**4.** *M***-Hazy Subspaces**

In this section, we introduce the concept of *M*-hazy subspaces of *M*-hazy vector space.

**Definition 12.** *Assume that* (*<sup>V</sup>*, ⊕, ◦, *F*) *is an M-hazy vector space over an M-hazy field* (*<sup>F</sup>*, +, •)*. A nonempty subset W of V is called an M-hazy subspace of V if W itself is an M-hazy vector space over F.*

**Theorem 2.** *Assume W is a nonempty subset of an M-hazy vector space* (*<sup>V</sup>*, ⊕, ◦, *F*) *over an M-hazy field* (*<sup>F</sup>*, +, •)*; then,* (*<sup>W</sup>*, ⊕, ◦, *F*) *is an M-hazy subspace of* (*<sup>V</sup>*, ⊕, ◦, *F*) *over* (*<sup>F</sup>*, +, •) *if and only if the following conditions hold:*


**Proof.** The proof is simple and omitted.

**Theorem 3.** *Assume W is a nonempty subset of an M-hazy vector space* (*<sup>V</sup>*, ⊕, ◦, *F*) *over an M-hazy field* (*<sup>F</sup>*, +, •)*; then,* (*<sup>W</sup>*, ⊕, ◦, *F*) *is an M-hazy subspace of* (*<sup>V</sup>*, ⊕, ◦, *F*) *over* (*<sup>F</sup>*, +, •) *if and only if the following conditions hold:*

$$(1)\qquad\forall\iota,\upsilon\in\mathcal{W},\text{ we have }\bigvee\_{p\in\mathcal{W}} (\iota\leftrightarrow(-\upsilon)) (p)\neq\bot,$$

*(2)* ∀*u* ∈ *W and* ∀*a* ∈ *F, we have p*∈*W* (*a* ◦ *u*)(*p*) = ⊥*.*

**Proof.** The proof is similar to the proof of Theorem 4.4 in [12] so it is omitted.

**Theorem 4.** *Assume W is a nonempty subset of an M-hazy vector space* (*<sup>V</sup>*, ⊕, ◦, *F*) *over an M-hazy field* (*<sup>F</sup>*, +, •)*; then,* (*<sup>W</sup>*, ⊕, ◦, *F*) *is an M-hazy subspace of* (*<sup>V</sup>*, ⊕, ◦, *F*) *over* (*<sup>F</sup>*, +, •) *if and only if* ∀*<sup>u</sup>*, *v*, *p*, *q*,*<sup>r</sup>* ∈ *W and* ∀*<sup>a</sup>*, *b* ∈ *F, we have*

$$\bigvee\_{r \in \mathcal{W}} ((a \diamond u) \oplus (b \diamond v))(r) \neq \bot$$

**Proof.** Assume that *W* is an *M*-hazy subspace of *V* over an *M*-hazy field *F*. Suppose that

$$\bigvee\_{r \in W} ((a \circ u) \oplus (b \circ v))(r) = \bot.$$

On the other hand, *p*∈*W* (*a* ◦ *u*)(*p*) = ⊥ and *q*∈*W* (*b* ◦ *v*)(*q*) = ⊥ by Theorem 3. Hence, *r*∈*W* (*p* ⊕ *q*)(*r*) = ⊥, which is a contradiction. Hence,

$$\bigvee\_{r \in W} ((a \circ u) \oplus (b \circ v))(r) \neq \bot$$

Conversely, suppose *r*∈*W* ((*a* ◦ *u*) ⊕ (*b* ◦ *v*))(*r*) = ⊥. Since *W* is a nonempty subset of *V* and we know that (MH1) and (MH2) holds in *V*, thus it holds in *W*. Hence, ∀*<sup>u</sup>*, *v* ∈ *W* and *e* ∈ *F*, we have *u*∈*W* (*e* ◦ *u*)(*u*) = ⊥ and *v*∈*W* (*e* ◦ *v*)(*v*) = ⊥. Hence, *r*∈*W* (*u* ⊕ *v*)(*r*) = ⊥. In addition, ∀*b* ∈ *F* and *o* ∈ *W*, we have (*b* ◦ *o*)(*o*) = ⊥ by statement (1) of Theorem 1. Since ∀*a* ∈ *F* and ∀*u* ∈ *W*, we have *p*∈*W* (*a* ◦ *u*)(*p*) = ⊥; thus, *r*∈*W* (*p* ⊕ *o*)(*r*) = ⊥. Hence, *r*∈*W* (*a* ◦ *u*)(*r*) = ⊥ by (MG2). Now, we already know that (MG3) holds in *V*, so it holds in *W*. Hence, ∀*u* ∈ *W*, we have −*u* ∈ *W*. Hence, *W* is an *M*-hazy subspace of *V* over *F*.

