$(\lambda ,\mu )$-Fuzzy linear subspaces

Abstract

In this paper, we first introduce the concepts of $(\lambda ,\mu )$-fuzzy subfields. Then we generalize the concepts of fuzzy linear spaces, we define $(\lambda ,\mu )$-fuzzy linear subspaces, and we obtain some of their fundamental properties. Lastly, we list some possible directions of the extending of the present work.

1 Introduction and preliminaries

The concept of fuzzy sets was first introduced by Zadeh [1] in 1965, and then the fuzzy sets have been used in the reconsideration of classical mathematics. Recently, Yuan [2] introduced the concept of fuzzy subgroup with thresholds. A fuzzy subgroup with thresholds λ and μ is also called a $(\lambda ,\mu )$-fuzzy subgroup. Yao continued to research $(\lambda ,\mu )$-fuzzy normal subgroups, $(\lambda ,\mu )$-fuzzy quotient subgroups and $(\lambda ,\mu )$-fuzzy subrings in [3–5].

Nanda [6] introduced the concepts of fuzzy field and fuzzy linear space and gave some results. Biswas [7] pointed out that Proposition 4.1 of [6] was incorrect and redefined fuzzy field and fuzzy linear space. Gu and Lu [8] listed two examples to show that Proposition 4.5 in [6] is also incorrect and redefined the concept of fuzzy linear space.

In this paper, we first introduce the concepts of $(\lambda ,\mu )$-fuzzy subfields. Next, we generalize the concepts of fuzzy linear spaces over fuzzy fields to $(\lambda ,\mu )$-fuzzy linear subspaces over $(\lambda ,\mu )$-fuzzy subfields, and give some fundamental properties.

Let us recall some definitions and notions.

By a fuzzy subset of a nonempty set X we mean a mapping from X to the unit interval $[0,1]$. If A is a fuzzy subset of X, then we denote $A_{\alpha }=\{x\in X|A(x)\ge \alpha \}$ for all $\alpha \in [0,1]$.

Throughout this paper, we always assume that $0\le \lambda <\mu \le 1$.

2 $(\lambda ,\mu )$-fuzzy subfields

Definition 1 A fuzzy subset F of a field F is said to be a $(\lambda ,\mu )$-fuzzy subfield of F if $\mathrm{\forall }k,l\in \mathbf{F}$, we have

• (1) $F(k+l)\vee \lambda \ge F(k)\wedge F(l)\wedge \mu$;

• (2) $F(-k)\vee \lambda \ge F(k)\wedge \mu$;

• (3) $F(kl)\vee \lambda \ge F(k)\wedge F(l)\wedge \mu$;

• (4) $F(k^{-1})\vee \lambda \ge F(k)\wedge \mu$, where $k\ne 0$.

From the previous definition, we can easily conclude that a fuzzy subfield defined by Gu [8, 9] or Biswas [7] is a $(0,1)$-fuzzy subfield.

Proposition 1 Let F be a $(\lambda ,\mu )$-fuzzy subfield of a field F, then $\mathrm{\forall }k\in \mathbf{F}$, we have

• (1) $F(0)\vee \lambda \ge F(k)\wedge \mu$;

• (2) $F(1)\vee \lambda \ge F(k)\wedge \mu$, where $k\ne 0$;

• (3) $F(0)\vee \lambda \ge F(1)\wedge \mu$.

Proof $F(0)\vee \lambda =F(k+(-k))\vee \lambda \vee \lambda \ge (F(k)\wedge F(-k)\wedge \mu )\vee \lambda =(F(k)\vee \lambda )\wedge (F(-k)\vee \lambda )$ ∧ $(\mu \vee \lambda )\ge F(k)\wedge (F(k)\wedge \mu )\wedge \mu =F(k)\wedge \mu$. Thus we complete the proof of (1).

