research article cyclic soft groups and their applications...

Research ArticleCyclic Soft Groups and Their Applications on Groups

HacJ AktaG1 and Ferif Oumlzluuml2

1 Department of Mathematics Faculty of Science Erciyes University 38039 Kayseri Turkey2 Kilis 7 Aralık University 79000 Kilis Turkey

Correspondence should be addressed to Hacı Aktas haktaserciyesedutr

Received 20 May 2014 Accepted 30 June 2014 Published 15 July 2014

Academic Editor Naim Cagman

Copyright copy 2014 H Aktas and S OzluThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In crisp environment the notions of order of group and cyclic group are well known due to many applications In this paper weintroduce order of the soft groups power of the soft sets power of the soft groups and cyclic soft group on a group We alsoinvestigate the relationship between cyclic soft groups and classical groups

1 Introduction

Most of our real life problems in economics engineeringenvironment social science and medical science involveimprecise data that contain uncertainties To solve these kindsof problems it is quite difficult to successfully use classicalmethods However there are some well-known theories(probability fuzzy sets [1] vague sets [2] rough sets [3] etc)which can be considered as mathematical tools for dealingwith uncertainties All these theories have their inherentdifficulties pointed out by Molodtsov [4]

The theory of soft sets proposed by Moldtsov [4] is anextension of set theory for the study of intelligent systemscharacterized by insufficient and incomplete informationHis pioneer paper has undergone tremendous growth andapplications in the last few years Maji et al [5] give anapplication of soft set theory in a decisionmaking problem byusing the rough sets and they conducted a theoretical studyon soft sets in a detailed way [6] Chen et al [7] proposeda reasonable definition of parameterizations reduction ofsoft sets and compared them with the concept of attributesreduction in rough set theory

The algebraic structures of set theories which deal withuncertainties have been studied by some authors Rosenfeld[8] proposed fuzzy groups to establish results for the algebraicstructures of fuzzy sets Rough groups are defined by Biswasand Nanda [9] and some others (ie Bonikowaski [10]

and Iwinski [11]) studied algebraic properties of rough setsFuzzification of algebraic structures was studied by manyauthors [8 12 13]

Many papers on soft algebras have been published sinceAktas and Cagman [14] introduced the notion of a soft groupin 2007 Recently Jun et al [15] studied soft ideals and idealis-tic soft BCKBCI-algebras Acar et al [16] introduced initialconcepts of soft rings Aygunoglu and Aygun [17] introducedthe concept of fuzzy soft group and in the meantime theystudied its properties and structural characteristics Atagunand Sezgin [18] introduced and studied the concepts of softsubrings soft ideal of a ring and soft subfields of a field

Our interest in this paper is to define order of thesoft group and cyclic soft group by using definition of thesoft group that was defined in [14] We then find out therelationships between cyclic soft groups and classical groupsFinally we conclude the study with suggestions for futurework

2 Preliminaries

The following definitions and preliminaries are required inthe sequel of our work and they are presented in brief

Throughout this work 119880 is an initial universe set 119864 is aset of parameters 119875(119880) is the power set of 119880 119860 sub 119864 and 119866denotes a group with identity 119890

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 437324 5 pageshttpdxdoiorg1011552014437324

2 The Scientific World Journal

Definition 1 (see [4]) A pair (119865 119860) is called a soft set over 119880where 119865 is a mapping given by

119865 119860 997888rarr 119875 (119880) (1)

In other words a soft set over119880 is a parameterized familyof subsets of the universe 119880

Definition 2 (see [6]) For two soft sets (119865 119860) and (119866 119861) over119880 (119865 119860) is called a soft subset of (119866 119861) if

(1) 119860 sub 119861 and(2) forall120576 isin 119860 119865(120576) and 119866(120576) are identical approximations

It is denoted by (119865 119860)sub(119866 119861)(119865 119860) is called a soft superset of (119866 119861) if (119866 119861) is a soft

subset of (119865 119860) It is denoted by (119865 119860)sup(119866 119861)

Definition 3 (see [6]) If (119865 119860) and (119866 119861) are two soft setsthen (119865 119860) AND (119866 119861) is denoted (119865 119860) and (119866 119861) (119865 119860) and(119866 119861) is defined as (119867 119860 times 119861) where119867(120572 120573) = 119865(120572) cap119866(120573)for all (120572 120573) isin 119860 times 119861

Definition 4 (see [6]) If (119865 119860) and (119866 119861) are two soft setsthen (119865 119860) OR (119866 119861) denoted by (119865 119860) or (119866 119861) is definedby (119865 119860)or (119866 119861) = (119867119860times119861) where119867(120572 120573) = 119865(120572)cup119866(120573)forall(120572 120573) isin 119860 times 119861

Definition 5 (see [6]) Union of two soft sets of (119865 119860) and(119866 119861) over 119880 is the soft set (119867 119862) where 119862 = 119860 cup 119861 andforall119890 isin 119862

119867(119890) =

119865 (119890) if 119890 isin 119860 minus 119861119866 (119890) if 119890 isin 119861 minus 119860119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(2)