**Proposition 6.** *The intersection of a family of M-hazy subspace of an M-hazy vector space* (*<sup>V</sup>*, ⊕, ◦, *F*) *over an M-hazy field* (*<sup>F</sup>*, +, •) *is an M-hazy subspace of* (*<sup>V</sup>*, +, ◦, *<sup>F</sup>*)*.*

**Proof.** Assume Λ is an index set and *Wi* is an *M*-hazy subspace of (*<sup>V</sup>*, ⊕, ◦, *<sup>F</sup>*). Assume *K* = *i*∈Λ*Wi*.

(1) Since *o* ∈ *Wi* for each *i* ∈ Λ and *K* is a nonempty subset of *V*, we have *o* ∈ *K*.

(2) For every *u*, *v* ∈ *K*, *a* ∈ *F* and for every *i* ∈ Λ, we obtain *u*, *v* ∈ *Wi*. Since *W* is an *M*-hazy subspace of *V*, we obtain *p*<sup>∈</sup>*Wi* (*u* ⊕ (−*<sup>v</sup>*))(*p*) = ⊥ and *p*<sup>∈</sup>*Wi* (*a* ◦ *u*)(*p*) = ⊥ by Theorem 3. Since *p*<sup>∈</sup>*Wi* (*a* ◦ *u*)(*p*) = ⊥. This implies that there exists *xau* ∈ *Wi* such that (*a* ◦ *u*)(*xau*) = ⊥. This implies, for all *i* ∈ Λ, *xau* ∈ *Wi*. Thus, we can obtain *xau* ∈ *i*=Λ*Wi*. Hence, *p*∈ *i*∈Λ *Wi* (*a* ◦ *u*)(*p*) ≥ (*a* ◦ *u*)(*xau*) = ⊥. Similarly, *p*∈ *i*∈Λ *Wi* (*u* ⊕ (−*<sup>v</sup>*))(*p*) = ⊥. Hence, *K* = *i*=Λ*Wi* is an *M*-hazy subspace of (*<sup>V</sup>*, ⊕, ◦, *<sup>F</sup>*).

**Proposition 7.** *The union of a nonempty up-directed family of M-hazy subspace of M-hazy vector space* (*<sup>V</sup>*, ⊕, ◦, *F*) *over an M-hazy field* (*<sup>F</sup>*, +, •) *is an M-hazy subspace of* (*<sup>V</sup>*, ⊕, ◦, *<sup>F</sup>*)*. In particular, the union of a nonempty chain of M-hazy subspace of M-hazy vector space* (*<sup>V</sup>*, ⊕, ◦, *F*) *is an M-hazy subspace of* (*<sup>V</sup>*, ⊕, ◦, *<sup>F</sup>*)*.*

**Proof.** Assume Λ is an index set and *Wi* is an *M*-hazy subspace of (*<sup>V</sup>*, ⊕, ◦, *<sup>F</sup>*), where {*Wi* | *i* ∈ Λ} is an up-directed subfamily of 2*V*. Let *N* = *Wi*.

*i*∈Λ

(1) Clearly, *N* is a nonempty subset of *V*.

(2) For every *u*, *v* ∈ *N* and *a* ∈ *F*, there exists *i*, *j* ∈ Λ such that *u* ∈ *Wi* and *v* ∈ *Wj*. Since *N* is an up-directed family, then there exists *m* ∈ Λ such that *Wi* ⊆ *Wm*; this implies that *u*, *v* ∈ *Wm*. As (*<sup>W</sup>*, ⊕, ◦, *F*) is an *M*-hazy subspace of (*<sup>V</sup>*, ⊕, ◦, *<sup>F</sup>*), we obtain *p*∈*Wm* (*u* ⊕ (−*<sup>v</sup>*))(*p*) = ⊥ and *p*∈*Wm*(*a* ◦ *u*)(*p*) = ⊥ by Theorem 3. Hence, *p*∈*N* (*u* ⊕ (−*<sup>v</sup>*))(*p*) = ⊥ and

$$\bigvee\_{p \in N} (a \diamond u)(p) \neq \bot. \text{ Hence, } \bigcup\_{i \in \Lambda} \mathcal{W}\_i \text{ is an } M\text{-hazy subspace of } (\stackrel{r \smile \cdots}{V} \circ\_{\prime} \circ\_{\prime} F). \qed$$

Based on the above results, we can draw an important and interesting conclusion; that is, we have the following result.

