The Dual of Generalized Fuzzy Subspaces

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 [1–4]. 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 𝑘-vector spaces. This notion was also studied and applied in many branches [2, 7–9], 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 𝑋 is defined as an object having the form 𝐴={⟨𝑥,𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)⟩∣𝑥∈𝑋}, where the functions 𝜇_𝐴∶𝑋→[0,1] and 𝜈_𝐴∶𝑋→[0,1] denote the degree of membership (namely, 𝜇_𝐴(𝑥)) and the degree of nonmembership (namely, ) of each element to the set , respectively, and 0≤𝜇_𝐴(𝑥)+𝜈_𝐴(𝑥)≤1 for each 𝑥∈𝑋. For the sake of simplicity, we shall use the symbol 𝐴=(𝜇_𝐴,𝜈_𝐴) for the intuitionistic fuzzy set 𝐴={⟨𝑥,𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)⟩∣𝑥∈𝑋}. The class of IFSs on a universe 𝑋 is denoted by IFS (𝑋).

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 𝐴=(𝜇_𝐴,𝜈_𝐴) be an intuitionistic fuzzy set of 𝑘-vector space 𝑉. For any 𝑥,𝑦∈𝑉 and 𝛼,𝛽∈𝑘, if it satisfies 𝜇_𝐴(𝛼𝑥+𝛽𝑦)≥min{𝜇_𝐴(𝑥),𝜇_𝐴(𝑦)} and 𝜈_𝐴(𝛼𝑥+𝛽𝑦)≤max{𝜈_𝐴(𝑥),𝜈_𝐴(𝑦)}, then 𝐴=(𝜇_𝐴,𝜈_𝐴) is called an intuitionistic fuzzy subspace of 𝑉.

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 𝐴=(𝐴^𝐿,𝐴^𝑈) on a universe 𝑋 (IVFS, for short) is a mapping 𝑋→Int([0,1]), where Int([0,1]) stands for the set of all closed subintervals of [0,1]. The class of all IVFSs on a universe 𝑋 is denoted by IVFS(𝑋).

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

Notations. For interval numbers 𝐷₁=[𝑎^𝐿1,𝑏^𝑈1],𝐷₂=[𝑎^𝐿2,𝑏^𝑈2]∈Int([0,1]). We define 𝑟𝑟mmaixn𝐷𝐷₁₁,𝐷₂=𝑟min𝑎^𝐿1,𝑏^𝑈1,𝑎^𝐿2,𝑏^𝑈2=min𝑎^𝐿1,𝑎^𝐿2,min𝑏^𝑈1,𝑏^𝑈2,,𝐷₂=𝑟max𝑎^𝐿1,𝑏^𝑈1,𝑎^𝐿2,𝑏^𝑈2=max𝑎^𝐿1,𝑎^𝐿2,max𝑏^𝑈1,𝑏^𝑈2,(1.1) and put (a)𝐷₁≤𝐷₂⇔𝑎^𝐿1≤𝑎^𝐿2 and 𝑏^𝑈1≤𝑏^𝑈2, (b)𝐷₁=𝐷₂⇔𝑎^𝐿1=𝑎^𝐿2 and 𝑏^𝑈1=𝑏^𝑈2, (c)𝐷₁<𝐷₂⇔𝐷₁≤𝐷₂ and 𝐷₁≠𝐷₂.
In [16], Deschrijver and Kerre presented that the mapping between the lattices IVFS(𝑋) and IFS(𝑋) 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 𝐴=(𝐴^𝐿,𝐴^𝑈) be an interval-valued fuzzy set of 𝑘-vector space 𝑉. For any 𝑥,𝑦∈𝑉 and 𝛼,𝛽∈𝑘, if it satisfies 𝐴(𝛼𝑥+𝛽𝑦)≥𝑟min{𝐴(𝑥),𝐴(𝑦)}, then 𝐴=(𝐴^𝐿,𝐴^𝑈) is called an interval-valued fuzzy subspace of 𝑉.