• (2) can be proved similarly and (3) is a corollary of (1). □

Theorem 1 Let F be a fuzzy subset of a field F. Then the following are equivalent:

• (1) F is a $(\lambda ,\mu )$-fuzzy subfield of F;

• (2) $F_{\alpha }$ is a subfield of F, for any $\alpha \in (\lambda ,\mu ]$, where $F_{\alpha }\ne \mathrm{\varnothing }$.

Proof It is a corollary of Proposition 2.4 of [4]. □

We use ${\mathbf{F}}_{\mathbf{1}}$, ${\mathbf{F}}_{\mathbf{2}}$ to stand for two fields in the following and define $sup\mathrm{\varnothing }=0$, where ∅ is the empty set.

Proposition 2 Let $f:{\mathbf{F}}_{\mathbf{1}}\to {\mathbf{F}}_{\mathbf{2}}$ be a homomorphism, and let $F_1$ be a $(\lambda ,\mu )$-fuzzy subfield of ${\mathbf{F}}_{\mathbf{1}}$. Then $f(F_1)$ is a $(\lambda ,\mu )$-fuzzy subfield of ${\mathbf{F}}_{\mathbf{2}}$, where

$$f(F_1)(y)=\underset{x\in {\mathbf{F}}_{\mathbf{1}}}{sup}\{F_1(x)|f(x)=y\},\mathrm{\forall }y\in {\mathbf{F}}_{\mathbf{2}}.$$

Proof Similar to the proof of Proposition 2.7 in [4]. □

Proposition 3 Let $f:{\mathbf{F}}_{\mathbf{1}}\to {\mathbf{F}}_{\mathbf{2}}$ be a homomorphism, and let $F_2$ be a $(\lambda ,\mu )$-fuzzy sublattice of ${\mathbf{F}}_{\mathbf{2}}$. Then $f^{-1}(F_2)$ is a $(\lambda ,\mu )$-fuzzy sublattice of ${\mathbf{F}}_{\mathbf{1}}$, where

$$f^{-1}(F_2)(x)=F_2(f(x)),\mathrm{\forall }x\in {\mathbf{F}}_{\mathbf{1}}.$$

Proof Similar to the proof of Proposition 2.8 in [4]. □

3 $(\lambda ,\mu )$-fuzzy linear subspaces

Definition 2 Let F be a $(\lambda ,\mu )$-fuzzy subfield of a field F, V be a linear space over F and V be a fuzzy subset of V. V is called a $(\lambda ,\mu )$-fuzzy linear subspace of V over F if for all $x,y\in \mathbf{V}$, $k\in \mathbf{F}$, we have

• (1) $V(x+y)\vee \lambda \ge V(x)\wedge V(y)\wedge \mu$;

• (2) $V(-x)\vee \lambda \ge V(x)\wedge \mu$;

• (3) $V(kx)\vee \lambda \ge F(k)\wedge V(x)\wedge \mu$;

• (4) $F(1)\vee \lambda \ge V(x)\wedge \mu$.

Obviously, the previous definition is a generalization of fuzzy linear space defined by Gu and Tu (see Definition 3.1 in [9]).

Theorem 2 Let F be a $(\lambda ,\mu )$-fuzzy subfield of a field F, let V be a linear space over F, and let V be a fuzzy subset of V. V is a $(\lambda ,\mu )$-fuzzy linear subspace of V over F if and only if, for all $x,y\in \mathbf{V}$, $k,l\in \mathbf{F}$, we have

• (1) $V(kx+ly)\vee \lambda \ge F(k)\wedge V(x)\wedge F(l)\wedge V(y)\wedge \mu$;

• (2) $F(1)\vee \lambda \ge V(x)\wedge \mu$.

