Cyclic Soft Groups and Their Applications on Groups

Abstract

In crisp environment the notions of order of group and cyclic group are well known due to many applications. In this paper, we introduce order of the soft groups, power of the soft sets, power of the soft groups, and cyclic soft group on a group. We also investigate the relationship between cyclic soft groups and classical groups.

1. Introduction

Most of our real life problems in economics, engineering, environment, social science, and medical science involve imprecise data that contain uncertainties. To solve these kinds of problems, it is quite difficult to successfully use classical methods. 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 dealing with uncertainties. All these theories have their inherent difficulties pointed out by Molodtsov [4].

The theory of soft sets, proposed by Moldtsov [4], is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. His pioneer paper has undergone tremendous growth and applications in the last few years. Maji et al. [5] give an application of soft set theory in a decision making problem by using the rough sets and they conducted a theoretical study on soft sets in a detailed way [6]. Chen et al. [7] proposed a reasonable definition of parameterizations reduction of soft sets and compared them with the concept of attributes reduction in rough set theory.

The algebraic structures of set theories which deal with uncertainties have been studied by some authors. Rosenfeld [8] proposed fuzzy groups to establish results for the algebraic structures of fuzzy sets. Rough groups are defined by Biswas and Nanda [9], and some others (i.e., Bonikowaski [10] and Iwinski [11]) studied algebraic properties of rough sets. Fuzzification of algebraic structures was studied by many authors [8, 12, 13].

Many papers on soft algebras have been published since Aktaş and Çaman [14] introduced the notion of a soft group in 2007. Recently, Jun et al. [15] studied soft ideals and idealistic soft BCK/BCI-algebras. Acar et al. [16] introduced initial concepts of soft rings. Aygünoǧlu and Aygün [17] introduced the concept of fuzzy soft group and, in the meantime, they studied its properties and structural characteristics. Atagün and Sezgin [18] introduced and studied the concepts of soft subrings, soft ideal of a ring, and soft subfields of a field.

Our interest, in this paper, is to define order of the soft group and cyclic soft group by using definition of the soft group that was defined in [14]. We then find out the relationships between cyclic soft groups and classical groups. Finally, we conclude the study with suggestions for future work.

2. Preliminaries

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

Throughout this work, is an initial universe set, is a set of parameters, is the power set of , , and denotes a group with identity .

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

In other words, a soft set over is a parameterized family of subsets of the universe .

Definition 2 (see [6]). For two soft sets and over , is called a soft subset of , if (1) and(2); and are identical approximations.
It is denoted by .
is called a soft superset of if is a soft subset of . It is denoted by .

Definition 3 (see [6]). If and are two soft sets, then AND is denoted . is defined as , where , for all .

Definition 4 (see [6]). If and are two soft sets, then OR , denoted by , is defined by , where , .

Definition 5 (see [6]). Union of two soft sets of and over is the soft set , where and , It is denoted by .

Definition 6 (see [14]). Let be a soft set over . Then is said to be a soft group over if and only if is subgroup of for all .

Definition 7 (see [14]). One considers the following.(1) is said to be an identity soft group over if for all , where is the identity element of .(2) is said to be an absolute soft group over if for all .

Definition 8 (see [14]). Let and be two soft groups over . Then is a soft subgroup of , written as , if(1),(2) is a subgroup of for all .

Definition 9 (see [14]). Let and be two soft groups over and , respectively, and let and be two functions. Then one says is a soft homomorphism and is soft homomorphic to , denoted by , if the following conditions are satisfied: (1) is a homomorphism from onto ,(2) is a mapping from onto ,(3) for all .

In this definition, if is an isomorphism from to and is a one-to-one mapping from onto , then we say that is a soft isomorphism and is soft isomorph to which is denoted by . The image of soft group under soft homomorphism will be denoted by .

Definition 10 (see [14]). Let and be two soft groups over and , respectively. The product of soft groups and is defined as , where , for all .

3. The Order of Soft Groups

Since the elements of a cyclic group are the powers of the element, properties of cyclic groups are closely related to the properties of the powers of an element. In this paper, we define order of the soft group by using definition of the soft group that was defined in [14]. We then investigate their properties.

Definition 11. A pair is called surjective soft set over , where is a surjective mapping given by .

Throughout this study, denotes a group and the soft set will be a surjective soft set. The element is used instead of the element of .

Definition 12. Let be a soft set over and for . Then is called -power of .

Example 13. Let be a soft set over , where , and let be a soft set over group . And the third power of is .

Of course, when the group is additive, the th power of will be written by .

Theorem 14. Let be a soft set over and for . Then, for all , (1), for all ,(2), for all ,(3), for all .

Proof. Let , for . From Definition 12   and and . This means that . This completes the proof. Theorem 14 and Theorem 14 can be proved similarly by using Definition 12.

In general, the opposite of Theorem 14(1) is not true. We illustrate an example of this situation.

Example 15. Let and let be a function such that and . The intersection is . So . On the other hand . Consequently .

Definition 16. Let be a soft set over and . If there is a positive integer such that , then the least such positive integer is called the order of . If no such exists, then has infinite order. The order of is denoted by .

If is a soft group over , then the order of coincides with the order of , which is subgroup of . Of course if there is any element in such that , then the order of is 1.

Example 17. In Example 13 the order of element is 3.
Let be a soft group over group of integer numbers , where is a mapping from natural numbers to such that for all . There is no any positive integer such that , so has infinite order for all .

Theorem 18. Let be finite group and a soft group over . Then, the orders of elements of are finite.

Proof. It is straightforward.

Theorem 19. Let be a soft set over finite group and for . Then, the order of is the least common multiple (LCM) of order of elements of .

