# Elements of Hyperstructure Theory in UWSN Design and Data Aggregation

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

#### Mathematical Background of the Model

- $EL$–
**hyperstructures**are constructed from pre- and partially-ordered semigroups, i.e., the hyperoperation is defined using an operation and a relation compatible with it; **Quasi-order****hypergroups**are constructed from pre-ordered sets, i.e., the hyperoperation is defined using a relation only;**Ordered****hyperstructures**are algebraic hyperstructures on which a relation compatible with the hyperoperation is defined.

**Definition**

**1.**

**Proposition**

**1.**

**Definition**

**2.**

**Proposition**

**2.**

**Definition**

**3.**

**Notation.**Further on, for some $a\in H$, by ${\left[a\right)}_{\le}$ means the set $\{x\in H\mid a\le x\}$. For this reason, closed intervals will not be denoted by $[a,b]$ but by $\langle a;b\rangle $.

## 2. Mathematical Model

**Lemma**

**2.**

**Lemma**

**3.**

**Lemma**

**4.**

**Lemma**

**5.**

## 3. Use of the Theory of Quasi-Automata

**Definition**

**4.**

- 1.
- There exists an element $e\in I$ such that $\delta (e,s)=s$ for any state $s\in S$;
- 2.
- $\delta (y,\delta (x,s\left)\right)=\delta (xy,s)$ for any pair $x,y\in I$ and any state $s\in S.$

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Example**

**1.**

**Lemma**

**6.**

**Proof.**

**Remark**

**2.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Remark**

**3.**

**Example**

**2.**

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Multipath approach to UWSN data aggregation. Notice the oriented communication between nodes.

**Figure 2.**Cluster based approach to UWSN data aggregation—idealized deployment. The tiers need not be horizontal, we usually regard distance towards sink instead of depth.

(Terrestrial) WSN | UWSN | |
---|---|---|

Communication Media | RF Waves | Acoustic Waves |

Frequency | High | Low |

Node size | Small | Large |

Deployment | Dense | Sparse |

Power | Low | High |

Energy consumption | Low | High |

Propagation delay | Low | High |

Bandwidth | High | Low |

Path loss | Low | High |

Cost | Inexpensive | Expensive |

Memory | Sensor nodes have low capacity | Sensor nodes require large capacity |

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**MDPI and ACS Style**

Novák, M.; Křehlík, Š.; Ovaliadis, K.
Elements of Hyperstructure Theory in UWSN Design and Data Aggregation. *Symmetry* **2019**, *11*, 734.
https://doi.org/10.3390/sym11060734

**AMA Style**

Novák M, Křehlík Š, Ovaliadis K.
Elements of Hyperstructure Theory in UWSN Design and Data Aggregation. *Symmetry*. 2019; 11(6):734.
https://doi.org/10.3390/sym11060734

**Chicago/Turabian Style**

Novák, Michal, Štepán Křehlík, and Kyriakos Ovaliadis.
2019. "Elements of Hyperstructure Theory in UWSN Design and Data Aggregation" *Symmetry* 11, no. 6: 734.
https://doi.org/10.3390/sym11060734