It is denoted by (119865 119860)cup(119866 119861) = (119867 119862)

Definition 6 (see [14]) Let (119865 119860) be a soft set over 119866 Then(119865 119860) is said to be a soft group over 119866 if and only if 119865(119909) issubgroup of 119866 for all 119909 isin 119860

Definition 7 (see [14]) One considers the following

(1) (119865 119860) is said to be an identity soft group over 119866 if119865(119909) = 119890 for all 119909 isin 119860 where 119890 is the identityelement of 119866

(2) (119865 119860) is said to be an absolute soft group over 119866 if119865(119909) = 119866 for all 119909 isin 119860

Definition 8 (see [14]) Let (119865 119860) and (119867119870) be two softgroups over 119866 Then (119867119870) is a soft subgroup of (119865 119860)written as (119867119870)lt(119865 119860) if

(1) 119870 sub 119860(2) 119867(119909) is a subgroup of 119865(119909) for all 119909 isin 119870

Definition 9 (see [14]) Let (119865 119860) and (119867 119861) be two softgroups over 119866 and 119870 respectively and let 119891 119866 rarr 119870 and119892 119860 rarr 119861 be two functions Then one says (119891 119892) is a soft

homomorphism and (119865 119860) is soft homomorphic to (119867 119861)denoted by (119865 119860) sim (119867 119861) if the following conditions aresatisfied

(1) 119891 is a homomorphism from 119866 onto119870(2) 119892 is a mapping from 119860 onto 119861(3) 119891(119865(119909)) = 119867(119892(119909)) for all 119909 isin 119860

In this definition if 119891 is an isomorphism from 119866 to 119870and 119892 is a one-to-one mapping from 119860 onto 119861 then we saythat (119891 119892) is a soft isomorphism and (119865 119860) is soft isomorphto (119867 119861) which is denoted by (119865 119860) ≃ (119867 119861) The imageof soft group (119865 119860) under soft homomorphism (119891 119892) will bedenoted by (119891(119865) 119892(119860))

Definition 10 (see [14]) Let (119865 119860) and (119867 119861) be two softgroups over119866 and119870 respectivelyThe product of soft groups(119865 119860) and (119867 119861) is defined as (119865 119860) times (119867 119861) = (119880 119860 times 119861)where 119880(119909 119910) = 119865(119909) times 119867(119910) for all (119909 119910) isin 119860 times 119861

3 The Order of Soft Groups

Since the elements of a cyclic group are the powers of theelement properties of cyclic groups are closely related tothe properties of the powers of an element In this paperwe define order of the soft group by using definition of thesoft group that was defined in [14] We then investigate theirproperties

Definition 11 A pair (119865 119860) is called surjective soft set over119880where 119865 is a surjective mapping given by 119865 119860 rarr 119875(119880)

Throughout this study 119866 denotes a group and the soft set(119865 119860) will be a surjective soft set The element 119865(119886) is usedinstead of the element (119886 119865(119886)) of (119865 119860)

Definition 12 Let (119865 119860) be a soft set over119866 and 119865(119909) isin (119865 119860)for 119909 isin 119860 Then 119865(119909)119899 = 119886119899 119886 isin 119865(119909) 119899 isin 119885 is called 119899-power of 119865(119909)

Example 13 Let (119865 119860) be a soft set over 1198783 where119860 = 119878

3 and

let(119865 119860) = 119865 (119890) = 119890 119865 (12) = 119890 (12)

119865 (13) = 119890 (13) 119865 (23) = 119890 (23)

119865 (123) = 119890 (123) (132)

(3)

be a soft set over group 1198783 And the third power of 119865(123) is

119865(123)

3

= 119890

3

(123)

3

(132)

3

= 119890 119890 119890 = 119890

Of course when the group is additive the 119899th power of119865(119909) will be written by 119899119865(119909) = 119899119886 119886 isin 119865(119909) 119899 isin 119885

Theorem 14 Let (119865 119860) be a soft set over 119866 and 119865(119909) 119865(119910) isin(119865 119860) for 119909 119910 isin 119860 Then for all 119899 isin 119885

(1) (119865(119909) cap 119865(119910))119899 sube 119865(119909)119899 cap 119865(119910)119899 for all 119899 isin 119885(2) (119865(119909) cup 119865(119910))119899 = 119865(119909)119899 cup 119865(119910)119899 for all 119899 isin 119885(3) (119865(119909) times 119865(119910))119899 = 119865(119909)119899 times 119865(119910)119899 for all 119899 isin 119885

The Scientific World Journal 3

Proof Let 119886119899 isin (119865(119909) cap 119865(119910))

119899 for 119899 isin 119885 FromDefinition 12 119886 isin 119865(119909)cap119865(119910) and 119886119899 isin 119865(119909)119899 and 119886119899 isin 119865(119910)119899Thismeans that 119886119899 isin 119865(119909)119899cap119865(119910)119899This completes the proofTheorem 14(2) andTheorem 14(3) can be proved similarly byusing Definition 12

In general the opposite of Theorem 14(1) is not true Weillustrate an example of this situation

