) For all g G, we have ( g) = ( ( g) I (( g) g)) So there eits g g G G such that ( g ) I (( g ) g), it follows that g A and ( g ) g A, and so g A, thus ( g) Therefore ( g) ( g) On the other hand, ( g) g A g G,( g ) g A I ( g ) (( g ) g) ( ( g ) (( g ) g)) g G I ( g) So we have ( g) ( g) Hence = (ii) Since is a soft subgroup of G, for all g G, we have ( g) = ( g ) = ( g) So = is hold Conversely, (i) g, g G, ( gg ) = ( gg ) = ( ( g ) I ( g )) (ii) g G g= gg ( g) I ( g) ( g ) = ( g) = ( g) Thus is a soft subgroup of G Theorem 3 Let, be soft subgroups of G Then () I is a soft subgroup of G ; () o is a soft subgroup of G if and only if o = o Proof () Let, G, (i) since, are soft subgroups of G, then ( I )( y) = ( y) I ( y) ( I( y)) I( I( y)) = ( I) I( ( y) I( y)) = ( I) I( I)( y) (ii) since, are soft subgroups of G, then ( I)( ) = ( ) I( ) I = ( I) Hence I is a soft subgroup of G () Suppose o is a soft subgroup of G Since, be soft subgroups of G, by Proposition and Theorem 3, we have o = o = ( o) = o Conversely, suppose o = o Then ( o) = ( o) = o = o and ( o) = ( o) o( o) = o( o) o = o( o) o = ( o) o( o) = o = o By Theorem 3, we know that o is a soft subgroup of G Theorem 33 Let f be a homomorphism from G to G Then () If is a soft subgroup of G,then f ( ) is a soft subgroup of G ; () If is a soft subgroup of G,then f ( ) is a soft subgroup of G Proof () (i) Let y, y G If f ( y ) = or f ( y) =, from Definition5, we can see that f ( y) I f( y) f( yy) If f ( y ) and f ( y), then g f( )( y) I f( )( y) g f( )( y), g f( )( y) g, g ( ) = ( ) = f y f y, G such that f( ) = y, f( ) = y and f( ) = yy, g, g ( ) g I ( ) ( ) and f( ) = yy since is a soft subgroup of G g = f( )( y y ) f = yy Thus f ( )( yy) f( )( y) f( )( y) I (ii) Let G If f ( y) =, then f( )( y) = Obviously f ( )( y ) f( )( y) If f ( y), then 03 ACADEMY PBLISHER

5 964 JORNAL OF COMPTERS, VOL 8, NO 4, APRIL 03 Thus g f( )( y) g f = y f = y 0 G such that f( 0) = y and g ( 0) g ( ) ( ) and f( ) = y since is a soft subgroup of G g = f( )( y ) f ( )( y ) f( )( y) Hence f ( ) is a soft subgroup of G () (i) Let, G Since is a soft subgroup of G, then f ( )( ) = ( f( )) = ( f f( )) ( f( )) I( f( )) I = f ( )( ) f ( )( ) (ii) Let G Since is a soft subgroup of G, then Hence f ( ) f ( )( ) = ( f( )) = (( f) ) ( f) = f ( ) is a soft subgroup of G Definition 33[7] Let be a soft subgroup of G Then is called a normal soft subgroup of G, denoted by %< G, if ( y) ( y) y G Remark 3 By proposition 3, the condition of Definition 33 can be replaced by ( y) = ( y) (or ( y) = ( y) ) G Lemma 3[7] () Let SP ( G) Then is a normal soft subgroup of G if and only if for all X, A is a normal subgroup of G () Let SP ( G) Then is a normal subgroup of G for all X if and only if A is a normal soft subgroup of G Theorem 34 Let, be normal soft subgroups of G Then () I is a normal soft subgroup of G ; () o is a normal soft subgroup of G Proof () since, are normal soft subgroups of G, then, are soft subgroups of G By Theorem 3 (), we know that I is a soft subgroup of G Let, G, then ( I)( y) = ( y) I( y) ( y) I( y) = ( I)( y) Thus I is a normal soft subgroup of G () since, are normal soft subgroups of G, then, are soft subgroups of G By Proposition 3, we know that o = o By Theorem 3 (), we know that o is a soft subgroup of G Let, G, then g ( o)( y) g ( ( a) I( b)) y= ab a, b G such that y = a b and g ( a ) I( b ) 0 0 since, are normal soft subgroups of G, we