#
Some Results on (Generalized) Fuzzy Multi-H_{v}-Ideals of H_{v}-Rings

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## Abstract

**:**

_{v}-structures; H

_{v}-ring; fundamental equivalence relation; H

_{v}-ideal; multiset; fuzzy multiset; fuzzy multi-H

_{v}-ideal

## 1. Introduction

## 2. Basic Definitions

#### 2.1. (Weak) Hyperstructure Theory

#### 2.2. Fuzzy Multisets

**Definition**

**1**

**Definition**

**2**

**Example**

**1.**

**Definition**

**3**

- 1.
- The image of A under f is denoted by $f\left(A\right)$ or$$C{M}_{f\left(A\right)}\left(y\right)=\left(\right)open="\{"\; close>\begin{array}{ll}{\bigvee}_{f\left(x\right)=y}C{M}_{A}\left(x\right)& \mathit{if}\phantom{\rule{4.pt}{0ex}}{f}^{-1}\left(y\right)\ne \varnothing \\ 0& \mathit{otherwise}.\end{array}$$
- 2.
- The inverse image of B under f is denoted by ${f}^{-1}\left(B\right)$ where $C{M}_{{f}^{-1}\left(B\right)}\left(x\right)=C{M}_{B}\left(f\left(x\right)\right)$.

**Example**

**2.**

## 3. Fuzzy Multi-${\mathit{H}}_{\mathit{v}}$-Ideal

**Definition**

**4.**

- 1.
- $C{M}_{A}\left(x\right)\wedge C{M}_{A}\left(y\right)\le inf\{C{M}_{A}\left(z\right):z\in x+y\}$;
- 2.
- for every $x,a\in R$ there exists $y\in R$ such that $x\in a+y$ and $C{M}_{A}\left(x\right)\wedge C{M}_{A}\left(a\right)\le C{M}_{A}\left(y\right)$;
- 3.
- for every $x,a\in R$ there exists $z\in H$ such that $x\in z+a$ and $C{M}_{A}\left(x\right)\wedge C{M}_{A}\left(a\right)\le C{M}_{A}\left(z\right)$;
- 4.
- $C{M}_{A}\left(x\right)\vee C{M}_{A}\left(y\right)\le C{M}_{A}\left(z\right)$ for all $z\in x\xb7y$.

**Remark**

**1.**

**Example**

**3.**

**Remark**

**2.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

**Proposition**

**1.**

**Proof.**

**Example**

**7.**

**Notation**

**1.**

- $C{M}_{A}\left(x\right)=0$ if ${\mu}_{A}^{1}\left(x\right)=0$,
- $C{M}_{A}\left(x\right)>0$ if ${\mu}_{A}^{1}\left(x\right)>0$,
- $C{M}_{A}\left(x\right)=\underline{1}$ if $C{M}_{A}\left(x\right)=\left(\underset{s\phantom{\rule{4.pt}{0ex}}\mathit{times}}{\underbrace{1,\dots ,1}}\right)$ where$$s=max\{k\in \mathbb{N}:C{M}_{A}\left(y\right)=({\mu}_{A}^{1}\left(y\right),{\mu}_{A}^{2}\left(y\right),\dots ,{\mu}_{A}^{k}\left(y\right)),{\mu}_{A}^{k}\left(y\right)\ne 0,y\in R\}.$$

**Definition**

**5.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Example**

**8.**

**Notation**

**2.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Definition**

**6.**

**Theorem**

**2.**

**Proof.**

**Definition**

**7.**

**Example**

**9.**

**Remark**

**3.**

**Remark**

**4.**

**Example**

**10.**

**Proposition**

**5.**

**Proof.**

**Example**

**11.**

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Definition**

**8.**

- 1.
- $C{M}_{A}\left(x\right)\wedge C{M}_{A}\left(y\right)\le C{M}_{A}(x+y)$ for all $x,y\in R$;
- 2.
- $C{M}_{A}(-x)\ge C{M}_{A}\left(x\right)$ for all $x\in R$;
- 3.
- $C{M}_{A}\left(x\right)\vee C{M}_{A}\left(y\right)\le C{M}_{A}(x\xb7y)$ for all $x,y\in R$.