Example 15 Let 119860 = 0 1 and let 119865 119860 rarr 119875(119885) bea function such that 119865(0) = 2119896 119896 isin 119885 and 119865(1) =2119896 + 1 119896 isin 119885 The intersection is 119865(0) cap 119865(1) = 0 So(119865(0) cap 119865(1))

2

= 0 On the other hand 119865(0)2 cap 119865(1)2 = 0Consequently (119865(119909) cap 119865(119910))119899 = 119865(119909)119899 cap 119865(119910)119899

Definition 16 Let (119865 119860) be a soft set over119866 and119865(119909) isin (119865 119860)If there is a positive integer 119899 such that 119865(119909)119899 = 119890 then theleast such positive integer 119899 is called the order of 119865(119909) If nosuch 119899 exists then 119865(119909) has infinite order The order of 119865(119909)is denoted by |119865(119909)|

If (119865 119860) is a soft group over 119866 then the order of 119865(119909) isin(119865 119860) coincides with the order of 119865(119909) which is subgroup of119866 Of course if there is any element 119909 in 119860 such that 119865(119909) =119890 then the order of 119865(119909) is 1

Example 17 In Example 13 the order of element 119865(123) is 3Let (119865119873) be a soft group over group of integer numbers

119885 where 119865 is a mapping from natural numbers 119873 to 119875(119885)such that 119865(119899) = 119899119885 for all 119899 isin 119873 There is no any positiveinteger119898 such that 119865(119899)119898 = 0 so 119865(119899) has infinite order forall 119899 isin 119873 minus 0

Theorem 18 Let119866 be finite group and (119865 119860) a soft group over119866 Then the orders of elements of (119865 119860) are finite

Proof It is straightforward

Theorem 19 Let (119865 119860) be a soft set over finite group 119866 and119865(119909) isin (119865 119860) for 119909 isin 119860 Then the order of 119865(119909) is the leastcommon multiple (LCM) of order of elements of 119865(119909)

Proof Let 119899 be the order of 119865(119909) Then 119865(119909)119899 = 119890 Thismeans that 119886119899 = 119890 for all 119886 isin 119865(119909) We know from classicalgroup theory that |119886| | 119899 namely 119886 divides 119899 for all 119886 isin119865(119909) Thus 119899 is common multiple of elements of 119865(119909) Let119898 be another commonmultiple of elements of 119865(119909)Then byreason of 119886119898 = 119890 for all 119886 isin 119865(119909)119865(119909)119898 = 119890 However since119899 is the least number that satisfies the condition 119865(119909)119899 = 119890hence 119899 | 119898 This completes the proof

Theorem 20 Let 119866 be finite group (119865 119860) a soft group over 119866and 119865(119909) and 119865(119910) the elements of (119865 119860) Then for all 119909 119910 isin119860 one has the following

(1) |119865(119909) cap 119865(119910)| le 119866119862119863(|119865(119909)| |119865(119910)|) for 119909 119910 isin 119860

(2) |119865(119909) cup 119865(119910)| = 119871119862119872(|119865(119909)| |119865(119910)|) for 119909 119910 isin 119860

(3) |119865(119909) times 119865(119910)| = |119865(119909)||119865(119910)| for 119909 119910 isin 119860

Proof We consider the following

(1) 119865(119909) cap 119865(119910) is subgroup of 119865(119909) and 119865(119910) so |119865(119909) cap119865(119910)| | |119865(119909)| and |119865(119909) cap 119865(119910)| | |119865(119910)| It follows|119865(119909) cap 119865(119910)| le 119866119862119863(|119865(119909)| |119865(119910)|)

(2) Let |119865(119909) cup 119865(119910)| = 119896 |119865(119909)| = 119898 and |119865(119910)| = 119899From Theorem 14(119865(119909) cup 119865(119910))119896 = 119865(119909)119896 cup 119865(119910)119896 =119890 This follows 119899 | 119896 and 119898 | 119896 Thus 119896 isa common multiple of 119898 and 119899 Let 119905 be anothercommon multiple of 119898 and 119899 Consider (119865(119909) cap119865(119910))

119905

= 119865(119909)

119905

cup 119865(119910)

119905

= 119890 cup 119890 = 119890 Since 119896 isthe least positive integer that satisfied the condition(119865(119909) cup 119865(119910))

119896

= 119890 119896 divides 119905 Hence 119896 is LCMorder of 119865(119909) and 119865(119910) This completes the proof

(3) Since 119865(119909) and 119865(119910) are subgroups of 119866 it is seeneasily

Definition 21 Let 119866 be group and (119865 119860) soft set over 119866 Theset

(119865 119860)

119899

= 119865(119909)

119899

119909 isin 119860 119899 isin 119885 (4)

is called 119899th power of soft set (119865 119860)

Example 22 Let (119865 119860) be a soft set over 1198783defined in

Example 13Then the second power of (119865 119860) is that (119865 119860)2 =119865(119890)

2

= 119865(119890) and 119865(12)2 = 119865(119890) 119865(13)2 = 119865(119890) 119865(23)2 =119865(119890) 119865(123)2 = 119865(132)

