#### Abstract

We define the dual of generalized fuzzy subspaces first. These concepts generalize the dual of fuzzy subspaces. And then we investigate the double dual of generalized fuzzy subspaces.

#### 1. Introduction and Preliminaries

In fuzzy algebra, fuzzy subspaces are basic concepts. They had been introduced by Katsaras and Liu [1] in 1977 as a generalization of the usual notion of vector spaces. Since then, many results of fuzzy subspaces had been obtained in the literature [1โ4]. Moreover, many researches in fuzzy algebra are closely related to fuzzy subspaces, such as fuzzy subalgebras of an associative algebra [5], fuzzy Lie ideals of a Lie algebra [6], fuzzy subcoalgebras of a coalgebra [7]. Hence fuzzy subspaces play an important role in fuzzy algebra. In 1996, Abdukhalikov [8] defined the dual of fuzzy subspaces as a generalization of the dual of -vector spaces. This notion was also studied and applied in many branches [2, 7โ9], especially in the fuzzy subcoalgebras [7] and fuzzy bialgebras [9].

After the introduction of fuzzy sets by Zadeh [10], there are a number of generalizations of this fundamental concept. So it is natural to study algebraic structures connecting with them. In this paper, we aim our attention at the dual of vector space in intuitionistic fuzzy sets, interval-valued fuzzy sets, and interval-valued intuitionistic fuzzy sets for our further researches.

##### 1.1. Atanassov's Intuitionistic Fuzzy Sets

In [11] intuitionistic fuzzy sets are defined as follows:

*Definition 1.1. *An intuitionistic fuzzy set (IFS, for short) on a universe is defined as an object having the form , where the functions and denote the degree of membership (namely, ) and the degree of nonmembership (namely, ) of each element to the set , respectively, and for each . For the sake of simplicity, we shall use the symbol for the intuitionistic fuzzy set . The class of IFSs on a universe is denoted by IFS .

In 1993, Gau and Buehrer [12] defined vague sets. Later, Bustince and Burillo [13] proved that the notion of vague sets is as same as that of intuitionistic fuzzy sets.

With the definition of intuitionistic fuzzy sets, we can give the following definition.

*Definition 1.2. *Let be an intuitionistic fuzzy set of -vector space . For any and , if it satisfies and , then is called an *intuitionistic fuzzy subspace* of .

##### 1.2. Interval-Valued Fuzzy Sets

The notion of interval-valued fuzzy sets was first introduced by Zadeh [14] as an extension of fuzzy sets in which the values of the membership degrees are intervals of numbers instead of the numbers.

*Definition 1.3. *An interval-valued fuzzy set on a universe (IVFS, for short) is a mapping , where stands for the set of all closed subintervals of . The class of all IVFSs on a universe is denoted by .

In [15] the interval-valued fuzzy sets are called grey sets.

*Notations. *For interval numbers . We define
and put (a) and , (b) and , (c) and .

In [16], Deschrijver and Kerre presented that the mapping between the lattices and is an isomorphism. Thus intuitionistic fuzzy sets and interval-valued fuzzy sets are same from mathematical viewpoints.

Similarly, we can define the following.

*Definition 1.4. *Let be an interval-valued fuzzy set of -vector space . For any and , if it satisfies , then is called an *interval-valued fuzzy subspace* of .

##### 1.3. Interval-Valued Intuitionistic Fuzzy Sets

The following definition generalizes the definitions of IFS and IVFS.

*Definition 1.5 (see [17]). *An interval-valued intuitionistic fuzzy set on a universe (IVIFS, for short) is an object of the form , where and satisfy for any . The class of all IVIFSs on a universe is denoted by .

*Definition 1.6. *Let be an interval-valued intuitionistic fuzzy set of -vector space . For any and , if it satisfies and , then is called an *interval-valued intuitionistic fuzzy subspace* of .

In this paper, intuitionistic fuzzy subspaces, interval-valued fuzzy subspaces and interval-valued intuitionistic fuzzy subspaces are called generalized fuzzy subspaces. In Section 2, we study the dual of generalized fuzzy subspaces. At first, we give the definitions of the dual of generalized fuzzy subspaces, and then discuss their properties and the relationship between them. In Section 3, we investigate the double dual of intuitionistic fuzzy subspaces. Other cases can be investigated similarly. At last a conclusion is presented.

#### 2. The Dual of Generalized Fuzzy Subspaces

In this paper, is denoted the dual space of , that is, the vector space of all linear maps from to . We recall the following.

*Definition 2.1 (see [8]). *Let be a fuzzy subspace of -vector space .

Define by
then is called the dual of fuzzy subspace .

Now, we study the dual of generalized fuzzy subspaces.

##### 2.1. The Dual of Intuitionistic Fuzzy Subspaces

Let be an intuitionistic fuzzy subspace of -vector space . Then for any , which implies . Hence we give the definition of the dual of intuitionistic fuzzy subspaces as follows.

*Definition 2.2. *Let be an intuitionistic fuzzy subspace of -vector space .

Define , where

Obviously, is an intuitionistic fuzzy set of and is called the dual of intuitionistic fuzzy subspace. The class of all the dual of intuitionistic fuzzy subspaces of is denoted by .