Proof ‘⇒’

For all $x,y\in \mathbf{V}$, $k,l\in \mathbf{F}$, we have

$$\begin{array}{rl}V(kx+ly)\vee \lambda & =V(kx+ly)\vee \lambda \vee \lambda \ge (V(kx)\wedge V(ly)\wedge \mu )\vee \lambda \\ =(V(kx)\vee \lambda )\wedge (V(ly)\vee \lambda )\wedge (\mu \vee \lambda )\\ \ge F(k)\wedge V(x)\wedge F(l)\wedge V(y)\wedge \mu .\end{array}$$

‘⇐’

From $F(1)\vee \lambda \ge V(x)\wedge \mu$, we know that $\lambda \ge V(x)\wedge \mu$ or $F(1)\ge V(x)\wedge \mu$. Two cases are possible:

Case 1. If $\lambda \ge V(x)\wedge \mu$, then

• (1) $V(x+y)\vee \lambda \ge \lambda \ge V(x)\wedge \mu \ge V(x)\wedge V(y)\wedge \mu$;

• (2) $V(-x)\vee \lambda \ge \lambda \ge V(x)\wedge \mu$;

• (3) $V(kx)\vee \lambda \ge \lambda \ge V(x)\wedge \mu \ge F(k)\wedge V(x)\wedge \mu$.

Case 2. $F(1)\ge V(x)\wedge \mu$, then

• (1) $V(x+y)\vee \lambda =V(1x+1y)\vee \lambda \ge F(1)\wedge V(x)\wedge F(1)\wedge V(y)\wedge \mu =V(x)\wedge V(y)\wedge \mu$;

• (2) $V(-x)\vee \lambda =V(-1x+0x)\vee \lambda \vee \lambda \ge (F(-1)\wedge V(x)\wedge F(0)\wedge V(x)\wedge \mu )\vee \lambda \ge (F(-1)\vee \lambda )$ ∧ $(V(x)\vee \lambda )\wedge (F(0)\vee \lambda )\wedge (\mu \vee \lambda )\ge (F(1)\wedge \mu )\wedge V(x)\wedge (F(1)\wedge \mu )\wedge \mu =V(x)\wedge \mu$.

• (3) $V(kx)\vee \lambda =V(kx+0x)\vee \lambda \vee \lambda \ge (F(k)\wedge V(x)\wedge F(0)\wedge V(x)\wedge \mu )\vee \lambda =(F(k)\vee \lambda )$ ∧ $(V(x)\vee \lambda )\wedge (F(0)\vee \lambda )\wedge (\mu \vee \lambda )\ge F(k)\wedge V(x)\wedge (F(1)\wedge \mu )\wedge \mu =F(k)\wedge V(x)\wedge \mu$.

 □

Theorem 3 Let F be a $(\lambda ,\mu )$-fuzzy subfield of a field F, let V be a linear space over $\mathbf{F},$ and let V be a fuzzy subset of V. Then the following are equivalent:

• (1) V is a $(\lambda ,\mu )$-fuzzy linear subspace over F;

• (2) $V_{\alpha }$ is a linear subspace over $F_{\alpha }$, for any $\alpha \in (\lambda ,\mu ]$, where $F_{\alpha }\ne \mathrm{\varnothing }$ and $V_{\alpha }\ne \mathrm{\varnothing }$.

Proof ‘(1) ⇒ (2)’

Let V be a $(\lambda ,\mu )$-fuzzy linear subspace over F.

Take any $\alpha \in (\lambda ,\mu ]$ such that $F_{\alpha }\ne \mathrm{\varnothing }$ and $V_{\alpha }\ne \mathrm{\varnothing }$; we need to show that $kx+ly\in V_{\alpha }$, $\mathrm{\forall }x,y\in V_{\alpha }$, $\mathrm{\forall }k,l\in F_{\alpha }$.

From $F(k)\ge \alpha$, $F(l)\ge \alpha$, $V(x)\ge \alpha$, $V(y)\ge \alpha$, and $\alpha \le \mu$, we conclude that $V(kx+ly)\vee \lambda \ge F(k)\wedge V(x)\wedge F(l)\wedge V(y)\wedge \mu \ge \alpha \wedge \mu =\alpha$. Note that $\lambda <\alpha$, we obtain that $V(kx+ly)\ge \alpha$. So $kx+ly\in V_{\alpha }$.

