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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.

In fuzzy algebra, fuzzy subspaces are basic concepts. They had been introduced by Katsaras and Liu [

After the introduction of fuzzy sets by Zadeh [

In [

An intuitionistic fuzzy set (IFS, for short) on a universe

In 1993, Gau and Buehrer [

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

Let

The notion of interval-valued fuzzy sets was first introduced by Zadeh [

An interval-valued fuzzy set

In [

For interval numbers

In [

Similarly, we can define the following.

Let

The following definition generalizes the definitions of IFS and IVFS.

An interval-valued intuitionistic fuzzy set on a universe

Let

In this paper, intuitionistic fuzzy subspaces, interval-valued fuzzy subspaces and interval-valued intuitionistic fuzzy subspaces are called generalized fuzzy subspaces. In Section

In this paper,

Let

Define

Now, we study the dual of generalized fuzzy subspaces.

Let

Let

Define

Obviously,

In [

(1) If

(2) If

Following the above remark, Definition

In the end, we give an explain of Definition

Let 10 experts vote to

In the voting model,

This is the idea that the

Let

Define

Then

In Definition

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.

The mapping

Let

Define

Then

The interval-valued intuitionistic fuzzy set

Since

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

Let

If

If

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

Let

Let

Let

Denote

Let

if

Hence for

On the other hand, let

if

Hence for

So

Let

We have

Let

Follows from Theorems

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 [

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).