have ( a ) ( a ), ( b ) ( b ), It follows that g ( a0) I ( b0)and ( a0)( b0) = y Thus y ab g ( ( a) I( b)) = ( o)( y) = Therefore ( o)( y) ( o )( y) It shows that o is a normal soft subgroup of G Theorem 35 Let f be a homomorphism from G onto G Then () If is a normal soft subgroup of G, then f ( ) is a normal soft subgroup of G ; () If is a normal soft subgroup of G, then f ( ) is a normal soft subgroup of G Proof () By Theorem33, we know that f ( ) is a soft subgroup of G For G, g G,since f is a homomorphism from G onto G, there eists h 0 G such that f( h0 ) = g Then z f( )( y) = f = y G such that f( ) = y and z ( ) z ( ) ( h h )since is a normal soft subgroup of G and f( h h ) = 0 0) ( 0) = f ( h h) = g yg f( h ) f( f h g yg z ( h h) = f ( )( g yg) Thus f ( )( g yg) f( )( y) Hence f ( ) is a normal soft subgroup of G () By Theorem33, we know that f ( ) is a soft subgroup of G For G, g G,Since is a normal soft subgroup of G, then f ( )( g g) = ( f( g g)) = ( f( g ) f f( g)) = (( f( g)) f f( g)) ( f) = f ( ) Hence f ( ) is a normal soft subgroup of G If, are soft subgroups of G and there eists u G such that = ( u u) G, then and are 03 ACADEMY PBLISHER

6 JORNAL OF COMPTERS, VOL 8, NO 4, APRIL called conjugate soft subgroups (with respect to u ) and u we write, =, where u = ( u u) G Theorem 36 Suppose S ( G) Let NG ( ) = { G, ( y ) = ( y ), y G } Then N ( ) is a subgroup of G and the restriction of G to NG ( ), denoted by N ( ) G, is a normal soft subgroup of NG ( ) Proof Obviously e NG ( ) Let, G, for any z G, ( z ( y)) = (( z ) y) = ( yz ( )) = ( z ( y )) = (( z y ) ) = (( y) z) So NG ( ), it follows that NG ( ) is a subgroup of G Clearly, NG ( ) is a soft subgroup of NG ( ) For y, NG ( ), we have N ( ) ( y ) G = N ( ) ( y ) G, thus NG ( ) is a normal soft subgroup of NG ( ) The subgroup N ( ) of G defined in Theorem 36 is G called the normalizer of in G Theorem 37 Suppose be a soft subgroup of G Then u the cardinal number of the set { u G} is equal to the inde G: NG ( ) of the normalizer NG ( ) in G Proof Let, G Then y y = ( z) = ( z), z G ( z) = ( y zy), z G (( yz ) ) = ( z ( y)), z G NG ( ) NG( ) = yng( ) Thus ϕ : a N G ( ) is a bijection from { G} to { NG ( ) G} Therefore the theorem holds Definition 34 Let be a soft subgroup of G and G Then the left coset and the right coset of respect to, written as and respectively, is defined as follows: : y a ( y) = ( y) G : y a ( y) = ( y ) G Proposition3 Let be a soft subgroup of G, then A = A, A = A Proof For all a X, we have g A ( a) a ( g) = ( g) g A ( a) g A ( a) Hence A = A Similarly, A = A Theorem 38 Let be a soft subgroup of G Then the following assertions are equivalent: () is a normal soft subgroup of G ; () = G ; (3) = G ; (4) NG ( ) = G Proof () () G, ( y) = ( y) = ( y), y G, thus = G () (3) G, ( y) = ( y) = ( y ) = ( ( y ) ) = ( y ) = ( y ) = ( )( y) y G, thus = G (3) (), G, ( y) = ( y) = ( y) = ( y ) = ( y), thus is a normal soft subgroup of G () (4) By Theorem 37, we have = G G: NG( ) = NG( ) = G According to Theorem 38, we know that if is a normal soft subgroup of G, then the left coset is just the right coset Thus in this case, we call a coset for short Theorem 39 Let be a soft subgroup of G Then is a normal soft subgroup of G if and only if ([, y]) G, where [, y ] is a commutator of G Proof Suppose is a normal soft subgroup of G Then for G, ([ y, ]) = ( y y) = ( ( y y)) ( ) I ( y y) = I = Conversely, suppose ([, y]) G Then ( y) = ( yy y) = ( yy ( y)) ( y) I ([ y, ]) ( y) I ( y) = ( y) G Therefore is a normal soft subgroup of G Corollary3 Let be a soft subgroup of G and ([, y]) = ( e) G, then is a normal soft subgroup of G Proposition3 Let be a normal soft subgroup of G Then = { = ( e), G} = I A is a X normal subgroup of G ; () = y if and only if = y for all, G Proof () Clearly, e y,, ( y) ( ) I( y) = I ( y) = ( e) 03 ACADEMY PBLISHER

