Journal of Mathematics

Volume 2013 (2013), Article ID 650480, 8 pages

http://dx.doi.org/10.1155/2013/650480

Research Article

Some Properties of Intuitionistic Fuzzy Soft Rings

B. A. Ersoy,¹ S. Onar,¹ K. Hila,² and B. Davvaz³

¹Department of Mathematics, Yildiz Technical University, 81270 Istanbul, Turkey

²Department of Mathematics & Computer Science, Faculty of Natural Sciences, University of Gjirokastra, Albania

³Department of Mathematics, Yazd University, Yazd, Iran

Received 8 November 2012; Revised 13 January 2013; Accepted 21 February 2013

Academic Editor: Jianming Zhan

Copyright © 2013 B. A. Ersoy 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.

Maji et al. introduced the concept of intuitionistic fuzzy soft sets, which is an extension of soft sets and intuitionistic fuzzy sets. In this paper, we apply the concept of intuitionistic fuzzy soft sets to rings. The concept of intuitionistic fuzzy soft rings is introduced and some basic properties of intuitionistic fuzzy soft rings are given. Intersection, union, AND, and OR operations of intuitionistic fuzzy soft rings are defined. Then, the deffinitions of intuitionistic fuzzy soft ideals are proposed and some related results are considered.

1. Introduction

Uncertain or imprecise data are inherent and pervasive in many important applications in the areas such as economics, engineering, environment, social science, medical science, and business management. Uncertain data in those applications could be caused by data randomness, information incompleteness, limitations of measuring instrument, delayed data updates, and so forth. Due to the importance of those applications and the rapidly increasing amount of uncertain data collected and accumulated, research on effective and efficient techniques that are dedicated to modeling uncertain data and tackling uncertainties has attracted much interest in recent years and yet remained challenging at large. There have been a great amount of research and applications in the literature concerning some special tools like probability theory, (intuitionistic) fuzzy set theory, rough set theory, vague set theory, and interval mathematics. However, all of these have theirs advantages as well as inherent limitations in dealing with uncertainties. One major problem shared by those theories is their incompatibility with the parameterizations tools. Soft set theory [1] was firstly proposed by a Russian researcher, Molodtsov, in 1999 to overcome these difficulties.

At present, work on the extension of soft set theory is progressing rapidly. Maji et al. proposed the concept of fuzzy soft set [2] and then gave its application. Roy and Maji presented a method of object recognition from an imprecise multiobserver data [3]. Yao et al. proposed the concept of fuzzy sets and defined some operations on fuzzy soft sets [4].

From the above discussion, we can see that all of these works are based on Zadeh’s fuzzy sets theory which was generalized to intuitionistic fuzzy sets by Atanassov [5]. The concept of intuitionistic fuzzy set was introduced and studied by Atanassov [5, 6] as a generalization of the notion of fuzzy set. Some applications of intuitionistic fuzzy sets are discussed in [7]. In [8–12], Davvaz et al. applied the concept of intuitionistic fuzzy sets to algebraic hyperstructures and some related properties are investigated.

The purpose of this paper is to deal with the algebraic structure of ring by applying intuitionistic fuzzy soft theory. We introduce the notion of intuitionistic fuzzy soft ring and study some of its characterization of operations and algebraic properties.

2. Preliminaries

In this section, for the sake of completeness, we first cite some useful definitions and results.

Definition 1 (see [13]). A fuzzy subset in a set is a function .

Definition 2 (see [1]). Let be an initial universe and a set of parameters. Let denote the power set of and . 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 3 (see [14]). Let be an initial universe, and a set of parameters and denotes the fuzzy power set of and . A pair is called a fuzzy soft set over , where is a mapping given by . A fuzzy soft set is a parameterized family of fuzzy subsets of .

Definition 4 (see [14]). An intuitionistic fuzzy set on the universe can be defined as follows:

where and with the following property:

The values and denote the degree of membership and nonmembership of to , respectively.

Lemma 5 (see [7, 15]). Let be an intuitionistic fuzzy set in and let such that . One defines

(1), (2), (3), (4).

Then, , , , and are intuitionistic fuzzy sets in .

Definition 6 (see [16]). Let be an initial universe and a set of parameters and denotes the intuitionistic fuzzy power set of and . A pair is called an intuitionistic fuzzy soft set over , where is a mapping given by .

An intuitionistic fuzzy soft set is a parameterized family of intuitionistic fuzzy subsets of ; a fuzzy soft set is a special case of an intuitionistic fuzzy soft set, because when all the intuitionistic fuzzy subset of degenerates into fuzzy subsets, the corresponding intuitionistic fuzzy soft set degenerates into a fuzzy soft set.

