Advances in Fuzzy Systems

Volume 2013 (2013), Article ID 319542, 8 pages

http://dx.doi.org/10.1155/2013/319542

Research Article

Applications of Soft Sets in -Algebras

N. O. Alshehri,¹ M. Akram,² and R. S. Al-ghamdi¹

¹Department of Mathematics, Faculty of Sciences (Girls), King Abdulaziz University, Jeddah, Saudi Arabia

²Punjab University College of Information Technology, University of the Punjab, Old Campus, Lahore 54000, Pakistan

Received 21 October 2012; Accepted 24 January 2013

Academic Editor: Martine de Cock

Copyright © 2013 N. O. Alshehri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In 1999, Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty and vagueness. In this paper, we apply the concept of soft sets to K-algebras and investigate some properties of Abelian soft K-algebras. We also introduce the concept of soft intersection K-algebras and investigate some of their properties.

1. Introduction

Most of the problems in engineering, medical science, economics, environments, and so forth, have various uncertainties. The problems in system identification involve characteristics which are essentially nonprobabilistic in nature. In response to this situation, Zadeh [1] introduced fuzzy set theory as an alternative to probability theory. Uncertainty is an attribute of information. In order to suggest a more general framework, the approach to uncertainty is outlined by Zadeh [2]. Molodtsov [3] initiated the concept of soft set theory as a new mathematical tool for dealing with uncertainties. In soft set theory, the problem of setting the membership function does not arise, which makes the theory easily applied to many different fields including game theory, operations research, Riemann integration, and Perron integration. At present, work on soft set theory is progressing rapidly. After Molodtsov’s work, some operations and application of soft sets were studied by many researchers including Ali et al. [4], Aktaş and Çağman [5], Chen et al. [6], and Maji et al. [7]. Maji et al. [7] gave first practical application of soft sets in decision making problems. To address decision making problems based on fuzzy soft sets, Feng et al. introduced the concept of soft level sets of fuzzy soft sets and initiated an adjustable decision making scheme using fuzzy soft sets [8]. It is interesting to see that soft sets are closely related to many other soft computing models such as rough sets and fuzzy sets. Feng et al. [9] first considered the combination of soft sets, fuzzy sets, and rough sets. Using soft sets as the granulation structures, Feng et al. [10] defined soft approximation spaces, soft rough approximations, and soft rough sets, which are generalizations of Pawlak’s rough set model based on soft sets. It has been proven that in some cases Feng’s soft rough set model could provide better approximations than classical rough sets. The algebraic structure of soft set theories has been studied increasingly in recent years. Aktaş and Çağman [5] defined the notion of soft groups. Feng et al. [11] initiated the study of soft semirings, and soft rings were defined by Acar et al. [12]. Jun [13] introduced soft -algebras, and Kazancı et al. [14] introduced soft -algebras. Along this direction, we apply soft set theory to -algebras and investigate some of their properties. We introduce the notion of Abelian soft -algebras and investigate some of their properties. We also introduce the concept of soft intersection soft -algebras and investigate some of their properties.

2. Review of the Literature

In this section, we recall some basic concepts that are necessary for subsequent discussion of -algebras.

The notion of a -algebra was first introduced by Dar and Akram [15] in 2003 and published in 2005. A -algebra is an algebra built on a group by adjoining an induced binary operation on group which is attached to an abstract -algebra. This system is, in general noncommutative and nonassociative with a right identity , if group is noncommutative.

Definition 1 (see [15]). Let be a group in which each nonidentity element is not of order 2. Then, a -algebra is a structure on a group , in which induced binary operation is defined by and satisfies the following axioms:

(K1), (K2), (K3), (K4), (K5)

for all , , .

Definition 2. A -algebra is called abelian if and only if for all , .

If a -algebra is abelian, then the axioms () and () can be written as

(),().

In what follows, we denote a -algebra by unless otherwise specified.

A nonempty subset of a -algebra is called a subalgebra [15] of the -algebra if for all , . Note that every subalgebra of a -algebra contains the identity of the group . Naturally, the mapping of -algebras is called a homomorphism [16] if for all . We refer the readers to the book [17] and research papers [16–20] for further information regarding -algebras.

