On properties of fuzzy subspaces of vectorspaces

H. Hedayati

Department of Mathematics, Faculty of Basic Science,

Babol University of Technology, Babol, Iran

E-mail: h.hedayati@nit.ac.ir, hedayati143@yahoo.com

Abstract

In this paper, we introduce the notion of normal fuzzy subspace of

vector spaces. By using it, we construct new fuzzy subspaces. We also

show that, under certain conditions, a fuzzy subspace of a vector space

is two-valued and takes 0 and 1.

Mathematics Subject Classification: 08A72

Keywords: vector space, fuzzy subspace, normal fuzzy subspace

1 Introduction

Zadeh in [17] introduced the notion of fuzzy set and started a generalized logic.

After that reconsideration of mathematics concepts begun. Also there have

been a number of generalizations of this fundamental concept. Fuzzy algebraic

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structures play a prominent role in mathematics with wide applications in

many other branches such as theoretical physics, computer sciences, control

engineering, information sciences, coding theory, topological spaces, logic, set

theory, group theory, groupoids, real analysis, measure theory etc (see [4], [5],

[10] and [12]). In 1977, Katsaras and Liu [7] formulated and studied the concept

of a fuzzy subspace of a vector space. Since then, a host of mathematicians are

involved in extending the basic concepts and results from the theory of crisp

vector spaces to the broader framework of the fuzzy setting. However, not all

the results can be fuzzified. In [8], among other concepts and results, the fuzzy

coset of a fuzzy subspace is defined and the algebraic nature of fuzzy subspaces

under homomorphism is studied. In [9], the fuzzy basis and dimension of a

fuzzy subspace are defined and studied. In [1] and [3], the fuzzy subspaces

over fuzzy fields are discussed.

In this paper, some properties of fuzzy subspaces of vector spaces are in-

vestigated. Specially, some ways are created to construct new fuzzy subspaces

from the old. Also the notion of normal fuzzy subspace of vector spaces is in-

troduced. We see some normal fuzzy subspaces can be constructed by a fuzzy

subspace. Finally, we show that, when a non-constant normal fuzzy subspace

be a maximal in the partial ordered set of normal fuzzy subspaces of a vector

space, then this fuzzy subspace is two-valued and takes the values 0 and 1.

2 Preliminaries

An abelian group (V, +) on a field F is called a vector space on F if there exists

a map . : V × F −→ V such that for all x, y ∈ V and a, b ∈ F the following

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conditions hold:

(i) 1x = x,

(ii) (ab)x = a(bx),

(iii) a(x + y) = ax + ay,

(v) (a + b)x = ax + by.

Also a non-empty subset W of a vector space V is called a subspace, if W

is a vector space on F .

Let X be an ordinary set. By a fuzzy set µ in X, we mean a function

µ : X −→ [0, 1] with the grade of membership µ(x) for x ∈ X. If t ∈ [0, 1),

then µt = {x ∈ X| µ(x) ≥ t} is called a level subset of µ.

3 Fuzzy normal subspaces

In what follows, V is a vector space on a field F , unless otherwise specified.

Definition 3.1. ([7], [11]) A fuzzy set µ of V is called a fuzzy subspace of

V, if for all x, y ∈ V and a ∈ F the following conditions hold:

(i) µ(x + y) ≥ µ(x) ∧ µ(y),

(ii) µ(−x) ≥ µ(x),

(iii) µ(ax) ≥ µ(x).

Clearly, if µ is a fuzzy subspace of V, then µ(0) ≥ µ(x) for all x ∈ V. Also,

µ is a fuzzy subspace of V if and only if µt is a subspace of V for all t ∈ [0, 1).

Lemma 3.2. If µ is a fuzzy subspace of V, then the set Vµ = {x ∈

V | µ(x) = µ(0)} is a subspace of V.

Proof. Let x, y ∈ Vµ. Then µ(x) = µ(y) = µ(0). Since µ is a fuzzy

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subspace, it follows that

µ(x − y) ≥ µ(x) ∧ µ(y) = µ(0) ∧ µ(0) = µ(0).

On the other hand µ(x − y) ≤ µ(0). Hence we have µ(x − y) = µ(0) and

so x − y ∈ Vµ. Also for any x ∈ Vµ and a ∈ F, we get µ(ax) ≥ µ(x) = µ(0).

