^{1}

^{*}

^{2}

^{2}

In this paper, we introduce the definition of
*L*-fuzzy vector subspace, define its dimension by an
*L*-fuzzy natural number. For a finite-dimensional
*L*-fuzzy vector subspace, we prove that the equality
holds without any restricted conditions. At the same time, we deduce that the formula
holds.

Firstly, fuzzy vector subspace was introduced by Katsaras and Liu [

is valid under certain conditions, where

In this paper, we generalize the results in [

Given a set

An element

The binary relation

In [

Throughout this paper,

If

Some properties of these cut sets can be found in [

In [

(3) For any

Definition 3.1. L-fuzzy vector subspace is a pair

In this definition, when

Let

We can obtain some properties of

Theorem 3.2. Let

(2) For any

The prove is trivial and omitted.

Remark: Since

Example 3.3. Let

Let

We can easily check

Theorem 3.4. Let

(1)

(3) For any

The prove is trivial and omitted.

In the following paper, the vector spaces we discuss are finite-dimensional. For their L-fuzzy vector subspaces, the following observation will be useful.

Remark: Let

In the fact, let

Theorem 3.5. Let

(1)

(2) For all

(3) For all

(4) For all

(5) For all

(6) For all

Proof. We prove

Since

Therefore

Therefore

Theorem 3.6. Let

(1)

(2) For all

Proof.

Therefore

We can define the operations between two L-fuzzy vector subspaces analogous to fuzzy vector subspaces.

Definition 3.7. Let

Definition 3.8. Let

Theorem 3.9. Let

(1)

(2)

The proof of the theorem is trivial and it is omitted.

Theorem 3.10. Let

(1) For all

(2) For all

(3) For any

(4) For any

Proof. The proofs of (1) and (2) are easy by the definition of

(3) For any

(4) By the definition of the sum of L-fuzzy vector subspaces, for any

Theorem 3.11. Let

The prove is trivial and omitted.

Definition 4.1. Let

is called the L-fuzzy dimensional function of the L-fuzzy vector subspace

Theorem 4.2. For each

Proof. For any

exists

This completes the proof.

Theorem 4.3. Let the pair

If

In particular,

Proof. In order to prove

In order to prove for any

Therefore

Theorem 4.4. Let

In particular,

Proof.

Let

Therefore for any

Theorem 4.5. Let

Proof. We denote the sum of

Therefore

Definition 4.6. Suppose that

(1)

(2) For all

Theorem 4.7. Suppose that

The prove is trivial and omitted.

Theorem 4.8. Suppose that

Proof. Suppose that

Since

Therefore

In this paper, L-fuzzy vector subspace is defined and showed that its dimension is an L-fuzzy natural number. Based on the definitions, some good properties of crisp vector spaces are hold in a finite-dimensional L-fuzzy vector subspace. In particular, the

equality

The authors would like to thank the reviewers for their valuable comments and sug- gestions.

The project is by the Science & Technology Program of Beijing Municipal Commission of Education (KM201611417007), the NNSF of China (11371002), the academic youth backbone project of Heilongjiang Education Department (1251G3036), the foundation of Heilongjiang Province (A201209).

Huang, C.E, Song, Y. and Wang, X.R. (2016) L-Fuzzy Vector Subspaces and Its Fuzzy Dimension. Advances in Linear Algebra & Matrix Theory, 6, 158-168. http://dx.doi.org/10.4236/alamt.2016.64015