Soft Set Theory. In 1999, Molodtsov [3] initiated soft set theory as a new approach for modelling uncertainties. Later on, Maji et al. [21] expanded this theory to fuzzy soft set theory. Based on the idea of parametrization, a soft set gives a series of approximate descriptions of a complicate object from various different aspects. Each approximate description has two parts, namely, predicate and approximate value set. A soft set can be determined by a set-valued mapping assigning to each parameter exactly one crisp subset of the universe. More specifically, we can define the notion of soft set in the following way. Let be an initial universe and a set of parameters. Let denote the power set of , and let be a nonempty subset of .

Definition 3. A pair is called a soft set over , where and is a set-valued mapping, called the approximate function of the soft set . It is easy to represent a soft set by a set of ordered pairs as follows:

It is clear to see that a soft set is a parameterized family of subsets of the set . Hereafter unless stated otherwise, we always identify a soft set with the corresponding soft set with for all . In this fashion, we can implicitly view any soft set as a soft set with the whole parameter set .

Definition 4. Let and be two soft sets over a common universe . is a said to be a soft subset of , denoted by , if for all .

Definition 5. The restricted intersection of a nonempty family soft sets over a common universe is defined as the soft set , where and for all .

Definition 6. The extended intersection of a nonempty family soft sets over a common universe is defined as the soft set , where and , for all .

Definition 7. The restricted union of a nonempty family soft sets over a common universe is defined as the soft set , where and , for all .

Definition 8. The -intersection of a nonempty family soft sets over a common universe is defined as the soft set , where and , for all .

Definition 9. The -union of a nonempty family soft sets over a common universe is defined as the soft set , where and , for all .

Definition 10. The Cartesian product of the nonempty family soft sets over a common universe is defined as the soft set , where and , for all .

Definition 11. For a soft set , the set Supp is called the support of the soft set , and the soft set is called nonnull if Supp .

We refer the readers to the papers [22–24] for further information regarding the application of soft set theory.

3. Applications of Soft Sets in -Algebras

If is a -algebra and a nonempty set, a set-valued function can be defined by , , where is an arbitrary binary relation from to ; that is, is a subset of unless otherwise specified. The pair is then a soft set over .

Definition 12 (see [19]). Let be a soft set over . Then, is called a soft -algebra over if is a -subalgebra of a -algebra for all .

Definition 13. Let be a nonnull soft set over . Then, is called a soft -algebra over if is a -subalgebra of for all Supp .

Example 14. Consider the -algebra = on the symmetric group , where , , , , , , and are given by the following Cayley table:

Let be a soft set over , where and is a set-valued function defined by , , , , and being -subalgebras of for all Supp . Therefore, is a soft -algebra over .

Example 15. Consider the -algebra on the Dihedral group , where , , , , , and are given by the following Cayley table:

Let be a soft set over , where and is set-valued function defined by , , , , , and being -subalgebras of . Therefore, is a soft -algebra over .

Lemma 16. Let be a soft -algebra over , then

(i)if for all ,(ii)if for all .

Proposition 17. Let be a nonempty family of soft -algebras over . Then, the bi-intersection is a soft -algebra over if it is nonnull.

Proof. Let be a nonempty family of soft -algebras over . By Definition 5, we can write , where and for all . Let . Then, , and so we have for all . Since is a nonempty family of soft -algebras over , it follows that is a -subalgebra of for all , and its intersection is also a -subalgebra of ; that is, is a -subalgebra of for all . Hence, is a soft -algebra over .

Proposition 18. Let be a nonempty family of soft -algebras over . Then, the extended intersection is a soft -algebra over .

Proof. Let be a nonempty family of soft -algebras over . By Definition 6, we can write , where and for all . Let . Then, , and so we have for all . Since is a nonempty family of soft -algebras over , it follows that is a -subalgebra of for all , and its intersection is also a -subalgebra of ; that is, is a -subalgebra of for all . Hence, is a soft -algebra over .

Proposition 19. Let be a nonempty family of soft -algebras over . If or for all , , , then the restricted union is a soft -algebra over .

Proof. Suppose that is a nonempty family of soft -algebras over . By Definition 7, we can write , where and for all . Let . Since Supp, for some . By assumption, is a -subalgebra of for all . Hence, restricted union is a soft -algebra over .

Proposition 20. Let be a nonempty family of soft -algebras over . Then, the -intersection is a soft -algebra over if it is nonnull.

Proof. Let be a nonempty family of soft -algebras over . By Definition 11, we can write , where and for all . Suppose that the soft set is nonempty. If , . Since is a nonempty family of soft -algebras over , nonempty set is a -subalgebra of for all . It follows that is a -subalgebra of for all . Hence -intersection is a soft -algebra over .

