A Novel Approach: Soft Groups

Transcription

1 International Journal of lgebra, Vol 9, 2015, no 2, HIKRI Ltd, wwwm-hikaricom Novel pproach: Soft Groups K Moinuddin Faculty of Mathematics, Maulana zad National Urdu University Gachibowli, Hyderabad, India D Madhusudana Rao Head & Faculty of Mathematics, VSR & NVR College Tenali, P, India Copyright 2014 K Moinuddin and D Madhusudana Rao This is an open access article distributed under the Creative Commons ttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited bstract In this paper, we construct a soft group and soft subgroups by assigning a group structure to the universe set X and investigate some interesting results in this context Keywords: Soft set, Soft Group, Soft Subgroup, Soft normal subgroup, Soft cyclic subgroup Introduction In 1999, Molodtsov [5] introduced the soft set theory as a general mathematical tool for dealing with uncertainty or vagueness There are many theories like Fuzzy set theory and Rough set theory to tackle the problem of imperfect knowledge Soft set theory is still a better approach to deal with problems involving uncertainty Molodtsov recognized the importance of the role of parameters and introduced the theory of Soft sets He has shown several applications of this theory in many fields like economics, engineering, medical sciences, etc Later, this theory became a very good source of research for many mathematicians and computer scientists of recent years because of its wide range of applicability The development in the fields of soft set theory and its application has been taking place in a rapid pace The notion of soft groups is a branch of soft set theory and it was first introduce by ktas and Cagman [1] in 2007 Later, some authors like Yin and Liao

2 80 K Moinuddin and D Madhusudana Rao [6] have studied various properties of soft groups The soft group theory has applications in the theory of computer science In this present work, we assign a group structure to the universe set X and introduce some soft group theoretical concepts like soft subgroup, soft normal subgroup, soft cyclic subgroup etc, in a different approach We investigate some interesting results in this context 1 Preliminaries In this section, we present some basic definitions which are needed in further Study of this paper Let U be an initial universe set and U (or simply ) be a collection of all possible parameters with respect tou, where parameters are the characteristics or properties of objects inu Let PU ( ) be the collection of all subsets ofu Definition 11: pair ( F, ) is said to be a soft set overu, if and F: PU ( ) We write F for ( F, ) Definition 12: Let F be soft sets over a common universe set U and, Then we say that (a) F Is a soft subset ofg, if (i) and (ii) Fa ( ) Ga ( ) a It is denoted by F G, (b) F quals tog, if F G F It is denoted as F = G In set theory, there is a unique empty (or null set), but in soft sets, null soft set and empty soft sets are different concepts, hence we have the following Definition 13: soft set F over U is said to be a null soft set if a, Fa ( ) = and it is denoted by Definition 14: Let U be an initial universe set, be the universe set of parameters and We shall denote by U the unique soft set over U with an empty parameter set which is called the empty soft set over U Definition 15: soft set F over U is said to be an absolute soft set ~ if a, Fa ( ) = U, and denoted by G Definition 16: The union of two soft sets F over a common universe U is the soft set H C, where C =, and for all a C, F( a) if a H ( a) = G( a) if a F( a) G( a) if a We write F G = H C Definition 17: The intersection of two soft sets F over a Common Universe U is the soft set H, where C =, and for all a C, C

3 novel approach: soft groups 81 Ha ( ) = Fa ( ) Ga ( ) We write F G = H C Definition 18: For a soft set F overu, the relative complement of F is c defined as F= F, where F : PU ( ) is a mapping given by F ( a) = U Fa ( ) for all a and is denoted by F c Definition 19: If F are two soft sets overu, then F ND G is defined as a soft set ( H, ) where H ( αβ, ) = F( α) G( β), ( αβ, ) It is denoted by F G Definition 110: If F are two soft sets overu, then F JOIN G is defined as ( H, ) where H ( αβ, ) = F( α) G( β), ( αβ, ) and is denoted by F G Definition 111: Suppose a binary operation denoted by, is defined for all subsets of U Let F be two soft sets over U Then the operation for the soft sets is defined in the following way: F H, H αβ, = F α G β, α and β G =( ) Where ( ) ( ) ( ) 2 Soft Groups and Related Concepts: In this section, we introduce the notions of soft subgroups, soft normal subgroups and then we define soft group These definitions are different from the concepts introduced by ktas and Cagman in their research article [1] In what follows G and PG ( ) stand for a group and the collection of all subsets of G Let be the set of parameters We denote the identity element in G by the symbol e G or simply e Definition 21: We say that a soft set F is a soft subgroup over G if Fa ( ) is a subgroup of G for every a G 1, 1, i, i H = 1, 1 Then G forms an xample 22: Let = { } and { } abelian group with respect to complex multiplication, H is a subgroup of G and any set of parameters with respect to G Then F is said to be soft subgroup over G provided F: PG ( ) and is defined by Fa ( ) = H for every a Definition 23: We define the identity soft subgroup over G to be a soft set I: PG ( ), such that Ia ( ) = { eg} for every a Definition 24: soft set F is said to be a soft normal subgroup over G if Fa ( ) is a normal subgroup of G for every a Definition 25: Let SG ( ) be the collection of all soft subgroups overg That is SG ( ) the collection of all soft sets F, then we say that SG ( ) is a soft group