Theorem 23 Let (119865 119860) and (119864 119861) be two soft sets over 119866Then

(1) ((119865 119860) or (119864 119861))119899 = (119865 119860)119899 or (119864 119861)119899

(2) if119860 sube 119861 and for all 119886 isin 119860 119865(119886) and 119864(119886) are identicalapproximations then ((119865 119860) and (119864 119861))119899 sube (119865 119860)

119899

and

(119864 119861)

119899


(1) Suppose that (119865 119860)or(119864 119861) = (119867119860times119861) and (119865 119860)119899or(119864 119861)

119899

= (119879 119860 times119861) We can write ((119865 119860) or (119864 119861))119899 =(119867119860 times 119861)

119899 Using Definition 21 and Theorem 14 wehave

(119867 119860 times 119861)

119899

= 119867(119886 119887)

119899

(119886 119887) isin 119860 times 119861

= (119865 (119886) cup 119864 (119887))

119899

(119886 119887) isin 119860 times 119861

= 119865(119886)

119899

cup 119864(119887)

119899

(119886 119887) isin 119860 times 119861

= 119879 (119886 119887) (119886 119887) isin 119860 times 119861

= (119865 119860)

119899

or (119864 119861)

119899

(5)


(2) Suppose that (119865 119860)and(119864 119861) = (119867119860times119861) and (119865 119860)119899and(119864 119861)

119899

= (119879 119860times119861) Using the same arguments in (1)we have

(119867119860 times 119861)

119899

= 119867(119886 119887)

119899

(119886 119887) isin 119860 times 119861

= (119865 (119886) cap 119864 (119887))

119899

(119886 119887) isin 119860 times 119861

sube 119865(119886)

119899

cap 119864(119887)

119899

(119886 119887) isin 119860 times 119861

= 119879 (119886 119887) (119886 119887) isin 119860 times 119861

= (119865 119860)

119899

and (119864 119861)

119899

(6)

If (119865 119860) and (119864 119861) are both soft groups then for all 119886 119887 isin119860 119865(119886) and 119865(119887) are all subgroups of119866 and so 119865(119886) and 119864(119887)contain the identity element 119890 of 119866 Thus the set 119865(119886) cap 119864(119887)contains at least 119890 hence (119865(119886)cap119864(119887))119899 = 0 It means that if wetake inTheorem 23(2) (119865 119860) and (119864 119861) as soft groups not softsets then the extra condition can be added inTheorem 23(2)

In classical groups the order of a group is defined as thenumber of elements it contains But in soft groups it differsfrom classical groups

Definition 24 Let (119865 119860) be soft group over 119866 If 119866 is a finitegroup then the least common multiple of orders of elementsof (119865 119860) is called order of (119865 119860) If119866 is an infinite group thenthe order of (119865 119860) is defined to be the number of elements of(119865 119860) and the order of soft group (119865 119860) is denoted by |(119865 119860)|

Of course if (119865 119860) is surjective then the number ofelements of (119865 119860) is the number of elements in 119860

Example 25 In Example 13 the order of (119865 119860) is 6 and inExample 17 if we chose 119873 = 119873

4= 0 1 2 3 4 and 119865(119899) =

119899119885 for 119899 isin 1198734 then the order of soft group (119865119873

4) is 5

We give the following results similar to Lagrange Theo-rem in group theory

Theorem 26 Let (119865 119860) be a soft group over a finite group 119866and 119865(119909) isin (119865 119860) Then one has the following

(1) The order of 119865(119909) divides the order of (119865 119860) Inparticular 119865(119909)|(119865119860)| = 119890

(2) The order of (119865 119860) divides the order of 119866

When 119866 is a finite group then the order (119865 119860) is a finiteand the orders of elements of (119865 119860) divide the order of (119865 119860)However when119866 is an infinite group then the order of (119865 119860)can be finite or infinite Hence Theorem 26 is not true forinfinite groups

Theorem 27 Let 119866 be a finite group and let (119865 119860) and (119864 119861)be two soft groups over 119866 Then |(119865 119860) and (119864 119861)| le |(119865 119860)| and|(119865 119860) and (119864 119861)| le |(119864 119861)|

Proof Suppose that (119865 119860)and(119864 119861) = (119867 119862) where119862 = 119860times119861Using fundamental theorems and definitions in group theoryand Definitions 24 and 3 we have|(119865 119860) and (119864 119861)| = LCM (119867 (119886

119894 119887

119894)) for (119886

119894 119887

119894) isin 119860 times 119861

= LCM (119865 (119886119894) cap 119864 (119887

119894))

le LCM (119865 (119886119894)) = |(119865 119860)|

(7)

The other inequality is shown similarly

4 Cyclic Soft Groups

The class of cyclic groups is an important class in grouptheory In this section we study soft groups which aregenerated by one element of119875(119866)We define cyclic soft groupand prove some of their properties which are analogous to thecrisp case

