# Breakable Semihypergroups

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Theorem**

**1.**

**Theorem**

**2.**

## 3. Breakable Semihypergroups

**Definition**

**2.**

**Definition**

**3.**

**Theorem**

**3.**

**Example**

**1.**

$\mathbf{\circ}$ | 1 | 2 | 3 |

1 | 1 | 1 | {1,3} |

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

3 | {1,3} | 3 | 3 |

**Example**

**2.**

$\mathbf{\circ}$ | 1 | 2 | 3 | 4 | 5 |

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

2 | 2 | 2 | {2,3} | 2 | {2,5} |

3 | 3 | {2,3} | 3 | 3 | {3,5} |

4 | 4 | 2 | 3 | 4 | 5 |

5 | 5 | {2,5} | {3,5} | 5 | 5 |

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

- (I)
- $({\mathcal{P}}^{*}\left(S\right),\star )$ is a semigroup, where the binary operation $\star $ is defined by:$$A\star B=\bigcup _{a\in A,b\in B}a\circ b,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}for\phantom{\rule{3.33333pt}{0ex}}all\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}A,B\in {\mathcal{P}}^{*}\left(S\right).$$
- (II)
- $(S,\circ )$ is breakable if and only if $({\mathcal{P}}^{*}\left(S\right),\star )$ is idempotent.

**Proof.**

- (I)
- The binary operation $\star $ is associative since, for every non empty subsets $A,B,C$ of S we have$$\begin{array}{ll}A\star (B\star C)& {\displaystyle =A\star (\bigcup _{b\in B,c\in C}b\circ c)}\\ & {\displaystyle =\bigcup _{a\in A,b\in B,c\in C}a\circ (b\circ c)}\\ & {\displaystyle =\bigcup _{a\in A,b\in B,c\in C}(a\circ b)\circ c}\\ & {\displaystyle =\left(\bigcup _{a\in A,b\in B}a\circ b\right)\star C}\\ & =(A\star B)\star C.\end{array}$$
- (II)
- Let $(S,\circ )$ be breakable and $A\subseteq S$. Then A is a subsemihypergroup of S, that is $A\star A\subseteq A$. On the other hand, for every $a\in A$ we have $a=a\circ a\subseteq A\star A$. Thus $A\star A=A$, so $({\mathcal{P}}^{*}\left(S\right),\star )$ is idempotent. Conversely, suppose that $({\mathcal{P}}^{*}\left(S\right),\star )$ is idempotent. Then, for every non empty subset A of S, we have $A\star A=A$, so A is a subsemihypergroup, meaning that S is breakable. □

**Proposition**

**1.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Remark**

**1.**

**Example**

**3.**

$\mathbf{\circ}$ | 1 | 2 | 3 | 4 |

1 | 1 | 1 | 3 | 4 |

2 | 2 | 2 | 3 | 4 |

3 | 3 | 3 | 3 | {3,4} |

4 | 4 | 4 | {3,4} | 4 |

**Example**

**4.**

$\mathbf{\circ}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

1 | 1 | {1,2} | {1,3} | 4 | 5 | 6 | 7 | 8 | 9 |

2 | {1,2} | 2 | {2,3} | 4 | 5 | 6 | 7 | 8 | 9 |

3 | {1,3} | {2,3} | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

4 | 4 | 4 | 4 | 4 | 5 | 6 | 7 | 8 | 9 |

5 | 5 | 5 | 5 | 4 | 5 | 6 | 7 | 8 | 9 |

6 | 6 | 6 | 6 | 6 | 6 | 6 | {6,7} | 8 | 9 |

7 | 7 | 7 | 7 | 7 | 7 | {6,7} | 7 | 8 | 9 |

8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |

9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |

**Example**

**5.**

**Definition**

**4.**

**Theorem**

**7.**

**Proof.**

**Corollary**

**1.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Heidari, D.; Cristea, I.
Breakable Semihypergroups. *Symmetry* **2019**, *11*, 100.
https://doi.org/10.3390/sym11010100

**AMA Style**

Heidari D, Cristea I.
Breakable Semihypergroups. *Symmetry*. 2019; 11(1):100.
https://doi.org/10.3390/sym11010100

**Chicago/Turabian Style**

Heidari, Dariush, and Irina Cristea.
2019. "Breakable Semihypergroups" *Symmetry* 11, no. 1: 100.
https://doi.org/10.3390/sym11010100