On the other hand µ(ax) ≤ µ(0). Hence, we obtain µ(ax) = µ(0), which shows

that ax ∈ Vµ. Consequently, the set Vµ is a subspace of V. �

Definition 3.3. A fuzzy subspace of V is said to be normal if there exists

x ∈ V such that µ(x) = 1. Note that if a fuzzy subspace of V is normal, then

µ(0) = 1. Hence µ is a normal fuzzy subspace if and only if µ(0) = 1.

Theorem 3.4. Let µ be a fuzzy subspace of V and let µ̃ be a fuzzy set in

V defined by µ̃(x) = µ(x) + 1 − µ(0) for all x ∈ V. Then µ̃ is a normal fuzzy

subspace of V containing µ.

Proof. Let x, y ∈ V and a ∈ F. Then

µ̃(x − y) = µ(x − y) + 1 − µ(0) ≥ (µ(x) ∧ µ(y)) + 1 − µ(0) =

(µ(x) + 1 − µ(0)) ∧ (µ(y) + 1 − µ(0)) = µ̃(x) ∧ µ̃(y).

Also we have

µ̃(ax) = µ(ax) + 1 − µ(0) ≥ µ(x) + 1 − µ(0) = µ̃(x).

Clearly, µ̃(0) = 1 and µ ⊆ µ̃. This completes the proof. �

Corollary 3.5. If µ is a fuzzy subspace of V satisfying µ̃(x) = 0 for some

x ∈ V, then µ(x) = 0.

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Lemma 3.6. Let χW be the characteristic function of a subset W ⊆ V.

Then W is a subspace of V if and only if χW is a fuzzy subspace of V.

Proof. It is directly followed from discussion after Definition 3.1. �

Theorem 3.7. For any subspace W of V, the characteristic function χW
is a normal fuzzy subspace of V and VχW = W.

Proof. Straightforward. �

Theorem 3.8. A fuzzy subspace µ of V is normal if and only if µ̃ = µ.

Proof. If µ̃ = µ, then it is obvious that µ is a normal fuzzy subspace

of V. Assume that µ is a normal fuzzy subspace of V and let x ∈ V. Then

µ̃(x) = µ(x) + 1 − µ(0) = µ(x), and hence µ̃ = µ. �

Theorem 3.9. If µ is a fuzzy subspace of V, then (̃µ̃) = µ̃.

Proof. Straightforward. �

Theorem 3.10. Let µ be a fuzzy subspace of V. If there exists a fuzzy

subspace ν of V satisfying ν̃ ⊆ µ, then µ is a normal fuzzy subspace of V.

Proof. Suppose there exists a fuzzy subspace ν of V such that ν̃ ⊆ µ.

Then 1 = ν̃(0) ≤ µ(0), and therefore µ(0) = 1. �

Corollary 3.11. Let µ be a fuzzy subspace of V. If there exists a fuzzy

subspace ν of V satisfying ν̃ ⊆ µ, then µ̃ = µ.

Proof. It is immediately obtained from Theorem 3.10 and definition of µ̃.

�

Theorem 3.12. Let µ be a fuzzy subspace of V and f : [0, µ(0)] −→ [0, 1]

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be an increasing map. Define a fuzzy set µf : V −→ [0, 1] by µf (x) = f (µ(x))

for all x ∈ V. Then µf is a fuzzy subspace of V. In particular, if f (t) ≥ t for

all t ∈ [0, µ(0)] then µ ⊆ µf .

Proof. Let x, y ∈ V. Then

µf (x − y) = f (µ(x − y)) ≥ f (µ(x) ∧ µ(y)) =

f (µ(x)) ∧ f (µ(y)) = µf (x) ∧ µf (y).

Also if a ∈ F and x ∈ V, then µf (ax) = f (µ(ax)) ≥ f (µ(x)) = µf (x).

Hence µf is a fuzzy subspace of V. Assume that f (t) ≥ t for all t ∈ [0, µ(0)].

Then µf (x) = f (µ(x)) ≥ µ(x) for all x ∈ V, which means µ ⊆ µf . �

Theorem 3.13. Let µ be a non-constant normal fuzzy subspace of V,

which is maximal in the partial ordered set of normal fuzzy subspaces of V

under fuzzy sets inclusion. Then µ is a two-valued fuzzy subspace and takes

the values 0 and 1.