Definition 28 Let (119865 119860) be a soft group over 119866 and 119883 anelement of 119875(119866) The set (119886 ⟨119909⟩) 119865(119886) = ⟨119909⟩ 119909 isin 119883

is called a soft subset of (119865 119860) generated by the set 119883 anddenoted by ⟨119883⟩ If (119865 119860) = ⟨119883⟩ then the soft group (119865 119860)is called the cyclic soft group generated by119883

If (119865 119860) is a cyclic soft group over 119866 then we can writeit in this form (119865 119860) = 119865(119886) = ⟨119909⟩ 119886 isin 119860 119909 isin 119866 where119909 isin 119866 is element of 119875(119866)That is to say if all the elements of(119865 119860) are generated by any elements of119883 of119875(119866) then (119865 119860)is a cyclic soft group over 119866

If119866 is a cyclic group then (119865 119860) is a soft cyclic group over119866 since all subgroups of cyclic group are cyclic but the reverseis not always true

As an illustration let us consider the following example

Example 29 Let 119866 = 119878

3be the symmetric group and

119860 = 119890 (12) (13) (23) (123) the set of parameters If weconstruct a soft set (119865 119860) over 119866 such that 119865(119909) = 119910 isin 119866 119910 = 119909

119899

119899 isin 119885 for all 119909 isin 119860 then one can easily show that(119865 119860) is a soft cyclic group over 119866 however 119866 is not a cyclicgroup

In the following theorem we can give some propertiesof cyclic soft groups that has similar features of the classicalcyclic groups

Theorem 30 One considers the following(1) If (119865 119860) is a finite cyclic soft group generated by119883 then|(119865 119860)| = 119871119862119872(|119909

119894|) where 119909

119894isin 119883

(2) If (119865 119860) is an infinite cyclic soft group generated by 119883then |(119865 119860)| = |119883|

(3) If (119865 119860) is an identity soft group then it is a cyclic softgroup generated by 119890

(4) Let (119865 119860) be an absolute soft group defined on119866Then(119865 119860) is a cyclic soft group if and only if 119866 is a cyclicgroup

(5) Let (119865 119860) be a soft group on119866 If the order of119866 is primethen (119865 119860) is a cyclic soft group

(6) A soft subgroup of a cyclic soft group is cyclic soft group


Proof It is easily seen from Definitions 24 7 and 8

Theorem 31 Let (119891 119892) be a soft homomorphism of the softgroup (119865 119860) over 119866 into the soft group (119867 119861) over 119870 If (119865 119860)is a cyclic soft group over 119866 then (119891(119865) 119892(119860)) is a cyclic softgroup over 119870

Proof First we show that (119891(119865) 119892(119860)) is a soft group over119870Since119891 is a homomorphism from119866 to119870119891(119865(119909)) = 119867(119892(119909))is a subgroup of119870 for all 119892(119909) isin 119892(119860) Thus (119891(119865) 119892(119860)) is asoft group over119870 Since 119865(119909) is cyclic subgroup for all 119909 in119860image of 119865(119909) under 119891 is cyclic that is 119891(119865(119909)) = 119867(119892(119909))is cyclic subgroup of 119870 for all 119892(119909) isin 119892(119860) Consequently(119891(119865) 119892(119860)) is a cyclic soft group over 119870

Theorem 32 Let (119865 119860) and (119867 119861) be two soft isomorphic softgroups over119866 and119870 respectively If (119865 119860) is a cyclic soft groupthen so is (119867 119861)

Proof First of all note that since (119865 119860) is a soft isomorphic to(119867 119861) there is an 119891 homomorphism from 119866 to 119870 such that119891(119865(119909)) = 119867(119892(119909)) for all 119909 isin 119860 where 119892 is a one-to-onemapping from 119860 onto 119861 Since 119865(119909) is a cyclic subgroup forall 119909 in119860 and 119891 is a homomorphism then119867(119892(119909)) is a cyclicsubgroup of 119870 Thus all elements of (119867 119861) are cyclic Thisresult completes the proof

Theorem 33 If (119865 119860) and (119867 119861) are two cyclic soft groupsover 119866 then (119865 119860) and (119867 119861) is a cyclic soft group over 119866

Proof Let (119865 119860) and (119867 119861) = (119864 119860 times 119861) where 119864(120572 120573) =119865(120572) cap119867(120573) forall(120572 120573) isin 119860 times 119861 Since 119865(120572) and119867(120573) are cyclicsubgroups of 119866 for all 120572 isin 119860 and 120573 isin 119861 and 119865(120572) cap 119867(120573)is a subgroup of both 119865(120572) and 119867(120573) 119865(120572) cap 119867(120573) is cyclicsubgroup of 119866 for all (120572 120573) isin 119860 times 119861 Hence (119867119860 times 119861) iscyclic soft group over 119866

Theorem 34 Let (119865 119860) and (119867 119861) be two cyclic soft groupsover119866 and119860cap119861 = 0Then (119865 119860)cup(119867 119861) is a cyclic soft groupover 119866

Proof It is trivial

Theorem 35 Let (119865 119860) and (119867 119861) be two cyclic soft groups offinite orders119898 and 119899 over 119866 and119870 respectively If119898 and 119899 arerelatively prime then the product (119865 119860)times (119867 119861) is a cyclic softgroup