‘(1) ⇐ (2)’

Conversely, let $V_{\alpha }$ be a linear subspace over $F_{\alpha }$, for any $\alpha \in (\lambda ,\mu ]$. If there exist $x,y\in V$ and $k,l\in F$ such that $V(kx+ly)\vee \lambda <\alpha =F(k)\wedge V(x)\wedge F(l)\wedge V(y)\wedge \mu$, then $\alpha \in (\lambda ,\mu ]$, $x,y\in V_{\alpha }$, $k,l\in F_{\alpha }$. But $kx+ly\notin A_{\alpha }$. This is a contradiction with that $V_{\alpha }$ is a linear subspace over $F_{\alpha }$.

Again, if there exists $x\in V$ such that $F(1)\vee \lambda <\alpha =V(x)\wedge \mu$, then $\alpha \in (\lambda ,\mu ]$, $x\in V_{\alpha }$ and $1\notin F_{\alpha }$. This is a contradiction to that $V_{\alpha }$ is a linear subspace over a subfield $F_{\alpha }$ of F. □

We use ${\mathbf{V}}_{\mathbf{1}}$, ${\mathbf{V}}_{\mathbf{2}}$ to stand for two linear spaces over the same field F in the following.

Proposition 4 Let F be a $(\lambda ,\mu )$-fuzzy subfield of a field F, let $f:{\mathbf{V}}_{\mathbf{1}}\to {\mathbf{V}}_{\mathbf{2}}$ be a linear transformation over F, and $V_1$ be a $(\lambda ,\mu )$-fuzzy linear subspace of ${\mathbf{V}}_{\mathbf{1}}$ over F. Then $f(V_1)$ is a $(\lambda ,\mu )$-fuzzy linear subspace of ${\mathbf{V}}_{\mathbf{2}}$ over F, where

$$f(V_1)(y)=\underset{x\in {\mathbf{V}}_{\mathbf{1}}}{sup}\{V_1(x)|f(x)=y\},\mathrm{\forall }y\in {\mathbf{V}}_{\mathbf{2}}.$$

Proof If $f^{-1}(y_1)=\mathrm{\varnothing }$ or $f^{-1}(y_2)=\mathrm{\varnothing }$ for any $y_1,y_2\in V_2$, then $(f(V_1)(k{y}_1+l{y}_2))\vee \lambda \ge 0=F(k)\wedge f(V_1)(y_1)\wedge F(l)\wedge f(V_1)(y_2)\wedge \mu$.

Suppose that $f^{-1}(y_1)\ne \mathrm{\varnothing }$, $f^{-1}(y_2)\ne \mathrm{\varnothing }$ for any $y_1,y_2\in V_2$, then $f^{-1}(k{y}_1+l{y}_2)\ne \mathrm{\varnothing }$. So for any $k,l\in F$, we have

$$\begin{array}{r}f(V_1)(k{y}_1+l{y}_2)\vee \lambda \\ =\underset{t\in {\mathbf{V}}_{\mathbf{1}}}{sup}\{A(t)|f(t)=k{y}_1+l{y}_2\}\vee \lambda \\ =\underset{t\in {\mathbf{V}}_{\mathbf{1}}}{sup}\{V_1(t)\vee \lambda |f(t)=k{y}_1+l{y}_2\}\\ \ge \underset{x_1,x_2\in {\mathbf{V}}_{\mathbf{1}}}{sup}\{(V_1(k{x}_1+l{x}_2))\vee \lambda |f(x_1)=y_1,f(x_2)=y_2\}\\ \ge \underset{x_1,x_2\in {\mathbf{V}}_{\mathbf{1}}}{sup}\{(F(k)\wedge V_1(x_1)\wedge F(l)\wedge V_1(x_2))\wedge \mu |f(x_1)=y_1,f(x_2)=y_2\}\\ =(\underset{x_1\in {\mathbf{V}}_{\mathbf{1}}}{sup}\{V_1(x_1)|f(x_1)=y_1\}\wedge \underset{x_2\in V_1}{sup}\{V_1(x_2)|f(x_2)=y_2\})\wedge F(k)\wedge F(l)\wedge \mu \\ =f(V_1)(y_1)\wedge f(V_1)(y_2)\wedge F(k)\wedge F(l)\wedge \mu .\end{array}$$