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 𝑋 (IVIFS, for short) is an object of the form 𝐴={(𝑥,𝑀_𝐴(𝑥),𝑁_𝐴(𝑥))∣𝑥∈𝑋}, where 𝑀_𝐴∶𝑋→Int([0,1]) and 𝑁_𝐴∶𝑋→Int([0,1]) 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 𝐴=(𝜇_𝐴,𝜈_𝐴) be an intuitionistic fuzzy set. Then □𝐴=(𝜇_𝐴,𝜇^𝑐𝐴) and ◊𝐴=(𝜈^𝑐𝐴,𝜈_𝐴). Using the notations in the dual of intuitionistic fuzzy subspaces, we have the following.

Remark 2.3. (1) If 𝐴=□𝐴, then 𝐴^∗=◊𝐴^∗.
(2) If 𝐴=◊𝐴, then 𝐴^∗=□𝐴^∗.
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. 𝐴^∗=(𝜇_𝐴^∗,𝜈_𝐴∗) is an intuitionistic fuzzy subspace of 𝑉^∗ if and only if □𝐴^∗ and ◊𝐴^∗ are intuitionistic fuzzy subspaces of 𝑉^∗.

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

Remark 2.5. Let 10 experts vote to 𝐴 and 𝐵 and require per expert to vote at most one. The result is that 𝐴 is 5 and 𝐵 is 3. That is, the number of the supporters of 𝐴 is 5 and the number of the supporters of 𝐵 is 3.
In the voting model, 𝐴 and 𝐵 can be regarded as the dual objects. The analysis indicates that the number of supporters of 𝐴 is equal to the number of objectors of 𝐵 and the number of supporters of 𝐵 is equal to the number of objectors of 𝐴. So if 𝐵 is the dual of 𝐴 and the numbers of supporters and objectors of 𝐴 are known, then we can calculate the supporters of 𝐵 by the objectors of 𝐴 and the objectors of 𝐵 by the supporters of 𝐴.
This is the idea that the 𝜇_𝐴^∗ is defined by 𝜈_𝐴 and the 𝜈_𝐴^∗ is defined by 𝜇_𝐴 in Definition 2.2.

2.2. The Dual of Interval-Valued Fuzzy Subspaces

Definition 2.6. Let 𝐴=(𝐴^𝐿,𝐴^𝑈) be an interval-valued fuzzy subspace of 𝑘-vector space 𝑉.
Define 𝐴^∗=(𝐴^∗𝐿,𝐴^∗𝑈), where 𝐴^∗∶𝑉^∗⟶Int([0,1])by𝐴^∗(𝑓)=[[11,,11]]−−sinufp{{𝐴𝐴((𝑥𝑥))∣∣𝑥𝑥∈∈𝑉𝑉,}𝑓(𝑥)≠0}iiff𝑓𝑓=≠00.(2.3)
Then 𝐴^∗=(𝐴^∗𝐿,𝐴^∗𝑈) is called the dual of an interval-valued fuzzy subspace. The class of all the dual of interval-valued fuzzy subspace of 𝑉^∗ is denoted by DIVFS(𝑉^∗).

Remark 2.7. In Definition 2.6, the definition of 𝐴^∗=(𝐴^∗𝐿,𝐴^∗𝑈) can be described in detail as follows: 𝐴^∗𝐿∶𝑉^∗⟶[0,1]by𝐴^∗𝐿(𝑓)=1−sup𝐴^𝐿1−inf𝐴_𝐿(𝑥)∣𝑥∈𝑉,𝑓(𝑥)≠0if𝑓≠0𝐴(𝑥)∣𝑥∈𝑉if𝑓=0,_∗𝑈∶𝑉^∗⟶[0,1]by𝐴^∗𝑈(𝑓)=1−sup𝐴^𝑈1−inf𝐴_𝑈(𝑥)∣𝑥∈𝑉,𝑓(𝑥)≠0if𝑓≠0(𝑥)∣𝑥∈𝑉if𝑓=0.(2.4)
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 𝜑∶DIVFS(𝑉^∗)→DIFS(𝑉^∗) by 𝜇↦𝐴 is an isomorphism between the lattices DIVFS(𝑉^∗) and DIFS(𝑉^∗), where 𝜇=(𝜇^𝐿,𝜇^𝑈)∶𝑉^∗→Int([0,1]) by 𝑓↦(𝜇^𝐿(𝑓),𝜇^𝑈(𝑓)) and 𝐴=(𝜇_𝐴,𝜈_𝐴)=(𝜇_𝐴=𝜇^𝑈,𝜈_𝐴=1−𝜇^𝐿).