Proposition 21. Let be a nonempty family of soft -algebras over . If or for all , , then the -union is a soft -algebra over .

Proof. Assume that is a nonempty family of soft -algebras over . By Definition 9, we can write , where and for all . Let . Then, , so we have for some . By assumption, is a -subalgebra of for all . Hence, -union is a soft -algebra over .

Proposition 22. Let be a nonempty family of soft -algebras over . Then, the cartesian product is a soft -algebra over .

Proof. Let be a nonempty family of soft -algebras over . By Definition 10, we can write , where and for all . Suppose that the soft set is nonnull. If , . Since is a nonempty family of soft -algebras over , the nonempty set is a -subalgebra of for all . It follows that is a -subalgebra of for all . Hence, -intersection is a soft -algebra over .

Definition 23. Let be a soft -algebra over .

(i) is called the trivial soft -algebra over if for all . (ii) is called the whole soft -algebra over if for all .

Definition 24. Let be a soft set over a -algebra . Then, the inverse of is denoted by and is defined as follows , where is called the inverse of and is defined as

Theorem 25. Let and be any two soft sets over . Then .

Theorem 26. If is a soft -algebra over , then .

The converse of above theorem is not true in general, and it can be seen in the following example.

Example 27. Consider the -algebra on the dihedral group which is given in Example 15. Let be a soft set over , where and is a set-valued function defined by , , , , , , , and . Therefore, we find that for all . Hence , but is not soft -algebra over because is not a -subalgebra of .

Definition 28. A soft -algebra over is said to be Abelian soft -algebra over if each is an Abelian -subalgebra of for all .

Example 29. Let be a soft -algebra over , which is given in Example 14. Then, it is easy to verify that each is an Abelian -subalgebra of for all . Hence, is an Abelian soft -algebra over .

Definition 30. Let be a soft -algebra over and a soft -subalgebra of . Then, we say that is an Abelian soft -subalgebra of if is an Abelian -subalgebra of for all .

Example 31. Let be a soft -algebra over which is given in Example 14, and let be a soft set over , where and is the set-valued function defined by , , and being Abelian -subalgebras of , , and , respectively. Hence, is an Abelian soft -subalgebra of .

Theorem 32. Let be an Abelian soft -algebra over and be a soft -algebras over . Then their restricted intersection is an Abelian soft -algebra over for all .

Definition 33. Let , be two -algebras and a mapping of -algebras. If and are soft sets over and , respectively, then is a soft set over where is defined by for all and is a soft set over where is defined by for all .

Definition 34. Let and be two soft sets over -algebras and , respectively, and let and be two functions. Then, we say that is a soft homomorphism, if the following conditions are satisfied:

(i) is a homomorphism from onto ; (ii) is mapping from onto ; (iii).

In this definition, if is an isomorphism to and is a one-to-one mapping from on to , then we say that is a soft isomorphism and that is soft isomorphic to . Notation, .

Example 35. Let be a Klein four-group. Consider a -algebra on with the following Cayley table:

This is an improper -algebra on Klein four-group since it is an elementary Abelian 2-group, that is, . Let be a soft set over , where and ; the set-valued function defined by where . Then, is a soft -algebra over . Let be a soft set over , where and is the set-valued function defined by . Then is a soft -algebra over . Let be the mapping defined by . It is clear that is a -homomorphism. Consider the mapping given by . Then, one can easily verify that for all . Hence, is a soft homomorphism from to .

Example 36. Consider the -algebra which is given in Example 14. Let be a soft set over , where and is the set-valued function defined by , where . Then, is a soft -algebra over . Let be a soft set over , where and is the set-valued function defined by . Then, is a soft -algebra over . Let be the mapping defined by . It is clear that is a -homomorphism. Consider the mapping given by . Then, one can easily verify that for all . Hence, is a soft homomorphism from to .

We state the following propositions without their proofs.

Proposition 37. Let be an onto homomorphism of -algebras and , two soft -algebras over and , respectively.

(i)The soft function from to is a soft homomorphism from to , where is the identity mapping and the set-valued function is defined by for all . (ii)If is an isomorphism, then the soft function from to is a soft homomorphism from to , where is the identity mapping and the set-valued function is defined by for all .