And for all $y\in {\mathbf{V}}_{\mathbf{2}}$, we have

$$\begin{array}{rcl}F(1)\vee \lambda & =& \underset{x\in {\mathbf{V}}_{\mathbf{1}}}{sup}\{F(1)\vee \lambda |V_1(x)=y\}\\ \ge & \underset{x\in {\mathbf{V}}_{\mathbf{1}}}{sup}\{V_1(x)\wedge \mu |V_1(x)=y\}\\ =& \underset{x\in {\mathbf{V}}_{\mathbf{1}}}{sup}\{V_1(x)|V_1(x)=y\}\wedge \mu \\ =& f(V_1)(y)\wedge \mu .\end{array}$$

Thus $f(V_1)$ is a $(\lambda ,\mu )$-fuzzy linear subspace of ${\mathbf{V}}_{\mathbf{2}}$ over F. □

Proposition 5 Let F be a $(\lambda ,\mu )$-fuzzy subfield of a field F, let $f:{\mathbf{V}}_{\mathbf{1}}\to {\mathbf{V}}_{\mathbf{2}}$ be a linear transformation over $\mathbf{F},$ and let $V_2$ be a $(\lambda ,\mu )$-fuzzy linear subspace of ${\mathbf{V}}_{\mathbf{2}}$ over F. Then $f^{-1}(V_2)$ is a $(\lambda ,\mu )$-fuzzy linear subspace of ${\mathbf{V}}_{\mathbf{1}}$ over F, where

$$f^{-1}(V_2)(x)=V_2(f(x)),\mathrm{\forall }x\in {\mathbf{V}}_{\mathbf{1}}.$$

Proof For any $x_1,x_2\in V_1$ and $k,l\in F$, we have

$$\begin{array}{rcl}{f}^{-1}(V_2)(k{x}_1+l{x}_2)\vee \lambda & =& V_2(f(k{x}_1+l{x}_2))\vee \lambda \\ =& V_2(kf(x_1)+lf(x_2))\vee \lambda \\ \ge & F(k)\wedge V_2(f(x_1))\wedge F(l)\wedge V_2(f(x_2))\wedge \mu \\ =& F(k)\wedge f^{-1}(V_2)(x_1)\wedge F(l)\wedge f^{-1}(V_2)(x_2)\wedge \mu .\end{array}$$

And for all $x\in V_1$, we have

$$f^{-1}(V_2)(x)\wedge \mu =V_2(f(x))\wedge \mu \le F(1)\vee \lambda .$$

So, $f^{-1}(V_2)$ is a $(\lambda ,\mu )$-fuzzy linear subspace of $V_1$ over F. □

4 Further research

The present work can be extended in several directions. Let us indicate some possibilities.

• 1. One can define $(\lambda ,\mu )$-fuzzy hypervector spaces and study their properties (definitions of fuzzy hypervector spaces can be found in [10]).

• 2. One may give the definition of $({\lambda }_1,{\mu }_1)$-fuzzy linear subspaces over $({\lambda }_2,{\mu }_2)$-fuzzy subfields, where $0\le {\lambda }_1<{\mu }_1\le 1$ and $0\le {\lambda }_2<{\mu }_2\le 1$. Then explore the properties of them.

• 3. One can investigate the interval-valued (type 2, lattice-valued, etc.) $({\lambda }_1,{\mu }_1)$-fuzzy linear subspaces.

• 4. One can also research $(\lambda ,\mu )$-anti-fuzzy linear subspaces [11].