In general, for all , is an intuitionistic fuzzy set on , which is called the intuitionistic fuzzy set of parameter . The intuitionistic value

denotes the degree that object holds parameter . can be written as following:

If for all , , then degenerates into a fuzzy set; if for all and for all , , then the intuitionistic fuzzy soft set degenerates into a fuzzy soft set.

Example 7. Let describe the character of the students with respect to the given parameter, for finding the best student of an academic year. Let the set of students under consideration be . Let and

Let

Then, the family of is an intuitionistic fuzzy soft set.

Definition 8 (see [16]). Let and be two intuitionistic fuzzy soft sets over . Then, is said to be an intuitionistic fuzzy soft subset of if

(1),(2) is an intuitionistic fuzzy subset of , for all .

We denote the above inclusion relationship by . Similarly, is called an intuitionistic fuzzy soft superset of if is an intuitionistic fuzzy soft subset of . We denoted the above relationship by . and over a universe are said to be intuitionistic fuzzy soft equals if and .

Definition 9 (see [16]). Let and be two intuitionistic fuzzy soft sets over a universe . Then, “ and ” is denoted by and is defined by , where , for all .

Definition 10 (see [16]). Let and be two intuitionistic fuzzy soft sets over a universe . Then, “ or ” is denoted by and is defined by , where , for all .

Definition 11 (see [16]). The intersection of two intuitionistic fuzzy soft sets and over a universe is an intuitionistic fuzzy soft set denoted by , where and

for all . This is denoted by .

Definition 12 (see [16]). The union of two intuitionistic fuzzy soft sets and over a universe is an intuitionistic fuzzy soft set denoted by , where and

for all . This is denoted by .

In contrast with the above definitions of union and intersection of intuitionistic fuzzy soft sets, we may sometimes adopt different definitions of union and intersection given as follows.

Definition 13 (see [16]). Let and be two intuitionistic fuzzy soft sets over a universe such that . The biunion of and is defined to be the intuitionistic fuzzy soft set , where and for all . This is denoted by .

Definition 14 (see [16]). Let and be two intuitionistic fuzzy soft sets over a universe such that . The bi-intersection and is defined to be the intuitionistic fuzzy soft set , where and for all . This is denoted by .

Definition 15. A fuzzy set in ring is called a fuzzy ideal of , if for all the following conditions hold:

(1), (2).

3. Intuitionistic Fuzzy Soft Rings

Definition 16. A pair is called an intuitionistic fuzzy soft ring over , where is a mapping given by

if for all the following conditions hold:

(1) and ,(2) and .

Definition 17. For two intuitionistic fuzzy soft rings and over a universe , we say that is an intuitionistic fuzzy soft subring of and write if

(i), (ii)for each , , and .

Definition 18. Two intuitionistic fuzzy soft rings and over a universe are said to be equal if and .

Theorem 19. Let and be two intuitionistic fuzzy soft rings over a universe . If for all , then is an intuitionistic fuzzy soft subring of .

Proof. The proof is straightforward.

Definition 20. The union of two intuitionistic fuzzy soft rings and over a universe is denoted by and is defined by an intuitionistic fuzzy soft ring such that for each ,

This is denoted by , where .

Theorem 21. If and are two intuitionistic fuzzy soft rings over a universe , then, so are .

Proof. For any and , we consider the following cases.

Case 1. Let . Then,

Case 2. Let . Then, analogous to the proof of Case  1, we have

Case 3. Let . In this case the proof is straightforward.

Thus, in any cases, we have

Therefore, is an intuitionistic fuzzy soft ring.

Definition 22. The intersection of two intuitionistic fuzzy soft rings and over a universe is denoted by and is defined by

This is denoted by , where .

Theorem 23. If and are two intuitionistic fuzzy soft rings over a universe , then, so are .

Proof. The proof is straightforward.

Definition 24. Let and be two intuitionistic fuzzy soft rings over a universe . Then, “ AND " is denoted by and is defined by , where and is defined as

Theorem 25. If and are two intuitionistic fuzzy soft rings over a universe , then, so are and .

Proof. For all and we have

In a similar way, we have

The case for can be similarly proved.

Example 26 (see [17]). Consider the ring with the following tables:

Let and be a set-valued function defined by

Obviously, is an intuitionistic fuzzy soft set over . Moreover, we see that is an intuitionistic fuzzy subring of , for all . Therefore, is an intuitionistic fuzzy soft ring over .

Definition 27. Let and be two intuitionistic fuzzy soft rings over a universe . Then, “ OR ” is denoted by and is defined by , where and is defined as

Theorem 28. If and are two intuitionistic fuzzy soft rings over a universe , then, so are and .

Proof. The proof is straightforward.

The following theorem is a generalization of previous results.

Theorem 29. Let be an intuitionistic fuzzy soft rings over a universe , and let be a nonempty family of intuitionistic fuzzy soft rings. Then, one has the following.