Proof. Let be the order of . Then . This means that for all . We know from classical group theory that , namely, , divides for all . Thus, is common multiple of elements of . Let be another common multiple of elements of . Then, by reason of for all , . However, since is the least number that satisfies the condition , hence . This completes the proof.

Theorem 20. Let be finite group, a soft group over , and and the elements of . Then, for all , one has the following: (1) for ,(2) for ,(3) for .

Proof. We consider the following.(1) is subgroup of and , so and . It follows .(2)Let , , and . From Theorem 14. This follows and . Thus is a common multiple of and . Let be another common multiple of and . Consider . Since is the least positive integer that satisfied the condition , divides . Hence is LCM order of and . This completes the proof.(3)Since and are subgroups of , it is seen easily.

Definition 21. Let be group and soft set over . The set
is called th power of soft set .

Example 22. Let be a soft set over defined in Example 13. Then, the second power of is that , and ,  ,  ,  .

Theorem 23. Let and be two soft sets over . Then, (1),(2)if and, for all , and are identical approximations, then .

Proof. We consider the following.(1)Suppose that and . We can write . Using Definition 21 and Theorem 14 we have (2)Suppose that and . Using the same arguments in (1), we have

If and are both soft groups, then, for all ,   and are all subgroups of , and so and contain the identity element of . Thus, the set contains at least ; hence . It means that if we take in Theorem 23(2) and as soft groups, not soft sets, then the extra condition can be added in Theorem 23(2).

In classical groups, the order of a group is defined as the number of elements it contains. But in soft groups, it differs from classical groups.

Definition 24. Let be soft group over . If is a finite group, then the least common multiple of orders of elements of is called order of . If is an infinite group, then the order of is defined to be the number of elements of and the order of soft group is denoted by .

Of course, if is surjective, then the number of elements of is the number of elements in .

Example 25. In Example 13 the order of is 6 and in Example 17 if we chose and , for , then the order of soft group is 5.

We give the following results similar to Lagrange Theorem in group theory.

Theorem 26. Let be a soft group over a finite group and . Then one has the following. (1)The order of divides the order of . In particular, .(2)The order of divides the order of .

When is a finite group, then the order is a finite and the orders of elements of divide the order of . However when is an infinite group, then the order of can be finite or infinite. Hence, Theorem 26 is not true for infinite groups.

Theorem 27. Let be a finite group and let and be two soft groups over . Then and .

Proof. Suppose that , where . Using fundamental theorems and definitions in group theory and Definitions 24 and 3, we have The other inequality is shown similarly.

4. Cyclic Soft Groups

The class of cyclic groups is an important class in group theory. In this section, we study soft groups which are generated by one element of . We define cyclic soft group and prove some of their properties which are analogous to the crisp case.

Definition 28. Let be a soft group over and an element of . The set is called a soft subset of generated by the set and denoted by . If , then the soft group is called the cyclic soft group generated by .

If is a cyclic soft group over , then we can write it in this form , where is element of . That is to say, if all the elements of are generated by any elements of of , then is a cyclic soft group over .

If is a cyclic group, then is a soft cyclic group over since all subgroups of cyclic group are cyclic but the reverse is not always true.

As an illustration, let us consider the following example.

Example 29. Let be the symmetric group and the set of parameters. If we construct a soft set over such that for all , then one can easily show that is a soft cyclic group over ; however is not a cyclic group.

In the following theorem, we can give some properties of cyclic soft groups that has similar features of the classical cyclic groups.

Theorem 30. One considers the following.(1)If is a finite cyclic soft group generated by , then , where .(2)If is an infinite cyclic soft group generated by , then .(3)If is an identity soft group, then it is a cyclic soft group generated by .(4)Let be an absolute soft group defined on . Then, is a cyclic soft group if and only if is a cyclic group.(5)Let be a soft group on . If the order of is prime, then 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 be a soft homomorphism of the soft group over into the soft group over . If is a cyclic soft group over , then is a cyclic soft group over .

Proof. First, we show that is a soft group over . Since is a homomorphism from to , is a subgroup of for all . Thus is a soft group over . Since is cyclic subgroup for all in , image of under is cyclic; that is, is cyclic subgroup of for all . Consequently, is a cyclic soft group over .

Theorem 32. Let and be two soft isomorphic soft groups over and , respectively. If is a cyclic soft group, then so is .

Proof. First of all note that since is a soft isomorphic to , there is an homomorphism from to such that for all , where is a one-to-one mapping from onto . Since is a cyclic subgroup for all in and is a homomorphism, then is a cyclic subgroup of . Thus, all elements of are cyclic. This result completes the proof.

Theorem 33. If and are two cyclic soft groups over , then is a cyclic soft group over .

Proof. Let , where , . Since and are cyclic subgroups of for all and and is a subgroup of both and , is cyclic subgroup of for all . Hence, is cyclic soft group over .

Theorem 34. Let and be two cyclic soft groups over and . Then, is a cyclic soft group over .

Proof. It is trivial.

Theorem 35. Let and be two cyclic soft groups of finite orders and over and , respectively. If and are relatively prime, then the product is a cyclic soft group.

Proof. Let . According to Lagrange Theorem, divides and divides for all and for all . Since , and are relatively prime. So is cyclic group for all . This completes the proof.

5. Conclusion

In this paper, we have expanded the soft set theory. We have focused on order of the soft groups and investigated relationship between the order of soft groups and the order of classical groups. Additionally, we have studied the algebraic properties of cyclic soft groups with respect to a group structure. Our future work will focus on the relationships between cyclic soft groups and other algebraic structures such as rings and fields.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.