In [11], Atanassov defined two operators and . Let be an intuitionistic fuzzy set. Then and . Using the notations in the dual of intuitionistic fuzzy subspaces, we have the following.

*Remark 2.3. *(1) If , then .

(2) If , then .

Following the above remark, Definition 2.1 is the special case of Definition 2.2. And we can give a characterization of the intuitionistic fuzzy subspaces.

Theorem 2.4. * is an intuitionistic fuzzy subspace of if and only if and are intuitionistic fuzzy subspaces of .*

In the end, we give an explain of Definition 2.2 intuitively.

*Remark 2.5. *Let 10 experts vote to and and require per expert to vote at most one. The result is that is 5 and is 3. That is, the number of the supporters of is 5 and the number of the supporters of is 3.

In the voting model, and can be regarded as the dual objects. The analysis indicates that the number of supporters of is equal to the number of objectors of and the number of supporters of is equal to the number of objectors of . So if is the dual of and the numbers of supporters and objectors of are known, then we can calculate the supporters of by the objectors of and the objectors of by the supporters of .

This is the idea that the is defined by and the is defined by in Definition 2.2.

##### 2.2. The Dual of Interval-Valued Fuzzy Subspaces

*Definition 2.6. *Let be an interval-valued fuzzy subspace of -vector space .

Define , where

Then is called the dual of an interval-valued fuzzy subspace. The class of all the dual of interval-valued fuzzy subspace of is denoted by .

*Remark 2.7. *In Definition 2.6, the definition of can be described in detail as follows:

The following theorem indicates that the dual of intuitionistic fuzzy subspaces and the dual of interval-valued fuzzy subspaces are same upon the lattice isomorphism.

Theorem 2.8. *The mapping by is an isomorphism between the lattices and , where by and .*

##### 2.3. The Dual of Interval-Valued Intuitionistic Fuzzy Subspaces

*Definition 2.9. *Let be an interval-valued intuitionistic fuzzy subspace of -vector space .

Define , where

Then is called the dual of interval-valued intuitionistic fuzzy subspace.

Theorem 2.10. *The interval-valued intuitionistic fuzzy set is an interval-valued intuitionistic fuzzy subspace of .*

*Proof. *Since is the upper bound of and is the lower bound of , it suffices to show that the nonempty sets and are subspaces of for all . The remainder proof can be imitated by Theorem 3.2 of [8].

*Remark 2.11. *The result is also true for intuitionistic fuzzy subspaces and interval-valued fuzzy subspaces.

*Example 2.12. *Let be a two dimensional vector space and be its dual space. We define , where for and

If , , , and , , , , then is an interval-valued intuitionistic fuzzy subspace. By Definition 2.9, , where if , and , if , and . Then is an interval-valued intuitionistic fuzzy subspace of .

If , , , and , , , , then is an intuitionistic fuzzy subspace. By Definition 2.6, , where if , and , if , and . Then is an intuitionistic fuzzy subspace of .

#### 3. The Double Dual of Generalized Fuzzy Subspaces

In this section, we mainly study the double dual of intuitionistic fuzzy subspaces. The double dual of interval-valued fuzzy subspaces and the double dual of interval-valued intuitionistic fuzzy subspaces can be investigated similarly.

Let be a -vector space and be the space of all linear maps from to . Then is the space of all linear maps from to , which is the double dual space for . There exists a canonical injection by . If , then the injection is an isomorphism.

Let be an intuitionistic fuzzy subspace of -vector space . Then the intuitionistic fuzzy subspace is defined by

Theorem 3.1. *Let be an intuitionistic fuzzy subspace of -vector space and be the canonical injection. Then for all .*

*Proof. *Let . Then let

Denote and let .

Let . Then if , we have and ; if , we have and , so .

Hence for , then . Since and , we have

On the other hand, let . Thenif , we have and ; if , we have and , so .

Hence for , then . Since and , we have

So and .

Theorem 3.2. *Let be an intuitionistic fuzzy subspace of -vector space . Then , where and .*

*Proof. *We have

Theorem 3.3. *Let be an intuitionistic fuzzy subspace of finite dimensional -vector space and and . Then the canonical map is an isomorphism between the intuitionistic fuzzy subspace and .*

*Proof. *Follows from Theorems 3.1 and 3.2.

#### 4. Conclusions

In this paper, we study the contents of the dual of generalized fuzzy subspaces. Generalized fuzzy subspaces, including intuitionistic fuzzy subspaces, interval-valued fuzzy subspaces and interval-valued intuitionistic fuzzy subspaces, are the basic contents for the further study of some algebras [18, 19]. Moreover, many algebras have the dual structures. Therefore it makes sense to investigate the dual of generalized fuzzy subspaces. In the future, we will consider the applications of the dual of generalized fuzzy subspaces in coalgebras and bialgebras.

#### Acknowledgments

The authors would like to show their sincere thanks to the referees. This project is supported by the National Natural Science Foundation of China (Grant no. 61070241, 11126301) and Promotive Research Fund for Young and Middle-aged Scientists of Shandong Province (no. BS2011SF002).