2.3. The Dual of Interval-Valued Intuitionistic Fuzzy Subspaces

Definition 2.9. Let 𝐴=(𝑀_𝐴,𝑁_𝐴) be an interval-valued intuitionistic fuzzy subspace of 𝑘-vector space 𝑉.
Define 𝐴^∗=(𝑀_𝐴^∗,𝑁_𝐴^∗), where 𝑀_𝐴^∗∶𝑉^∗⟶Int([0,1])by𝑀_𝐴^∗(𝑓)=inf𝑁^𝐴sup𝑁_𝐴(𝑥)∣𝑥∈𝑉,𝑓(𝑥)≠0if𝑓≠0𝑁(𝑥)∣𝑥∈𝑉if𝑓=0,_𝐴∗∶𝑉^∗⟶Int([0,1])by𝑁_𝐴^∗(𝑓)=sup𝑀^𝐴inf𝑀_𝐴(𝑥)∣𝑥∈𝑉,𝑓(𝑥)≠0if𝑓≠0(𝑥)∣𝑥∈𝑉if𝑓=0.(2.5)
Then 𝐴^∗=(𝑀_𝐴^∗,𝑁_𝐴^∗) is called the dual of interval-valued intuitionistic fuzzy subspace.

Theorem 2.10. The interval-valued intuitionistic fuzzy set 𝐴^∗=(𝑀_𝐴^∗,𝑁_𝐴^∗) is an interval-valued intuitionistic fuzzy subspace of 𝑉^∗.

Proof. Since 𝑀_𝐴^∗(0) is the upper bound of 𝑀_𝐴^∗(𝑉^∗) and 𝑁_𝐴^∗(0) is the lower bound of 𝑁_𝐴^∗(𝑉^∗), it suffices to show that the nonempty sets 𝑈(𝑀_𝐴^∗,[𝛿₁,𝛿₂])={𝑓∈𝑉^∗∣𝑀_𝐴^∗(𝑓)≥[𝛿₁,𝛿₂]} and 𝐿(𝑁_𝐴^∗,[𝜉₁,𝜉₂])={𝑓∈𝑉^∗∣𝑁_𝐴^∗(𝑓)≤[𝜉₁,𝜉₂]} are subspaces of 𝑉^∗ for all [𝛿₁,𝛿₂],[𝜉₁,𝜉₂]∈[𝑁_𝐴^∗(0),𝑀_𝐴^∗(0)]. 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 𝑉={𝑥=(0,𝑎,𝑏)∣𝑎,𝑏∈𝑅} be a two dimensional vector space and 𝑉^∗ be its dual space. We define 𝐴=(𝑀_𝐴,𝑁_𝐴), where for 𝑘∈𝑅 and 𝑥∈𝑉𝑀_𝐴(𝑥)=⎧⎪⎨⎪⎩𝑎¹𝑏if𝑥=(0,𝑘,0)₁𝑐if𝑥=(0,0,𝑘)₁𝑑if𝑥=(0,0,0)₁otherwise,𝑁^𝐴(𝑥)=⎧⎪⎨⎪⎩𝑎²𝑏if𝑥=(0,𝑘,0)₂𝑐if𝑥=(0,0,𝑘)₂𝑑if𝑥=(0,0,0)₂otherwise,(2.6)
If 𝑎₁=[0.3,0.4], 𝑏₁=[0.2,0.5], 𝑐₁=[1,1], 𝑑₁=[0.2,0.4] and 𝑎₂=[0.5,0.6], 𝑏₂=[0.4,0.5], 𝑐₂=[0,0], 𝑑₂=[0.5,0.6], then 𝐴 is an interval-valued intuitionistic fuzzy subspace. By Definition 2.9, 𝐴^∗=(𝑀_𝐴^∗,𝑁_𝐴^∗), where if 𝑓≠0, 𝑀_𝐴^∗(𝑓)=[0.4,0.5] and 𝑁_𝐴^∗(𝑓)=[0.3,0.5], if 𝑓=0, 𝑀_𝐴^∗(0)=[0.5,0.6] and 𝑁_𝐴^∗(0)=[0.2,0.4]. Then 𝐴^∗=(𝑀_𝐴^∗,𝑁_𝐴^∗) is an interval-valued intuitionistic fuzzy subspace of 𝑉^∗.
If 𝑎₁=0.3, 𝑏₁=0.4, 𝑐₁=1, 𝑑₁=0.3 and 𝑎₂=0.5, 𝑏₂=0.2, 𝑐₂=0, 𝑑₁=0.5, then 𝐴 is an intuitionistic fuzzy subspace. By Definition 2.6, 𝐴^∗=(𝑀_𝐴^∗,𝑁_𝐴^∗), where if 𝑓≠0, 𝑀_𝐴^∗(𝑓)=0.2 and 𝑁_𝐴^∗(𝑓)=0.4, if 𝑓=0, 𝑀_𝐴^∗(0)=0.5 and 𝑁_𝐴^∗(0)=0.3. Then 𝐴^∗=(𝑀_𝐴^∗,𝑁_𝐴^∗) is an intuitionistic fuzzy subspace of 𝑉^∗.