(1) is an intuitionistic fuzzy soft ring over .(2) is an intuitionistic fuzzy soft ring over .(3)If , for all , then is an intuitionistic fuzzy soft ring over .

Theorem 30. Let be an intuitionistic fuzzy soft ring over a universe , and let such that . Then the following are intuitionistic fuzzy soft rings: over , where

Proof. Suppose that is an intuitionistic fuzzy soft ring over . Then,

For all ,

The proof of other cases is similar.

Definition 31. Let and be two intuitionistic fuzzy soft rings over a universe . Then, the product of and is defined to be the intuitionistic fuzzy soft ring , where and

for all and . This is denoted by .

Theorem 32. If and are intuitionistic fuzzy soft rings over a universe , then so is .

Proof. Let and be two intuitionistic fuzzy soft rings over . Then, for any and , we consider the following cases.

Case  1. Let . Then,

Case  2. Let . Then, analogous to the proof of Case  1, the proof is straightforward.

Case  3. Let . Then,

Similarly, we have , and so

In a similar way, we prove that

and the rest of the other cases can be similarly proved. Therefore, is an intuitionistic fuzzy soft ring over .

Definition 33. Let be an ideal of and let be an intuitionistic fuzzy soft set over . Then, is an intuitionistic fuzzy soft ideal of , which will be denoted by , if for each and the following conditions hold:

(i) and .

Theorem 34. One has

(ii) and ,(iii) and .

Theorem 35. For any intuitionistic fuzzy soft ideals and over , where , one has .

Proof. Since , then . By Definition  2.5, we can write , where , and for all . Then, for all and ,

Since and , we know that and are intuitionistic fuzzy soft ideals of . For all , we have

We obtain a similar result for . Hence, is an intuitionistic fuzzy soft ideal. Therefore, .

Theorem 36. For any fuzzy soft ideals and over , in which and are disjoint, one has .

Proof. Assume that and . By means of definition we can write , where and for every

Since , either or for all . If , the is an ideal of since . If , then is an ideal of since . Thus, is an ideal of for all , and so .

Theorem 37. Let be an intuitionistic fuzzy soft ideal over . If is a nonempty family of intuitionistic fuzzy soft ideals of , where is an index set, then,

(1),(2),(3), where for all .

Definition 38. Let and be two functions, where and are parameter sets for intuitionistic fuzzy soft sets and , respectively. Then, the pair is called an intuitionistic fuzzy soft function from to .

Definition 39. Let and be two intuitionistic fuzzy soft rings over and , respectively, let be a homomorphism of rings, and let be a mapping of sets. Then, we say that is an intuitionistic fuzzy soft homomorphism of intuitionistic fuzzy soft rings and define by , if the following conditions are satisfied:

Definition 40. If there exists an intuitionistic fuzzy soft ring homomorphism between and , we say that is intuitionistic fuzzy soft homomorphic to and is denoted by . Moreover, if is an isomorphism, then is intuitionistic fuzzy soft isomorphic to , which is denoted by .

Now, we show that the homomorphic image and preimage of an intuitionistic fuzzy soft ring are also intuitionistic fuzzy soft rings.

Definition 41. Let and be two intuitionistic fuzzy soft rings over and . Let be an intuitionistic fuzzy soft function from to.

(1)The image of under the intuitionistic fuzzy soft function denoted by is the intuitionistic fuzzy soft ring over defined by , where

for all and for all . (2)The preimage of under the intuitionistic fuzzy soft function denoted by is the intuitionistic fuzzy soft ring over defined by , where , for all and for all .

If and are injective (surjective), then is said to be injective (surjective).

Definition 42. Let be an intuitionistic fuzzy soft function from to . If is a homomorphism function from to, then is said to be intuitionistic fuzzy soft homomorphism. If is an isomorphism from to and is a one-to-one mapping from onto, then is said to be an intuitionistic fuzzy soft isomorphism.

Theorem 43. Let be an intuitionistic fuzzy soft ring over and an intuitionistic fuzzy soft homomorphism from to . Then, is an intuitionistic fuzzy soft ring over .

Proof. The proof is straightforward.

Theorem 44. Let be an intuitionistic fuzzy soft ring over and be an intuitionistic fuzzy soft homomorphism from to . Then, is an intuitionistic fuzzy soft ring over .

Proof. The proof is straightforward.

4. Conclusion

In this work the theoretical point of view of intuitionistic fuzzy soft sets in ring and ideal is discussed. The work is focused on intuitionistic fuzzy soft rings and fuzzy soft ideals. These concepts are basic structures for improvement of fuzzy soft set theory. One can extend this work by studying other algebraic structures.

Acknowledgment

The authors are highly grateful to the referees for their valuable comments and suggestions for improving the paper.

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