Proposition 38. Let , , and be -algebras and , and soft -algebras over , , and , respectively. Let the soft function from to be a soft homomorphism from to , and the soft function ( from to a soft homomorphism from to . Then, the soft function from to is a soft homomorphism from to .

Theorem 39. Let and be -algebras and , soft sets over and , respectively. If is a soft -algebra over and , then is a soft -algebra over .

Definition 40. Let and be two soft -algebras over and , respectively. Then the Cartesian product of soft -algebras and is denoted by and is defined as for all .

Theorem 41. Let and be two soft -algebras over and , respectively. Then, (1) the Cartesian product is a soft -algebra over , and (2) is soft isomorphic to .

Proof. We will prove second part. We show that is a soft isomorphism, that is, is a soft isomorphism where is defined as . We prove the following three conditions:

(i)we show that is an isomorphism. Let be a function defined by . Then obviously is an isomorphism.(ii)We now show that is a bijective mapping. The mapping is defined by then obviously is a bijective mapping.(iii) Consider

for all . This implies that is a soft isomorphism. Hence, .

We now introduce the concept of soft intersection soft -algebras and investigate some of their properties

Definition 42. Let be a -algebra and let be a subset of . Let be a soft set over . Then, is called a soft intersection -subalgebra over if it satisfies the following condition:

for all , .

Example 43. Assume that is the universal set. Let be the cyclic group of order . Then, is a -algebra , and is given by the following Cayley table:

Let be a soft set over . Then, and . It is easy to see that is a soft intersection -subalgebra over .

From now on, we will always assume that unless otherwise specified.

Proposition 44. Let be a -algebra and let and be -subalgebras of . If and are soft intersection -subalgebras over . Then, is a soft intersection -subalgebra over , where is defined by for all .

Proof. Let . Then,

Hence, is a soft intersection -subalgebra over .

Theorem 45. Let be a family of soft intersection -subalgebras (resp., soft intersection ideals) over . Then, is a soft intersection -subalgebra over .

Proposition 46. Let be a -algebra and let be a -subalgebra of . If and are soft intersection -subalgebras over . Then, is a soft intersection -algebra over , where is defined by for all .

Proof. Let . Then,

Hence, is a soft intersection -subalgebras over .

Theorem 47. Let be a family of soft intersection -subalgebras over . Then, is a soft intersection -subalgebra over .

Proposition 48. Let be a -algebra and let and , be -subalgebras of . If and are soft intersection -subalgebras over . Then , is a soft intersection -algebra over , where is defined by for all .

Proof. Let . Then

Hence, is a soft intersection -subalgebras over .

Theorem 49. Let be a family of soft intersection -subalgebras over . Then, is a soft intersection -subalgebra over .

Proposition 50. Let be a -algebra and let , and be -subalgebras of . If , and are soft intersection -subalgebras over , , and , then over .

Proof. Straightforward.

Definition 51 (see [25]). Let and be two soft sets over the common on universe and let be a function from to . Then, soft image of under denoted by is a soft set over by

for all , and soft preimage (or soft inverse image) of under denoted by is a soft set over by for all .

Proposition 52. Let be a -algebra and let be ideal of . If is a soft intersection -subalgebra over , then is a soft intersection -subalgebra over .

Theorem 53. Let be a -algebra and and ideals of . Let be a -homomorphism from to . If is soft intersection -subalgebra over , then is a soft intersection -subalgebra over .

Proof. Let , , and . Then,

Hence, is a soft intersection -subalgebra over .

Theorem 54. Let be a -algebra and and ideals of , and let be a -isomorphism from to . If is a soft intersection -subalgebra over , then is a soft intersection -subalgebra over .

Proof. Since is surjective, there exist , such that and for all , . Then,

Hence, is a soft intersection -subalgebra over .

4. Conclusions

Presently, science and technology are featured with complex processes and phenomena for which complete information is not always available. For such cases, mathematical models are developed to handle various types of systems containing elements of uncertainty. A large number of these models are based on an extension of the ordinary set theory, namely, soft sets. In 1999, Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty and vagueness, and many researchers have created some models to solve problems in decision making and medical diagnosis. We have applied the concept of soft set theory to -algebras and have investigated some of their properties. The natural extension of this research work is connected with the study of (i) fuzzy soft intersection -algebras and (ii) roughness in -algebras.

Acknowledgment

The authors are highly thankful to the referees for their valuable comments and suggestions.

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