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 𝑉 be a 𝑘-vector space and 𝑉^∗ be the space of all linear maps from 𝑉 to 𝑘. Then 𝑉^∗∗=(𝑉^∗)^∗ is the space of all linear maps from 𝑉^∗ to 𝑘, which is the double dual space for 𝑉. There exists a canonical injection 𝑖∶𝑉→𝑉^∗∗ by 𝑖(𝑥)(𝑓)=𝑓(𝑥). If dim𝑉<∞, then the injection 𝑖 is an isomorphism.

Let 𝐴=(𝜇_𝐴,𝜈_𝐴) be an intuitionistic fuzzy subspace of 𝑘-vector space 𝑉. Then the intuitionistic fuzzy subspace 𝐴^∗∗=(𝜇_𝐴^∗∗,𝜈_𝐴^∗∗)∶𝑉^∗∗→[0,1] is defined by 𝜇_𝐴^∗∗(𝑥)=inf𝜈^𝐴∗sup𝜈_𝐴∗(𝑓)∣𝑓∈𝑉^∗,𝑥(𝑓)≠0if𝑥≠0𝜈(𝑓)∣𝑓∈𝑉^∗if𝑥=0,_𝐴∗∗(𝑥)=sup𝜇^𝐴∗inf𝜇_𝐴∗(𝑓)∣𝑓∈𝑉^∗,𝑥(𝑓)≠0if𝑥≠0(𝑓)∣𝑓∈𝑉^∗if𝑥=0.(3.1)

Theorem 3.1. Let 𝐴=(𝜇_𝐴,𝜈_𝐴) be an intuitionistic fuzzy subspace of 𝑘-vector space 𝑉 and 𝑖∶𝑉→𝑉^∗∗ be the canonical injection. Then 𝐴^∗∗(𝑖(𝑥))=𝐴(𝑥) for all 𝑥∈𝑉⧵{0}.