Proof Let (119898 119899) = 1 According to LagrangeTheorem |119865(119909)|divides 119898 and |119867(119910)| divides 119899 for all 119909 isin 119860 and for all 119910 isin119861 Since (119898 119899) = 1 |119865(119909)| and |119867(119910)| are relatively primeSo 119865(119909) times 119867(119910) is cyclic group for all (119909 119910) isin 119860 times 119861 Thiscompletes the proof

5 Conclusion

In this paper we have expanded the soft set theory Wehave focused on order of the soft groups and investigatedrelationship between the order of soft groups and the order ofclassical groups Additionally we have studied the algebraic

properties of cyclic soft groups with respect to a groupstructure Our future work will focus on the relationshipsbetween cyclic soft groups and other algebraic structures suchas rings and fields

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[3] Z Pawlak ldquoRough setsrdquo International Journal of Computer andInformation Sciences vol 11 no 5 pp 341ndash356 1982

[4] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[5] P K Maji A R Roy and R Biswas ldquoAn application of soft setsin a decision making problemrdquo Computers ampMathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[6] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[7] D Chen E C C Tsang D S Yeung and X Wang ldquoTheparameterization reduction of soft sets and its applicationsrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 757ndash763 2005

[8] A Rosenfeld ldquoFuzzy groupsrdquo Journal of Mathematical Analysisand Applications vol 35 pp 512ndash517 1971

[9] R Biswas and S Nanda ldquoRough groups and rough subgroupsrdquoBulletin of the Polish Academy of Sciences Mathematics vol 42no 3 pp 251ndash254 1994

[10] Z Bonikowski ldquoAlgebraic structures of rough setsrdquo in RoughSets Fuzzy Sets and Knowledge Discovery W P Ziarko Ed pp242ndash247 Springer Berlin Germany 1994

[11] T B Iwinski ldquoAlgebraic approach to rough setsrdquo Bulletin of thePolish Academy of Sciences Mathematics vol 35 no 9-10 pp673ndash683 1987

[12] S Abdullah M Aslam T A Khan and M Naeem ldquoA newtype of fuzzy normal subgroups and fuzzy cosetsrdquo Journal ofIntelligent amp Fuzzy Systems vol 25 no 1 pp 37ndash47 2013

[13] S Abdullah M Aslam T A Khan and M Naeem ldquoA noteon ordered semigroups characterized by their ((120598 120598119881119902))rdquo UPBScientific Bulletin A vol 75 pp 41ndash44 2013

[14] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[15] Y B Jun andCH Park ldquoApplications of soft sets in ideal theoryof BCKBCI-algebrasrdquo Information Sciences vol 178 no 11 pp2466ndash2475 2008

[16] U Acar F Koyuncu and B Tanay ldquoSoft sets and soft ringsrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3458ndash3463 2010

[17] A Aygunoglu and H Aygun ldquoIntroduction to fuzzy softgroupsrdquo Computers and Mathematics with Applications vol 58no 6 pp 1279ndash1286 2009

[18] A O Atagun and A Sezgin ldquoSoft substructures of rings fieldsandmodulesrdquoComputers ampMathematics with Applications vol61 no 3 pp 592ndash601 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Definition 1 (see [4]) A pair (119865 119860) is called a soft set over 119880where 119865 is a mapping given by

119865 119860 997888rarr 119875 (119880) (1)

In other words a soft set over119880 is a parameterized familyof subsets of the universe 119880

Definition 2 (see [6]) For two soft sets (119865 119860) and (119866 119861) over119880 (119865 119860) is called a soft subset of (119866 119861) if

(1) 119860 sub 119861 and(2) forall120576 isin 119860 119865(120576) and 119866(120576) are identical approximations

It is denoted by (119865 119860)sub(119866 119861)(119865 119860) is called a soft superset of (119866 119861) if (119866 119861) is a soft

subset of (119865 119860) It is denoted by (119865 119860)sup(119866 119861)

Definition 3 (see [6]) If (119865 119860) and (119866 119861) are two soft setsthen (119865 119860) AND (119866 119861) is denoted (119865 119860) and (119866 119861) (119865 119860) and(119866 119861) is defined as (119867 119860 times 119861) where119867(120572 120573) = 119865(120572) cap119866(120573)for all (120572 120573) isin 119860 times 119861

Definition 4 (see [6]) If (119865 119860) and (119866 119861) are two soft setsthen (119865 119860) OR (119866 119861) denoted by (119865 119860) or (119866 119861) is definedby (119865 119860)or (119866 119861) = (119867119860times119861) where119867(120572 120573) = 119865(120572)cup119866(120573)forall(120572 120573) isin 119860 times 119861

Definition 5 (see [6]) Union of two soft sets of (119865 119860) and(119866 119861) over 119880 is the soft set (119867 119862) where 119862 = 119860 cup 119861 andforall119890 isin 119862

119867(119890) =

119865 (119890) if 119890 isin 119860 minus 119861119866 (119890) if 119890 isin 119861 minus 119860119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(2)

It is denoted by (119865 119860)cup(119866 119861) = (119867 119862)