7 966 JORNAL OF COMPTERS, VOL 8, NO 4, APRIL 03 So ( y) = ( e) It follows that Gy,, ( y) = ( y) = ( e) It follows that y So is a normal subgroup of G Furthermore, we have g ( g) = ( e) = X X, ( g) X, g A g I A Thus = I A X X () Suppose that = y Then ( z) = ( y z) z GChoosing z = y yields ( y) = ( y y) = () e and thus Hence = y Conversely, suppose that = y Then and y Hence ( z) = ( yy z) = (( y)( y z)) ( y) I ( y z) = () e I ( y z) = ( y z) z G Similarly, we have ( y z) ( z) z G Therefore ( z) = ( y z) z G, which shows that = y Theorem 30 Suppose is a normal soft subgroup of G, let G/ = { G} Then the following assertions hold: () o ( y) = ( y) G ; () ( G /, o ) is a group; (3) G/ G/, where = { = ( e), G} ; (4) Define ( ) ( by ) = G, then ( ) is a normal soft subgroup of G / Proof () Let, G Then we have [ o( y)]( z) = (( u) I( y)( u z)) = u G ( ( u) I ( y u z)) u G ( ) I ( y z) = ( )( e) I ( )( y z) = ( y z) = (( y) z) = (( y) )( z) z G On the other hand, (( y) )( z) = (( y) z) = ( y z) = (( y z)( y y)) = ( y ( zy ) y) = ( zy ) = ( ( uu ) zy ) = (( u)( u zy )) ( u) I ( u zy ) = ( u) I ( y u z) = ( u) I ( y)( u z) u G It follows that (( y) )( z) (( u) I ( y)( u z)) u G = [ o ( y)]( z) Thus o ( y) = ( y) () By (), the binary operation o on G / is closed Also, o satisfies the associative low Now o = ( e) o = ( e) = G and ( ) o = ( ) = e = G Hence ( G /, o ) is a group (3) Since is a normal soft subgroup of G, it follows by Proposition 3 that is a normal subgroup of G Hence G / is a group and the mapping f : G/ G/ given by a G is an isomorphism Therefore G/ G/ (4) Let, G Suppose = y, then = ( y y) = (( y ) y) ( y ) I ( y) = ( y) I( y) = I( y) = ( ) I( y) = ( e) I( y) = ( y) Similarly, ( y) Hence = ( y), that is ( ) = ( y) So is single-valued Since G ( ) ( ) ( ) = ( ) = ( ) ( ) = = and ( ) ( ) ( o ( y)) = (( y) ) = ( y) I ( y) ( ) ( ) = I ( y) G, it follows that ( ) is a soft subgroup of G / Moreover ( o( y)) = (( y) ) = ( y) = ( y) = (( y) ) = (( y) o) 03 ACADEMY PBLISHER

8 JORNAL OF COMPTERS, VOL 8, NO 4, APRIL G, we have that of G / () IV CONCLSION is a normal soft subgroup In this paper, we review the basic concepts of soft sets and give a detailed theoretical study on soft sets We introduce the concepts of soft groups which are different from the concepts in previous papers This work focuses on soft subgroups, normal soft subgroups and investigates the algebraic properties of soft sets in a group structure To etend this work, one could study the theory of soft subgroups compare with the classical group theory ACKNOWLEDGMENT This work was supported by grants from the National Natural Science Foundation of China (No 0009), the Education Science Foundation of Jiangsu Province during the -th Five-Year Plan Period(D/006/0/7) REFERENCES [] L A Zadeh, Fuzzy sets, Inf Control, Vol 8, pp , 965 [] X H Yuan, C Zhang, Y H Ren, Generalized fuzzy groups and many-valued Implications, Fuzzy Sets and Systems, Vol 38, pp 05-, 003 [3] Z Pawlak, Rough