Proof. We know µ(0) = 1. Let x ∈ V be such that µ(x) 6= 1. It is enough to

show that µ(x) = 0. Assume that there exists x′ ∈ V such that 0 < µ(x′) < 1.

Define a fuzzy set ν : V −→ [0, 1] by ν(x) = 1/2(µ(x) + µ(x′)) for all x ∈ V.

Then clearly ν is well-defined. Let x, y ∈ V. Then

ν(x − y) = 1/2(µ(x − y) + µ(x′)) ≥ 1/2((µ(x) ∧ µ(y)) + µ(x′)) =

(1/2(µ(x) + µ(x′))) ∧ (1/2(µ(y) + µ(x′))) = ν(x) ∧ ν(y).

Also if a ∈ F and x ∈ V, then

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ν(ax) = 1/2(µ(ax) + µ(x′)) ≥ 1/2(µ(x) + µ(x′)) = ν(x).

Hence ν is a fuzzy subspace of V. Now we have

ν̃(x) = ν(x) + 1 − ν(0) =

1/2(µ(x) + µ(x′)) + 1 − 1/2(µ(0) + µ(x′)) = 1/2(µ(x) + 1).

So ν̃(0) = 1/2(µ(0) + 1) = 1. Thus ν̃ is a normal fuzzy subspace of V. Also

ν̃(0) = 1 > ν̃(x′) = 1/2(µ(x′) + 1) > µ(x′). We know that ν̃ is non-constant.

So by ν̃(x′) > µ(x′), it follows that µ is not maximal, which is a contradiction.

Therefore µ takes only the values 0 and 1. �

Theorem 3.14. Let µ be a fuzzy subspace of V and let µ be a fuzzy set

in V defined by µ(x) = µ(x)/µ(0) for all x ∈ V. Then µ is a normal fuzzy

subspace of V containing µ.

Proof. For any x, y ∈ V, we have

µ(x − y) = µ(x − y)/µ(0) ≥ (1/µ(0))(µ(x) ∧ µ(y)) =

(µ(x)/µ(0)) ∧ (µ(y)/µ(0)) = µ(x) ∧ µ(y).

Also if a ∈ F and x ∈ V we get

µ(ax) = (µ(ax)/µ(0)) ≥ (µ(x)/µ(0)) = µ(x).

Hence µ is a fuzzy subspace of V. Clearly µ(0) = 1 and µ ⊆ µ. �

Corollary 3.15. If µ is a fuzzy subspace of V satisfying µ(x) = 0 for some

x ∈ V, then µ(x) = 0.

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Proof. Obvious. �

Theorem 3.16. Let µ be a non-constant fuzzy subspace of V such that µ̃

is a maximal in the partial ordered set of normal fuzzy subspace of V under

fuzzy sets inclusion. Then

(1) µ is normal.

(2) µ takes only the values 0 and 1.

(3) χVµ = µ.

(4) Vµ is a maximal subspace of V.

Proof. Since µ is non-constant, so µ̃ is non-constant maximal. Also µ̃

is normal, which implies µ̃ takes only values 0 and 1 by Theorem 3.13. . If

µ̃(x) = 1, then µ(x) = µ(0) and if µ̃(x) = 0, then µ(x) = µ(0)−1. By Corollary

3.5, we have µ(x) = 0 which implies µ(0) = 1. Therefore µ is normal, and also

µ̃ = µ by Theorem 3.8, which proves (1) and (2).

(3) Clearly χVµ ⊆ µ and χVµ takes only the values 0 and 1. Let x ∈ V and

µ(x) = 0, then µ ⊆ χVµ . If µ(x) = 1 then x ∈ Vµ and so χVµ (x) = 1. In any

case µ ⊆ χVµ .

(4) Since µ is non-constant, Vµ is a proper subspace of V. Let W be a

subspace of V such that Vµ ⊆ W. Then we obtain µ = χVµ ⊆ χW. Since µ

and χW are normal and µ = µ̃ is maximal in the partial ordered set of normal

fuzzy subspaces under fuzzy sets inclusion, we have µ = χW or χW(x) = 1

for all x ∈ V, so W = V. If µ = χW then Vµ = VχW = W by Theorem 3.7 .

Therefore Vµ is a maximal subspace of V. �

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