**Proposition 8.** *All of the M-hazy subspaces of M-hazy vector space and the empty set form a convex structure.*

#### **5. Linear Transformation of** *M***-Hazy Vector Spaces**

In this section, we introduce the linear transformation of *M*-hazy vector spaces. We have also shown that *M*-fuzzifying convex spaces are induced by *M*-hazy subspace of *M*-hazy vector space.

**Definition 13.** *Assume that* (*<sup>V</sup>*, ⊕, ◦, *F*) *and* (*<sup>W</sup>*, -, , *F*) *are two M-hazy vector spaces over an M-hazy field* (*<sup>F</sup>*, +, •)*. Then, the mapping τ* : *V* −→ *W is called a linear transformation if the following conditions hold:*

*(1)* ∀*<sup>u</sup>*, *v* ∈ *V*, *<sup>τ</sup>*→*M*(*u* ⊕ *v*)=(*τ*(*u*) - *<sup>τ</sup>*(*v*)),

*(2)* ∀*u* ∈ *V*, ∀*a* ∈ *F*, *<sup>τ</sup>*→*M* (*a* ◦ *u*)=(*a <sup>τ</sup>*(*u*)).

**Definition 14.** *Assume V and W are two M-hazy vector spaces over an M-hazy field F, τ* : *V* −→ *W is a linear transformation, and o is the additive identity element of W. Then, the kernel of τ,* Ker*τ is determined by*

$$\text{Ker}\tau = \tau^{\leftarrow}(\{\sigma'\}) = \{p \in V \mid \tau(p) = \sigma'\}.$$

**Example 3.** *(1) Assume that* (*<sup>V</sup>*, ⊕, ◦, *F*) *is an M-hazy vector space over an M-hazy field* (*<sup>F</sup>*, +, •)*, the set* {*o*} *and the whole M-hazy vector space V are M-hazy subspaces of V; they are called the trivial M-hazy subspaces of V.*

*(2) Assume Rn and Rm are the Euclidean spaces and τ* : *Rn* −→ *Rm is a linear transformation.The image*

$$\tau^{\rightarrow}(p) = \{\tau(p) : p \in \mathcal{R}^n\}$$

*of τ is an M-hazy subspace of Rm, and the inverse image*

$$\pi^{\leftarrow}(\{o'\}) = \{p \in \mathbb{R}^n \mid \pi(p) = o'\}$$

*is an M-hazy subspace of Rm.*

**Proposition 9.** *Assume the mapping τ* : *V* −→ *W is a linear transformation,* (*<sup>V</sup>*, ⊕, ◦, *F*) *and* (*<sup>W</sup>*, -, , *F*) *are two M-hazy vector spaces over an M-hazy field* (*<sup>F</sup>*, +, •)*. Then, the following statements are valid:*


**Proof.** (1) For all *u*, *v* ∈ *J*, we have *p*<sup>∈</sup>*J*(*u* ⊕ (−*<sup>v</sup>*))(*p*) = ⊥. Then,

$$\begin{array}{c} \bigvee\_{\begin{subarray}{c}\tau(p)\in\mathsf{T}^{\rightarrow}(f)\end{subarray}} (\tau(\mathsf{u})\boxplus(-\tau(\mathsf{v})))(\tau(p)) \\ =\ \bigvee\_{\begin{subarray}{c}\tau(p)\in\mathsf{T}^{\rightarrow}(f)\end{subarray}} (\tau(\mathsf{u})\boxplus\tau(-\mathsf{v}))(\tau(p)) \\ =\ \bigvee\_{\begin{subarray}{c}\tau(p)\in\mathsf{T}^{\rightarrow}(f)\end{subarray}} \tau\_{\mathsf{M}}^{\rightarrow}(\mathsf{u}\oplus(-\mathsf{v}))(\tau(p)) \\ =\ \bigvee\_{\begin{subarray}{c}\tau(p)\in\mathsf{T}^{\rightarrow}(f)\ \tau(\mathsf{x})=\mathsf{T}(p) \\ =\ \bigvee\_{\begin{subarray}{c}\tau(p)\in\mathsf{T}^{\rightarrow}(f)\end{subarray}} (\mathsf{u}\oplus(-\mathsf{v}))(\mathsf{x}) \\ =\ \bigvee\_{\begin{subarray}{c}\tau(p)\in\mathsf{T}^{\rightarrow}(f)\end{subarray}} (\mathsf{u}\oplus(-\mathsf{v}))(p) \\ \geq\ \bigvee\_{\begin{subarray}{c}\tau(p)\in\mathsf{T}^{\rightarrow}(f)\end{subarray}} (\mathsf{u}\oplus(-\mathsf{v}))(p) \\ \neq\ \quad\bigvee\_{\begin{subarray}{c}p\in f\end{subarray}} (\mathsf{u}\oplus(-\mathsf{v}))(p) \\ \neq\ \quad\bigsqcup\_{\begin{subarray}{c}\tau(p)\in\mathsf{T}^{\rightarrow}(f)\end{subarray}} \end{array}$$