4 82 K Moinuddin and D Madhusudana Rao over G if F: PG ( ) is defined by Fa ( ) is a subgroup of G for every a Definition 26: We define an absolute soft subgroup to be a soft set F: PG ( ) such that Fa ( ) = G for every a We denote it byg Remark 27: We call the soft sets I andg, improper soft subgroups overg Proposition 28: The intersection of any two soft subgroups over G is also a soft subgroup overg Proposition 29: The intersection of any two soft normal subgroups over G is also a soft normal subgroup overg Proposition 210: The union of two soft subgroups is a soft subgroup over G if one is contained in the other Remark 211: One can notice that sometimes the union of two soft subgroups can be a soft subgroup without the condition as mentioned in the above proposition 210 Hence the condition is sufficient but not necessary It can be observed from the following example xample 212: Let G be a group and e 0, a fixed parameter in Let F: PG ( ) and F: PG ( ) 1 2 be two soft subgroup over G defined by { e G } if e = e0 H if e = e0 F1 = and F2 = H if e e0 { eg} if e e0 Where H is a proper subgroup of G Then F 1 F 2 F 3 and F: PG ( ) 3 is given by F () e H 3 = for every e Proposition 213: The union of two soft normal subgroups is a soft normal subgroup over G if one is contained in the other Proposition 214: very soft subgroup over an abelian group is soft normal Definition 215: We say that a soft set F is a soft cyclic subgroup over G if Fe () is a cyclic subgroup of G for every e If ae () G is a generator of Fe () then we say that the soft set F : ( ) ( ) PG defined by { } Fae ( )() e = ae () is a soft generator of F Proposition 216: very soft subgroup over a cyclic group is soft cyclic ae = 3 Order of a Soft Group In this section we introduce the notation of the order of a soft group Definition 31: Let G be a finite group Then the number of elements of G is called the order of G and it is denoted by G

5 novel approach: soft groups 83 Definition 32: Let G be a finite group The order of a soft subgroup F: PG ( ) is defined by F = min { Fe ( ) : e } where Fe () is the number of elements in Fe () Definition 33: We define the order of SG ( ) as follows SG ( ) = max F : F SG ( ) { } xample 34: Consider the finite abelian group G { 1, 1, i, i} multiplication Then { 1, 1} {1} if e = e0 by Fe () = H if e e F = min{{1}, H} = min{1, 2} = 1 Define φ : PG ( ) by then φ = min{ H, G} = min{2,4} = 2 Proposition 35: SG ( ) = under complex H = is a subgroup of G Define F: PG ( ) 0 where e0 is a fixed parameter Then = G H if e = e0 φ() e = G if e e Proposition 36: If G is a finite group and if then F divides SG ( ) 0 F is any soft subgroup over G References [1] H ktas and N Cagman, Soft sets and Soft groups, Inf Sci, 177(2007), , [2] I N Herstein, Topics in lgebra, 2 nd edition, Wiley & Sons, New York, 975 [3] F Li, Notes on the Soft Operations, rpn Journal of System and Software, 66 (2011), [4] P K Maji, R iswas and R Roy, Soft set theory, Comp Math ppl, 45 (2008), [5] D Molodtsov, Soft Set Theory- First Results, Comp Math ppl, 37 (1999) [6] X Yin and Z Liao, Study on Soft Groups, Journal of Computers, 8 (2013), Received: December 27, 2014; Published: February 20, 2015