Proof. Let 0≠𝑥∈𝑉. Then let 𝜇_𝐴^∗∗(𝑖(𝑥))==inf𝜈in_𝐴^∗f(𝑓)∣𝑓_{𝑓∈𝑉∗,𝑖(𝑥)(𝑓)=𝑓(𝑥)≠0}∈𝑉^∗,𝑖(𝑥)(𝑓)≠0𝜈sup𝜇_𝐴(𝑦)∣𝑦∈𝑉,𝑓(𝑦)≠0_𝐴∗∗,(𝑖(𝑥))==sup𝜇sup_𝐴^∗(𝑓)∣𝑓_{𝑓∈𝑉∗,𝑖(𝑥)(𝑓)=𝑓(𝑥)≠0}∈𝑉^∗,𝑖(𝑥)(𝑓)≠0inf𝜈_𝐴(𝑦)∣𝑦∈𝑉,𝑓(𝑦)≠0.(3.2)
Denote 𝑆={𝑓∈𝑉^∗∣𝑓(𝑥)≠0} and let 𝜇_𝐴(𝑥)=𝑡₁,𝜈_𝐴(𝑥)=𝑡₂.
Let 𝑓∈𝑆. Then if 𝑦∈𝑈^𝑡1, we have 𝑓(𝑦)≠0 and sup_𝑦∈𝑈^𝑡1_{,𝑓(𝑦)≠0}{𝜇_𝐴(𝑦)}≥𝑡₁; if 𝑦∉𝑈^𝑡1, we have 𝑓(𝑦)≠0 and 𝜇_𝐴(𝑦)<𝑡₁, so sup_{𝑦∈𝑉⧵𝑈}^𝑡1_{,𝑓(𝑦)≠0}{𝜇_𝐴(𝑦)}<𝑡₁.
Hence for 𝑦∈𝑉⧵{0}, then sup_{𝑓(𝑦)≠0}{𝜇_𝐴(𝑦)}≥𝑡₁. Since 𝑥∈𝑉⧵{0} and 𝑓(𝑥)≠0, we have inf_{𝑓∈𝑉∗,𝑖(𝑥)(𝑓)=𝑓(𝑥)≠0}sup𝜇_𝐴(𝑦)∣𝑦∈𝑉,𝑓(𝑦)≠0=𝑡₁=𝜇_𝐴(𝑥).(3.3)
On the other hand, let 𝑓∈𝑆. Thenif 𝑦∈𝐿^𝑡2, we have 𝑓(𝑦)≠0 and inf_𝑦∈𝐿^𝑡2_{,𝑓(𝑦)≠0}{𝜈_𝐴(𝑦)}≤𝑡₂; if 𝑦∉𝐿^𝑡2, we have 𝑓(𝑦)≠0 and 𝜈_𝐴(𝑦)>𝑡₂, so inf_{𝑦∈𝑉⧵𝐿}^𝑡2_{,𝑓(𝑦)≠0}{𝜈_𝐴(𝑦)}>𝑡₂.
Hence for 𝑦∈𝑉⧵{0}, then inf_{𝑓(𝑦)≠0}{𝜈_𝐴(𝑦)}≤𝑡₂. Since 𝑥∈𝑉⧵{0} and 𝑓(𝑥)≠0, we have sup_{𝑓∈𝑉∗,𝑖(𝑥)(𝑓)=𝑓(𝑥)≠0}inf𝜈_𝐴(𝑦)∣𝑦∈𝑉,𝑓(𝑦)≠0=𝑡₂=𝜈_𝐴(𝑥).(3.4)
So 𝜇_𝐴^∗∗(𝑖(𝑥))=𝜇_𝐴(𝑥) and 𝜈_𝐴^∗∗(𝑖(𝑥))=𝜈_𝐴(𝑥).

Theorem 3.2. Let 𝐴=(𝜇_𝐴,𝜈_𝐴) be an intuitionistic fuzzy subspace of 𝑘-vector space 𝑉. Then 𝐴^∗∗(0)=(𝜇_𝐴^∗∗(0),𝜈_𝐴^∗∗(0)), where 𝜇_𝐴^∗∗(0)=sup{𝜇_𝐴(𝑥)∣𝑥∈𝑉⧵{0}} and 𝜈_𝐴^∗∗(0)=inf{𝜈_𝐴(𝑥)∣𝑥∈𝑉⧵{0}}.

Proof. We have 𝜇_𝐴^∗∗(0)==ssuupp𝜈𝜈_𝐴^∗_𝐴∗(𝑓)∣𝑓∈𝑉^∗=sup_{0≠𝑓∈𝑉∗}(𝑓)∣0≠𝑓∈𝑉^∗=sup𝜇_𝐴sup𝜇_𝐴(𝑥)∣𝑥∈𝑉,𝑓(𝑥)≠0𝜈(𝑥)𝑥∈𝑉⧵{0}_𝐴∗∗,(0)==iinnff𝜇𝜇_𝐴^∗_𝐴∗(𝑓)∣𝑓∈𝑉^∗=inf_{0≠𝑓∈𝑉∗}(𝑓)∣0≠𝑓∈𝑉^∗=inf𝜈_𝐴inf𝜈_𝐴(𝑥)∣𝑥∈𝑉,𝑓(𝑥)≠0(𝑥)∣𝑥∈𝑉⧵{0}.(3.5)

Theorem 3.3. Let 𝐴=(𝜇_𝐴,𝜈_𝐴) be an intuitionistic fuzzy subspace of finite dimensional 𝑘-vector space 𝑉 and 𝜇_𝐴(0)=sup{𝜇_𝐴(𝑉⧵{0})} and 𝜈_𝐴(0)=inf{𝜈_𝐴(𝑉⧵{0})}. Then the canonical map 𝑖∶𝑉→𝑉^∗∗ is an isomorphism between the intuitionistic fuzzy subspace 𝐴 and 𝐴^∗∗.

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).