sets, Int J Inf Comput Sci, Vol, pp , 98 [4] Z Pawlak, A Skowron, Rudiments of rough sets, Inf Sci, Vol 77, pp 3-7, 007 [5] Y Q Zhang, X B Yang, Intuitionistic fuzzy dominance based rough set approach: model and attribute reductions, Journal of Software, Vol 7, NO 3, pp , 0 [6] M L Yan, Dominance based Rough Interval valued Fuzzy Set in Incomplete Fuzzy Information System, Journal of Software, Vol 7, NO 6, pp , 0 [7] W L Gau, D J Buehrer, Vague sets, IEEE Trans Syst Man Cybern, Vol 3, No, pp 60-64, 993 [8] M B Gorzalzany, A method of inference in approimate reasoning based on interval-valued fuzzy sets, Fuzzy Sets and Systems, Vol, pp -7, 987 [9] D Molodtsov, Soft set theory-first results, Comput Math Appl, Vol 37, pp 9-3, 999 [0] P K Maji, A R Roy, R Biswas, An application of soft sets in a decision making problem, Comput Math Appl, Vol 44, pp , 00 [] A R Roy, P K Maji, A fuzzy soft set theoretic approach to decision making problems, J Comput Appl Math, Vol 03, pp 4-48, 007 [] D Chen, E C C Tsang, D S Yeung, X Wang, The parameter reduction of soft sets and its applications, Comput Math Appl, Vol 49, pp , 005 [3] Z Kong, L Gao, L Wang, S Li, The parameter reduction of soft sets and its algorithm, Comput Math Appl, Vol 56, No, pp , 008 [4] P K Maji, R Biswas, A R Roy, Soft set theory, Comput Math Appl, Vol 45, , 003 [5] M I Ali, F Feng, X Liu, W K Min, On some new operations in soft set theory, Comput Math Appl, Vol 57, No 9, pp , 009 [6] Y B Jun, Soft BCK/BCI-algebras, Comput Math Appl, Vol 56, pp , 008 [7] Y B Jun, C H Park, Applications of soft sets in ideal theory of BCK/BCI-algebras, Inf Sci, Vol 78, pp , 008 [8] Y B Jun, K J Lee, J Zhan, Soft p-ideals of soft BCIalgebras, Comput Math Appl, Vol 58, pp , 009 [9] H Aktas, N Cagman, Soft sets and soft groups, Inf Sci, Vol 77, pp , 007 [0] A Sezgin, A O Atagün, Soft groups and normalistic soft groups, Comput Math Appl, Vol 6, pp , 0 [] A O Atagün, A Sezgin, Soft substructures of rings, fields and modules, Comput Math Appl, Vol 6, pp 59-60, 0 [] A Sezgin, A O Atagün, On operations of soft sets, Comput Math Appl, Vol 6, pp , 0 [3] J H Park, O H Kima, Y C Kwunb, Some properties of equivalence soft set relations, Comput Math Appl, Vol 63, pp , 0 [4] A Rosenfeld, Fuzzy groups, J Math Anal Appl, Vol 35, pp 5-57, 97 [5] Z H Liao, L H Yi, M H Hu, (, q( λ, ))-fuzzy k- ideals of semigroups, J Math, Vol 3, pp 9-05, 0 [6] R Biswas, S Nanda, Rough groups and rough subgroups, Bull Polish Acad Math, Vol 4, pp 5-54, 994 [7] Y C Wen, Study on soft sets, Liaoning: Liaoning Normal niversity, 008(in Chinese) Xia Yin was born in Wui, Jiangsu Province She received her MSc degree in Applied Mathematics from School of Mathematical Sciences of Soochow niversity, Suzhou, China, in 006 She received her BSc degree in Basic Mathematics from Nanjing Normal niversity, Nanjing, China, in 997 Her research interests include finite group theory, fuzzy algebra and rough sets Zuhua Liao was born in Nanchang, Jiangi Province He is now a Professor in Department of Mathematics, School of Science, Jiangnan niversity, Wui, China With more than 3 decades research and teaching eperience, he authored more than 00 papers in various national and international journals and produced about 0 masters in applied mathematics His research interests are in the areas of fuzzy algebra, rough sets, theory of generalized inverses and artificial intelligence 03 ACADEMY PBLISHER