# A Comparison of Complete Parts on m-Idempotent Hyperrings

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries on the ${\mathit{\xi}}_{\mathit{m}}$-Relation on Hyperrings

**Definition**

**1.**

- (1)
- (general) hyperring, if $(R,+)$ is a hypergroup, $(R,\xb7)$ is a semihypergroup, and the hypermultiplication · is distributive with respect to the hyperaddition +. If $(R,+)$ is a semihypergroup, then $(R,+,\xb7)$ is called a semihyperring.
- (2)
- Krasner hyperring, if $(R,+)$ is a canonical hypergroup and $(R,\xb7)$ is a semigroup such that 0 is a zero element (called also absorbing element), that is, for all $x\in R$, we have $x\xb70=0$, and the multiplication · is distributive over the hyperaddition +.

## 3. ${\mathit{\xi}}_{\mathit{m}}$-Parts and Transitivity of the ${\mathit{\xi}}_{\mathit{m}}$-Relation

**Definition**

**2.**

**Proposition**

**1.**

- (i)
- A nonempty subset M of R is a ${\xi}_{m}$-part.
- (ii)
- For any $x\in M$ with the property $x{\xi}_{m}y$ it follows that $y\in M$.
- (iii)
- For any $x\in M$ with the property $x{\xi}_{m}^{*}y$ it follows that $y\in M$.

**Proof.**

**Example**

**1.**

**Proposition**

**2.**

**Proof.**

**Example**

**2.**

**Proposition**

**3.**

**Proof.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

**Theorem**

**1.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**1.**

- (1)
- The ${\xi}_{m}$-relation is transitive;
- (2)
- ${\xi}_{m}^{*}\left(x\right)={\mathcal{A}}_{m}\left(x\right)$, for all $x\in R$;
- (3)
- The set ${\mathcal{A}}_{m}\left(x\right)$ is a ${\xi}_{m}$-part of R, for every $x\in R$.

## 4. ${\mathit{\xi}}_{\mathit{m}}$-Complete Hyperrings

**Definition**

**3.**

**Example**

**7.**

**Example**

**8.**

**Corollary**

**2.**

**Proof.**

**Example**

**9.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Example**

**10.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Adineh Zadeh, A.; Norouzi, M.; Cristea, I.
A Comparison of Complete Parts on *m*-Idempotent Hyperrings. *Symmetry* **2020**, *12*, 554.
https://doi.org/10.3390/sym12040554

**AMA Style**

Adineh Zadeh A, Norouzi M, Cristea I.
A Comparison of Complete Parts on *m*-Idempotent Hyperrings. *Symmetry*. 2020; 12(4):554.
https://doi.org/10.3390/sym12040554

**Chicago/Turabian Style**

Adineh Zadeh, Azam, Morteza Norouzi, and Irina Cristea.
2020. "A Comparison of Complete Parts on *m*-Idempotent Hyperrings" *Symmetry* 12, no. 4: 554.
https://doi.org/10.3390/sym12040554