Similarly,

$$\bigvee\_{\pi(p)\in\pi^{\to}(f)} (a\odot\pi(\mathfrak{u}))(\pi(p)) \ge \bigvee\_{p\in f} (a\diamond\mathfrak{u})(p) \ne \bot\bot$$

Then, by Theorem 3, it follows that *τ*<sup>→</sup>(*J*) is an *M*-hazy subspace of *W*. 

(2) For all *u*, *v* ∈ *<sup>τ</sup>*<sup>←</sup>(*K*), we have *<sup>τ</sup>*(*u*), *τ*(*v*) ∈ *K*. We find that *τ*(*p*)∈*<sup>K</sup>* (*τ*(*u*) - ⊥and since*K*is

(−*<sup>τ</sup>*(*v*)))(*τ*(*p*)) = *τ*(*p*)∈*<sup>K</sup>* (*a τ*(*u*))(*τ*(*p*)) = ⊥, an *M*-hazy subspace

of *W*. Furthermore,

$$\begin{array}{lcl} & \bigvee & (\tau(u)\boxplus(-\tau(v)))(\tau(p)) \\ = & \bigvee & (\tau(u)\boxplus\tau(-v))(\tau(p)) \\ = & \bigvee\limits\_{\tau(p)\in K} \tau\_{\mathcal{M}}(u\oplus(-v))(\tau(p)) \\ = & \bigvee\limits\_{\tau(p)\in K} \tau\_{\mathcal{M}}(u\oplus(-v))(\tau(p)) \\ = & \bigvee\limits\_{\tau(p)\in K} \bigvee\limits\_{\tau(x)=\tau(p)} (u\oplus(-v))(x) \\ = & \bigvee\limits\_{\tau(p)\in K} (u\oplus(-v))(p) \\ = & \bigvee\limits\_{p\in \tau^{+}(K)} (u\oplus(-v))(p) \\ \neq \bot. \end{array}$$

Similarly,

$$\bigvee\_{\pi(p)\in K} (a \odot \pi(u))(\pi(p)) = \bigvee\_{p\in \pi^+(K)} (a \diamond u)(p) \neq \bot.$$

Consequently, *τ*<sup>←</sup>(*K*) is an *M*-hazy subspace of *V*.

Now, assume *p* ∈ ker *τ*. Since *K* is an *M*-hazy subspace of *W*, then *τ*(*p*) = *o* ∈ *K*, and so *p* ∈ *<sup>τ</sup>*<sup>←</sup>(*K*). Hence, ker *τ* ⊆ *<sup>τ</sup>*<sup>←</sup>(*K*).

**Proposition 10.** *Assume V and W are two M-hazy vector spaces over an M-hazy field F and τ* : *V* −→ *W is a linear transformation. Then,* Ker*τ is an M-hazy subspace of V.*

**Proof.** It is easy to see that {*o*} is an *M*-hazy subspace of *W*. Then, by Proposition 9, we have that Ker*τ* is an *M*-hazy subspace of *V*.

The theorems below give an approach to induce *M*-fuzzifying convex spaces using *M*-hazy subspace of *M*-hazy vector space.

**Theorem 5.** *Assume* (*<sup>V</sup>*, ⊕, ◦, *F*) *is an M-hazy vector space over an M-hazy field* (*<sup>F</sup>*, +, •) *and define* S : 2*<sup>V</sup>* → *M as follows:*

$$\forall A \in \mathcal{2}^V, \mathcal{P}(A) = \bigwedge\_{p \in V} \left( \left( \left( \bigvee\_{u,v \in A} (u \oplus (-v))(p) \right) \to A(p) \right) \land \left( \left( \bigvee\_{a \in F, u \in A} (a \circ u)(p) \right) \to A(p) \right) \right).$$

*Then,* (*<sup>V</sup>*, S ) *is an M-fuzzifying convex space.*

**Proof.** The proof is similar to the proof of Theorem 7 in [13] so it is omitted.

**Theorem 6.** *Assume that the mapping τ* : *V* −→ *W is a linear transformation,* (*<sup>V</sup>*, ⊕, ◦, *F*) *and* (*<sup>W</sup>*, -, , *F*) *are two M-hazy vector spaces over an M-hazy field* (*<sup>F</sup>*, +, •)*; then, τ* : (*<sup>V</sup>*, S*V*) −→ (*<sup>W</sup>*, S*W*) *is an M-CP mapping.*