**Proposition**

**8.**

**Proof.**

**Example**

**12.**

## 4. Generalized Fuzzy Multi-${\mathit{H}}_{\mathit{v}}$-Ideal

**Notation**

**3.**

- 1.
- ${x}_{t}\in CM$ when $CM\left(x\right)\ge t$,
- 2.
- ${x}_{t}\in qCM$ when $CM\left(x\right)+t\ge \underline{1}$,
- 3.
- ${x}_{t}\in \vee qCM$ when ${x}_{t}\in CM$ or ${x}_{t}\in qCM$,
- 4.
- $\underline{0.5}=\left(\underset{s\phantom{\rule{4.pt}{0ex}}\mathit{times}}{\underbrace{0.5,\dots ,0.5}}\right)$ where$$s=max\{k\in \mathbb{N}:C{M}_{A}\left(y\right)=({\mu}_{A}^{1}\left(y\right),{\mu}_{A}^{2}\left(y\right),\dots ,{\mu}_{A}^{k}\left(y\right)),{\mu}_{A}^{k}\left(y\right)\ne 0,y\in R\}.$$

**Definition**

**9.**

- ${x}_{t}\in CM$, ${y}_{r}\in CM$ implies ${z}_{t\wedge r}\in \vee qCM$ for all $z\in x+y$;
- ${x}_{t}\in CM$, ${a}_{r}\in CM$ implies ${y}_{t\wedge r}\in \vee qCM$ for some $y\in R$ with $x\in a+y$;
- ${x}_{t}\in CM$, ${a}_{r}\in CM$ implies ${z}_{t\wedge r}\in \vee qCM$ for some $z\in R$ with $x\in z+a$;
- ${y}_{t}\in CM$, $x\in R$ implies ${z}_{t}\in \vee qCM$ for all $z\in x\xb7y$(${x}_{t}\in CM$, $y\in R$ implies ${z}_{t}\in \vee qCM$ for all $z\in x\xb7y$).

**Remark**

**5.**

**Example**

**13.**

**Example**

**14.**

**Example**

**15.**

**Proposition**

**9.**

**Proof.**

**Theorem**

**3.**

- (a)
- $CM\left(x\right)\wedge CM\left(y\right)\wedge \underline{0.5}\le CM\left(z\right)$ for all $z\in x+y$;
- (b)
- For all $x,a\in R$ there exists $y\in R$ such that $x\in a+y$ and$$CM\left(a\right)\wedge CM\left(x\right)\wedge \underline{0.5}\le CM\left(y\right).$$
- (c)
- For all $x,a\in R$ there exists $z\in R$ such that $x\in z+a$ and$$CM\left(a\right)\wedge CM\left(x\right)\wedge \underline{0.5}\le CM\left(z\right).$$
- (d)
- For all $z\in x\xb7y$, $CM\left(y\right)\wedge \underline{0.5}\le CM\left(z\right)$ and $CM\left(x\right)\wedge \underline{0.5}\le CM\left(z\right)$.

**Proof.**

**(1) → (a):**Let $x,y\in R$. Since each of $CM\left(x\right),CM\left(y\right)$ are comparable with $\underline{0.5}$, we can consider the cases: $CM\left(x\right)\wedge CM\left(y\right)<\underline{0.5}$ and $CM\left(x\right)\wedge CM\left(y\right)\ge \underline{0.5}$.

**(2) → (b):**Let $x,a\in R$. Since each of $CM\left(x\right),CM\left(a\right)$ are comparable with $\underline{0.5}$, we can consider the cases: $CM\left(x\right)\wedge CM\left(a\right)<\underline{0.5}$ and $CM\left(x\right)\wedge CM\left(a\right)\ge \underline{0.5}$.

**(3) → (c):**This case is done in a similar manner to that of

**(2) → (b)**.

**(4) → (d):**Let $x,y\in R$. Since $CM\left(y\right)$ is comparable with $\underline{0.5}$, we can consider the cases: $CM\left(y\right)<\underline{0.5}$ and $CM\left(y\right)\ge \underline{0.5}$.

**Remark**

**6.**

**Remark**

**7.**

**Example**

**16.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Al Tahan, M.; Hoskova-Mayerova, S.; Davvaz, B.
Some Results on (Generalized) Fuzzy Multi-*H _{v}*-Ideals of

*H*-Rings.

_{v}*Symmetry*

**2019**,

*11*, 1376. https://doi.org/10.3390/sym11111376

**AMA Style**

Al Tahan M, Hoskova-Mayerova S, Davvaz B.
Some Results on (Generalized) Fuzzy Multi-*H _{v}*-Ideals of

*H*-Rings.

_{v}*Symmetry*. 2019; 11(11):1376. https://doi.org/10.3390/sym11111376

**Chicago/Turabian Style**

Al Tahan, Madeline, Sarka Hoskova-Mayerova, and Bijan Davvaz.
2019. "Some Results on (Generalized) Fuzzy Multi-*H _{v}*-Ideals of

*H*-Rings"

_{v}*Symmetry*11, no. 11: 1376. https://doi.org/10.3390/sym11111376