# Linear Diophantine Fuzzy Subspaces of a Vector Space

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition 1**

**Definition 2**

**Definition 3**

**Remark 1.**

**Definition 4**

- (1)
- The union of ${D}^{*}$ and ${D}^{**}$ is denoted as ${D}^{*}\cup {D}^{**}$, where for all $x\in E$,$({D}^{*}\cup {D}^{**})\left(x\right)$ is given as follows:$(<{U}^{*}\left(x\right)\vee {U}^{**}\left(x\right),{V}^{*}\left(x\right)\wedge {V}^{**}\left(x\right)>,<{\alpha}^{*}\left(x\right)\vee {\alpha}^{**}\left(x\right),{\beta}^{*}\left(x\right)\wedge {\beta}^{**}\left(x\right)>)$;
- (2)
- The intersection of ${D}^{*}$ and ${D}^{**}$ is denoted as ${D}^{*}\cap {D}^{**}$, where for all $x\in E$,$({D}^{*}\cap {D}^{**})\left(x\right)$ is given as follows:$(<{U}^{*}\left(x\right)\wedge {U}^{**}\left(x\right),{V}^{*}\left(x\right)\vee {V}^{**}\left(x\right)>,<{\alpha}^{*}\left(x\right)\wedge {\alpha}^{**}\left(x\right),{\beta}^{*}\left(x\right)\vee {\beta}^{**}\left(x\right)>)$.

**Definition 5**

**Notation 1.**

- (1)
- $D\left(x\right)\wedge D\left(y\right)=(<u,v>,<\alpha ,\beta >)$ where $u=U\left(x\right)\wedge U\left(y\right)$, $v=V\left(x\right)\vee V\left(y\right)$, $\alpha =\alpha \left(x\right)\wedge \alpha \left(y\right)$, $v=\beta \left(x\right)\vee \beta \left(y\right)$.
- (2)
- $D\left(x\right)\vee D\left(y\right)=(<u,v>,<\alpha ,\beta >)$ where $u=U\left(x\right)\vee U\left(y\right)$, $v=V\left(x\right)\wedge V\left(y\right)$, $\alpha =\alpha \left(x\right)\vee \alpha \left(y\right)$, $v=\beta \left(x\right)\wedge \beta \left(y\right)$.
- (3)
- $D\left(x\right)\le D\left(y\right)$ means $U\left(x\right)\le U\left(y\right)$, $V\left(x\right)\ge V\left(y\right)$, $\alpha \left(x\right)\le \alpha \left(y\right)$, $\beta \left(x\right)\ge \beta \left(y\right)$.

**Proposition 1.**

## 3. LDF Subfields of a Field

**Definition 6.**

- (1)
- $F(a+b)\ge F\left(a\right)\wedge F\left(b\right)$;
- (2)
- $F\left(ab\right)\ge F\left(a\right)\wedge F\left(b\right)$;
- (3)
- $F(-a)\ge F\left(a\right)$;
- (4)
- $F\left({a}^{-1}\right)\ge F\left(a\right)$.

**Proposition 2.**

- (1)
- $F(-a)=F\left(a\right)$.
- (2)
- $F\left({a}^{-1}\right)=F\left(a\right)$.
- (3)
- $F\left(0\right)\ge F\left(a\right)$ for all $a\in K$.
- (4)
- $F\left(1\right)\ge F\left(a\right)$ for all $a\in K\setminus \left\{0\right\}$.
- (5)
- $F\left(0\right)\ge F\left(1\right)$.

**Proof.**

**Theorem 1.**

- (1)
- $F(a-b)\ge F\left(a\right)\wedge F\left(b\right)$ for all $a,b\in K$;
- (2)
- $F\left(a{b}^{-1}\right)\ge F\left(a\right)\wedge F\left(b\right)$ for all $a\in K,b\in K\setminus \left\{0\right\}$.

**Example 1.**

**Theorem 2.**

**Proof.**

**Corollary 1.**

**Proof.**

**Corollary 2.**

**Proof.**

## 4. LDF Subspaces of a Vector Space

**Definition 7.**

- (1)
- $D(x+y)\ge D\left(x\right)\wedge D\left(y\right)$;
- (2)
- $D\left(ax\right)\ge F\left(a\right)\wedge D\left(x\right)$.

**Example 2.**

**Example 3.**

**Proposition 3.**

**Proof.**

**Example 4.**

**Proposition 4.**

**Proof.**

**Theorem 3.**

**Proof.**

**Remark 2.**

**Example 5.**

**Example 6.**

**Theorem 4.**

**Proof.**

**Corollary 3.**

**Proposition 5.**

**Proof.**

**Proposition 6.**

**Theorem 5.**

**Proof.**

**Definition 8.**

**Theorem 6.**

**Proof.**

**Proposition 7.**

**Proof.**

**Theorem 7.**

**Proof.**

**Proposition 8.**

**Proof.**

**Definition 9.**

**Lemma 1.**

**Proof.**

**Lemma 2.**

**Proof.**

**Corollary 4.**

**Proof.**

**Proposition 9.**

**Proof.**

**Theorem 8.**

**Proof.**

**Example 7.**

**Proposition 10.**

**Proof.**

**Proposition 11.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Al-Tahan, M.; Hoskova-Mayerova, S.; Al-Kaseasbeh, S.; Tahhan, S.A.
Linear Diophantine Fuzzy Subspaces of a Vector Space. *Mathematics* **2023**, *11*, 503.
https://doi.org/10.3390/math11030503

**AMA Style**

Al-Tahan M, Hoskova-Mayerova S, Al-Kaseasbeh S, Tahhan SA.
Linear Diophantine Fuzzy Subspaces of a Vector Space. *Mathematics*. 2023; 11(3):503.
https://doi.org/10.3390/math11030503

**Chicago/Turabian Style**

Al-Tahan, Madeleine, Sarka Hoskova-Mayerova, Saba Al-Kaseasbeh, and Suha Ali Tahhan.
2023. "Linear Diophantine Fuzzy Subspaces of a Vector Space" *Mathematics* 11, no. 3: 503.
https://doi.org/10.3390/math11030503