**Proof.** The proof is similar to the proof of Theorem 8 in [13] so it is omitted.

**Theorem 7.** *Assume the mapping τ* : *V* −→ *W is a linear transformation,* (*<sup>V</sup>*, ⊕, ◦, *F*) *and* (*<sup>W</sup>*, -, , *F*) *are two M-hazy vector spaces over an M-hazy field* (*<sup>F</sup>*, +, •)*; then, τ* : (*<sup>V</sup>*, S*V*) −→ (*<sup>W</sup>*, S*W*) *is an M-CC mapping.*

**Proof.** Since the mapping *τ* : *V* −→ *W* is a linear transformation if the following conditions hold:

(1) ∀*<sup>u</sup>*, *v* ∈ *V*, *<sup>τ</sup>*→*M* (*u* ⊕ *v*)=(*τ*(*u*) - *<sup>τ</sup>*(*v*)), (2) ∀*u* ∈ *V*, ∀*a* ∈ *F*, *<sup>τ</sup>*→*M* (*a* ◦ *u*)=(*a <sup>τ</sup>*(*u*)).

> Then, for all *A* ∈ <sup>2</sup>*V*, we have

S*W*(*τ*<sup>→</sup>(*A*)) = *p*∈*W* - *<sup>τ</sup>*(*u*),*<sup>τ</sup>*(*v*)<sup>∈</sup>*τ*<sup>→</sup>(*A*)(*τ*(*u*) - *<sup>τ</sup>*(−*<sup>v</sup>*))(*τ*(*p*)) → (*τ*<sup>→</sup>(*A*))(*τ*(*p*)) *<sup>a</sup>*∈*F*,*<sup>τ</sup>*(*u*)<sup>∈</sup>*τ*<sup>→</sup>(*A*) (*a τ*(*u*))(*τ*(*p*)) → (*τ*<sup>→</sup>(*A*))(*τ*(*p*)) ≥ *p*∈*W* - *<sup>τ</sup>*(*u*),*<sup>τ</sup>*(*v*)<sup>∈</sup>*τ*<sup>→</sup>(*A*) *τ*→*M* · *<sup>τ</sup>*←*M* (*τ*(*u*) - *<sup>τ</sup>*(−*<sup>v</sup>*))(*τ*(*p*)) → (*τ*<sup>→</sup>(*A*))(*τ*(*p*)) *<sup>a</sup>*∈*F*,*<sup>τ</sup>*(*u*)<sup>∈</sup>*τ*<sup>→</sup>(*A*) *τ*→*M* · *<sup>τ</sup>*←*M* (*a τ*(*u*))(*τ*(*p*)) → (*τ*<sup>→</sup>(*A*))(*τ*(*p*)) = *p*∈*W* - *<sup>τ</sup>*(*u*),*<sup>τ</sup>*(*v*)<sup>∈</sup>*τ*<sup>→</sup>(*A*) *τ*→*M* (*u* ⊕ (−*<sup>v</sup>*))(*τ*(*p*)) → (*τ*<sup>→</sup>(*A*))(*τ*(*p*)) *<sup>a</sup>*∈*F*,*<sup>τ</sup>*(*u*)<sup>∈</sup>*τ*<sup>→</sup>(*A*) *τ*→*M* (*a* ◦ *u*)(*τ*(*p*))<sup>→</sup> (*τ*<sup>→</sup>(*A*))(*τ*(*p*)) = *x*∈*V* - *<sup>τ</sup>*(*p*)<sup>∈</sup>*τ*→*M* (*A*) *τ*(*x*)=*τ*(*p*) *u* ⊕ − *v*)(*x*)) → (*A*)(*x*) *<sup>a</sup>*∈*F*,*<sup>τ</sup>*(*p*)<sup>∈</sup>*τ*→*<sup>M</sup>* (*A*) *τ*(*x*)=*τ*(*p*) (*a* ◦ *u*)(*x*) → (*A*)(*x*) ≥ *p*∈*V* - *<sup>u</sup>*,<sup>−</sup>*v*∈*<sup>A</sup>*(*<sup>u</sup>* ⊕ (−*<sup>v</sup>*))(*p*)→(*A*)(*p*) *<sup>a</sup>*∈*F*,*u*∈*A* (*a u*)(*p*) → (*A*)(*p*) = S*V*(*A*).

This implies that *τ* : (*<sup>V</sup>*, S*V*) −→ (*<sup>W</sup>*, S*W*) is an *M*-CC mapping.
