# Weak Embeddable Hypernear-Rings

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- 1.
- $(R,+)$ is a quasicanonical hypergroup (named also polygroup [6]), meaning that:
- (a)
- $x+(y+z)=(x+y)+z$ for any $x,y,z\in R$,
- (b)
- there exists $0\in R$ such that, for any $x\in R,x+0=0+x=\{x\}$,
- (c)
- for any $x\in R$ there exists a unique element $-x\in R,$ such that $0\in x+(-x)\cap (-x)+x$,
- (d)
- for any $x,y,z\in R,z\in x+y$ implies that $x\in z+(-y),y\in (-x)+z$.

- 2.
- $(R,\xb7)$ is a semigroup endowed with a two-sided absorbing element 0, i.e., for any $x\in R,x\xb70=0\xb7x=0.$
- 3.
- The operation “·” is distributive with respect to the hyperoperation “+” from the left-hand side: for any $x,y,z\in R$, there is $x\xb7(y+z)=x\xb7y+x\xb7z.$

**Definition**

**2.**

**Definition**

**3.**

- 1.
- $\rho (x+y)\subseteq \rho (x)+\rho (y)$
- 2.
- $\rho (x\xb7y)\subseteq \rho (x)\xb7\rho (y)$ for all $x,y\in {R}_{1}$.

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

## 3. Weak Embeddable Hypernear-Rings

**Theorem**

**4.**

**Proof.**

**Definition**

**4.**

**Remark**

**1.**

**Definition**

**5.**

**Theorem**

**5.**

**Proof.**

**Remark**

**2.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

- 1.
- $0+0=\{0\}$ and
- 2.
- $0\xb7r=0$, for all $r\in R$.

**Proof.**

**Example**

**4.**

**Remark**

**3.**

**Remark**

**4.**

**Proposition**

**1.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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· | 0 | 1 | 2 | 3 |
---|---|---|---|---|

0 | 0 | 0 | 0 | 0 |

1 | 0 | 1 | 2 | 3 |

2 | 0 | 1 | 2 | 3 |

3 | 0 | 1 | 2 | 3 |

${+}_{\le}$ | 0 | 1 | 2 | 3 |
---|---|---|---|---|

0 | R | R | R | R |

1 | R | {1,2,3} | {1,2,3} | {1,2,3} |

2 | R | {1,2,3} | {2,3} | {2,3} |

3 | R | {1,2,3} | {2,3} | {3} |

+ | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

0 | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

1 | 1 | 2 | 3 | 4 | 5 | {0, 6} | 1 |

2 | 2 | 3 | 4 | 5 | {0, 6} | 1 | 2 |

3 | 3 | 4 | 5 | {0, 6} | 1 | 2 | 3 |

4 | 4 | 5 | {0, 6} | 1 | 2 | 3 | 4 |

5 | 5 | {0, 6} | 1 | 2 | 3 | 4 | 5 |

6 | 6 | 1 | 2 | 3 | 4 | 5 | 0 |

· | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1 | 0 | 5 | 4 | 3 | 2 | 1 | 0 |

2 | 0 | 1 | 2 | 3 | 4 | 5 | 0 |

3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

4 | 0 | 5 | 4 | 3 | 2 | 1 | 0 |

5 | 0 | 1 | 2 | 3 | 4 | 5 | 0 |

6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

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**MDPI and ACS Style**

Dakić, J.; Jančić-Rašović, S.; Cristea, I. Weak Embeddable Hypernear-Rings. *Symmetry* **2019**, *11*, 964.
https://doi.org/10.3390/sym11080964

**AMA Style**

Dakić J, Jančić-Rašović S, Cristea I. Weak Embeddable Hypernear-Rings. *Symmetry*. 2019; 11(8):964.
https://doi.org/10.3390/sym11080964

**Chicago/Turabian Style**

Dakić, Jelena, Sanja Jančić-Rašović, and Irina Cristea. 2019. "Weak Embeddable Hypernear-Rings" *Symmetry* 11, no. 8: 964.
https://doi.org/10.3390/sym11080964