Definition 6 (see [14]) Let (119865 119860) be a soft set over 119866 Then(119865 119860) is said to be a soft group over 119866 if and only if 119865(119909) issubgroup of 119866 for all 119909 isin 119860

Definition 7 (see [14]) One considers the following

(1) (119865 119860) is said to be an identity soft group over 119866 if119865(119909) = 119890 for all 119909 isin 119860 where 119890 is the identityelement of 119866

(2) (119865 119860) is said to be an absolute soft group over 119866 if119865(119909) = 119866 for all 119909 isin 119860

Definition 8 (see [14]) Let (119865 119860) and (119867119870) be two softgroups over 119866 Then (119867119870) is a soft subgroup of (119865 119860)written as (119867119870)lt(119865 119860) if

(1) 119870 sub 119860(2) 119867(119909) is a subgroup of 119865(119909) for all 119909 isin 119870

Definition 9 (see [14]) Let (119865 119860) and (119867 119861) be two softgroups over 119866 and 119870 respectively and let 119891 119866 rarr 119870 and119892 119860 rarr 119861 be two functions Then one says (119891 119892) is a soft

homomorphism and (119865 119860) is soft homomorphic to (119867 119861)denoted by (119865 119860) sim (119867 119861) if the following conditions aresatisfied

(1) 119891 is a homomorphism from 119866 onto119870(2) 119892 is a mapping from 119860 onto 119861(3) 119891(119865(119909)) = 119867(119892(119909)) for all 119909 isin 119860

In this definition if 119891 is an isomorphism from 119866 to 119870and 119892 is a one-to-one mapping from 119860 onto 119861 then we saythat (119891 119892) is a soft isomorphism and (119865 119860) is soft isomorphto (119867 119861) which is denoted by (119865 119860) ≃ (119867 119861) The imageof soft group (119865 119860) under soft homomorphism (119891 119892) will bedenoted by (119891(119865) 119892(119860))

Definition 10 (see [14]) Let (119865 119860) and (119867 119861) be two softgroups over119866 and119870 respectivelyThe product of soft groups(119865 119860) and (119867 119861) is defined as (119865 119860) times (119867 119861) = (119880 119860 times 119861)where 119880(119909 119910) = 119865(119909) times 119867(119910) for all (119909 119910) isin 119860 times 119861

3 The Order of Soft Groups

Since the elements of a cyclic group are the powers of theelement properties of cyclic groups are closely related tothe properties of the powers of an element In this paperwe define order of the soft group by using definition of thesoft group that was defined in [14] We then investigate theirproperties

Definition 11 A pair (119865 119860) is called surjective soft set over119880where 119865 is a surjective mapping given by 119865 119860 rarr 119875(119880)

Throughout this study 119866 denotes a group and the soft set(119865 119860) will be a surjective soft set The element 119865(119886) is usedinstead of the element (119886 119865(119886)) of (119865 119860)

Definition 12 Let (119865 119860) be a soft set over119866 and 119865(119909) isin (119865 119860)for 119909 isin 119860 Then 119865(119909)119899 = 119886119899 119886 isin 119865(119909) 119899 isin 119885 is called 119899-power of 119865(119909)

Example 13 Let (119865 119860) be a soft set over 1198783 where119860 = 119878

3 and

let(119865 119860) = 119865 (119890) = 119890 119865 (12) = 119890 (12)

119865 (13) = 119890 (13) 119865 (23) = 119890 (23)

119865 (123) = 119890 (123) (132)

(3)

be a soft set over group 1198783 And the third power of 119865(123) is

119865(123)

3

= 119890

3

(123)

3

(132)

3

= 119890 119890 119890 = 119890

Of course when the group is additive the 119899th power of119865(119909) will be written by 119899119865(119909) = 119899119886 119886 isin 119865(119909) 119899 isin 119885

Theorem 14 Let (119865 119860) be a soft set over 119866 and 119865(119909) 119865(119910) isin(119865 119860) for 119909 119910 isin 119860 Then for all 119899 isin 119885

(1) (119865(119909) cap 119865(119910))119899 sube 119865(119909)119899 cap 119865(119910)119899 for all 119899 isin 119885(2) (119865(119909) cup 119865(119910))119899 = 119865(119909)119899 cup 119865(119910)119899 for all 119899 isin 119885(3) (119865(119909) times 119865(119910))119899 = 119865(119909)119899 times 119865(119910)119899 for all 119899 isin 119885






2











(1) |119865(119909) cap 119865(119910)| le 119866119862119863(|119865(119909)| |119865(119910)|) for 119909 119910 isin 119860

(2) |119865(119909) cup 119865(119910)| = 119871119862119872(|119865(119909)| |119865(119910)|) for 119909 119910 isin 119860

(3) |119865(119909) times 119865(119910)| = |119865(119909)||119865(119910)| for 119909 119910 isin 119860




119905

= 119865(119909)

119905

cup 119865(119910)

119905


119896




(119865 119860)

119899

= 119865(119909)

119899

119909 isin 119860 119899 isin 119885 (4)




2

= 119865(119890) and 119865(12)2 = 119865(119890) 119865(13)2 = 119865(119890) 119865(23)2 =119865(119890) 119865(123)2 = 119865(132)


(1) ((119865 119860) or (119864 119861))119899 = (119865 119860)119899 or (119864 119861)119899


119899

and

(119864 119861)

119899



119899



(119867 119860 times 119861)

119899

= 119867(119886 119887)

119899

(119886 119887) isin 119860 times 119861

= (119865 (119886) cup 119864 (119887))

119899

(119886 119887) isin 119860 times 119861

= 119865(119886)

119899

cup 119864(119887)

119899

(119886 119887) isin 119860 times 119861

= 119879 (119886 119887) (119886 119887) isin 119860 times 119861

= (119865 119860)

119899

or (119864 119861)

119899

(5)



119899


(119867119860 times 119861)

119899

= 119867(119886 119887)

119899

(119886 119887) isin 119860 times 119861

= (119865 (119886) cap 119864 (119887))

119899

(119886 119887) isin 119860 times 119861

sube 119865(119886)

119899

cap 119864(119887)

119899

(119886 119887) isin 119860 times 119861

= 119879 (119886 119887) (119886 119887) isin 119860 times 119861

= (119865 119860)

119899

and (119864 119861)

119899

(6)






4= 0 1 2 3 4 and 119865(119899) =


4) is 5








119894 119887

119894)) for (119886

119894 119887

119894) isin 119860 times 119861

= LCM (119865 (119886119894) cap 119864 (119887

119894))

le LCM (119865 (119886119894)) = |(119865 119860)|

(7)









Example 29 Let 119866 = 119878



119899




119894|) where 119909

119894isin 119883















Proof It is trivial



5 Conclusion





References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














2











(1) |119865(119909) cap 119865(119910)| le 119866119862119863(|119865(119909)| |119865(119910)|) for 119909 119910 isin 119860

(2) |119865(119909) cup 119865(119910)| = 119871119862119872(|119865(119909)| |119865(119910)|) for 119909 119910 isin 119860

(3) |119865(119909) times 119865(119910)| = |119865(119909)||119865(119910)| for 119909 119910 isin 119860




119905

= 119865(119909)

119905

cup 119865(119910)

119905


119896




(119865 119860)

119899

= 119865(119909)

119899

119909 isin 119860 119899 isin 119885 (4)




2

= 119865(119890) and 119865(12)2 = 119865(119890) 119865(13)2 = 119865(119890) 119865(23)2 =119865(119890) 119865(123)2 = 119865(132)


(1) ((119865 119860) or (119864 119861))119899 = (119865 119860)119899 or (119864 119861)119899


119899

and

(119864 119861)

119899



119899



(119867 119860 times 119861)

119899

= 119867(119886 119887)

119899

(119886 119887) isin 119860 times 119861

= (119865 (119886) cup 119864 (119887))

119899

(119886 119887) isin 119860 times 119861

= 119865(119886)

119899

cup 119864(119887)

119899

(119886 119887) isin 119860 times 119861

= 119879 (119886 119887) (119886 119887) isin 119860 times 119861

= (119865 119860)

119899

or (119864 119861)

119899

(5)



119899


(119867119860 times 119861)

119899

= 119867(119886 119887)

119899

(119886 119887) isin 119860 times 119861

= (119865 (119886) cap 119864 (119887))

119899

(119886 119887) isin 119860 times 119861

sube 119865(119886)

119899

cap 119864(119887)

119899

(119886 119887) isin 119860 times 119861

= 119879 (119886 119887) (119886 119887) isin 119860 times 119861

= (119865 119860)

119899

and (119864 119861)

119899

(6)






4= 0 1 2 3 4 and 119865(119899) =


4) is 5








119894 119887

119894)) for (119886

119894 119887

119894) isin 119860 times 119861

= LCM (119865 (119886119894) cap 119864 (119887

119894))

le LCM (119865 (119886119894)) = |(119865 119860)|

(7)









Example 29 Let 119866 = 119878



119899




119894|) where 119909

119894isin 119883















Proof It is trivial



5 Conclusion





References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899


(119867119860 times 119861)

119899

= 119867(119886 119887)

119899

(119886 119887) isin 119860 times 119861

= (119865 (119886) cap 119864 (119887))

119899

(119886 119887) isin 119860 times 119861

sube 119865(119886)

119899

cap 119864(119887)

119899

(119886 119887) isin 119860 times 119861

= 119879 (119886 119887) (119886 119887) isin 119860 times 119861

= (119865 119860)

119899

and (119864 119861)

119899

(6)






4= 0 1 2 3 4 and 119865(119899) =


4) is 5








119894 119887

119894)) for (119886

119894 119887

119894) isin 119860 times 119861

= LCM (119865 (119886119894) cap 119864 (119887

119894))

le LCM (119865 (119886119894)) = |(119865 119860)|

(7)









Example 29 Let 119866 = 119878



119899




119894|) where 119909

119894isin 119883















Proof It is trivial



5 Conclusion





References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















Proof It is trivial



5 Conclusion





References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article cyclic soft groups and their applications...

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