# Elements of Hyperstructure Theory in UWSN Design and Data Aggregation

^{1}

^{2}

^{3}

^{*}

## 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

- Novák, M.; Ovaliadis, K.; Křehlík, Š. A hyperstructure model of Underwater Wireless Sensor Network (UWSN) design. In Proceedings of the International Conference on Numerical Analysis and Applied Mathematics (ICNAAM 2017), Thessaloniki, Greece, 25–30 September 2017. [Google Scholar]
- Wahid, A.; Dongkyun, K. Analyzing Routing Protocols for Underwater Wireless Sensor Networks. IJCNIS
**2010**, 2, 253–261. [Google Scholar] - Domingo, M.C.; Prior, R. A distributed clustering scheme for underwater wireless sensor networks. In Proceedings of the 2007 IEEE 18th International Symposium on Personal, Indoor and Mobile Radio Communications, Athens, Greece, 3–7 September 2007. [Google Scholar]
- Ayaz, M.; Baig, I.; Abdullah, A.; Faye, I. A survey on routing techniques in underwater wireless sensor networks. J. Netw. Comput. Appl.
**2011**, 34, 1908–1927. [Google Scholar] [CrossRef] - Ovaliadis, K.; Savage, N. Cluster protocols in underwater sensor networks: A research review. J. Eng. Sci. Technol. Rev.
**2014**, 7, 171–175. [Google Scholar] [CrossRef] - Rault, T.; Abdelmadjid, B.; Yacine, C. Energy efficiency in wireless sensor networks: A top-down survey. Comput. Netw.
**2014**, 67, 104–122. [Google Scholar] [CrossRef] - Abougamila, S.; Elmorsy, M.; Elmallah, E.S. A graph theoretic approach to localization under uncertainty. In Proceedings of the 2018 IEEE International Conference on Communications, ICC 2018, Kansas City, MO, USA, 20–24 May 2018. [Google Scholar]
- Domingo, M.C. Optimal placement zones of wireless nodes in Underwater Wireless Sensor Networks with shadow zones. In Proceedings of the 2009 2nd IFIP Wireless Days (WD), Paris, France, 15–17 December 2009. [Google Scholar]
- Jiang, P.; Wang, X.; Jiang, L. Node deployment algorithm based on connected tree for underwater sensor networks. Sensors
**2015**, 15, 16763–16785. [Google Scholar] [CrossRef] - Corsini, P.; Leoreanu, V. Applications of Hyperstructure Theory; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2003. [Google Scholar]
- Comer, S.D. Combinatorial aspects of relation. Algebr. Universalis
**1984**, 18, 77–94. [Google Scholar] [CrossRef] - Corsini, P. Binary relations and hypergroupoids. Ital. J. Pure Appl. Math.
**2000**, 7, 11–18. [Google Scholar] - Corsini, P. Hyperstructures associated with ordered sets. Bull. Greek Math. Soc.
**2003**, 48, 7–18. [Google Scholar] - Cristea, I.; Ştefănescu, M. Hypergroups and n-ary relations. Eur. J. Combin.
**2010**, 31, 780–789. [Google Scholar] [CrossRef] - De Salvo, M.; Lo Faro, G. Hypergroups and binary relations. Multi. Val. Logic
**2002**, 8, 645–657. [Google Scholar] - Pickett, H.E. Homomorphisms and subalgebras of multialgebras. Pac. J. Math.
**1967**, 21, 327–342. [Google Scholar] [CrossRef] - Chvalina, J. Functional Graphs, Quasi-ordered Sets and Commutative Hypergroups; Masaryk University: Brno, Czech Republic, 1995. (In Czech) [Google Scholar]
- Chvalina, J.; Křehlík, Š.; Novák, M. Cartesian composition and the problem of generalising the MAC condition to quasi-multiautomata. An. Şt. Univ. Ovidius Constanţa
**2016**, 24, 79–100. [Google Scholar] - Křehlík, Š.; Novák, M. From lattices to H
_{v}–matrices. An. Şt. Univ. Ovidius Constanţa**2016**, 24, 209–222. [Google Scholar] [CrossRef] - Novák, M. Some basic properties of EL–hyperstructures. Eur. J. Combin.
**2013**, 34, 446–459. [Google Scholar] [CrossRef] - Novák, M.; Cristea, I. Composition in EL–hyperstructures. Hacet. J. Math. Stat.
**2019**, 48, 45–58. [Google Scholar] [CrossRef] - Novák, M.; Křehlík, Š. EL–hyperstructures revisited. Soft Comput.
**2018**, 22, 7269–7280. [Google Scholar] [CrossRef] - Novák, M.; Křehlík, Š.; Cristea, I. Cyclicity in EL–hypergroups. Symmetry
**2018**, 10, 611. [Google Scholar] [CrossRef] - Chvalina, J. Commutative hypergroups in the sense of Marty and ordered sets. In Proceedings of the Summer School on General Algebra and Ordered Sets, Olomouc, Czech Republic, 12–15 August 1994; pp. 19–30. [Google Scholar]
- Chvalina, J.; Chvalinová, L. State hypergroups of automata. Acta Math. et Inform. Univ. Ostraviensis
**1996**, 4, 105–120. [Google Scholar] - Comer, S.D. Some problems on hypergroups. In Algebraic Hyperstructures and Applications; Vougiouklis, T., Ed.; World Scientific Publishing: Singapore, 1991; pp. 67–74. [Google Scholar]
- Massouros, G.G.; Mittas, J.D. Languages, Automata and Hypercompositional Structures. In Algebraic Hyperstructures and Applications; Vougiouklis, T., Ed.; World Scientific Publishing: Singapore, 1991; pp. 137–147. [Google Scholar]
- Heidari, D.; Davvaz, B. On ordered hyperstructures. UPB Sci. Bull. Ser. A
**2011**, 73, 85–96. [Google Scholar] - Li, N.; Martínez, J.F.; Meneses Chaus, J.; Eckert, M. A survey on underwater acoustic sensor network routing protocols. Sensors
**2016**, 16, 414. [Google Scholar] [CrossRef] - Novák, M.; Křehlík, Š.; Staněk, D. n–ary Cartesian composition of automata. Soft Comput.
**2019**. [Google Scholar] [CrossRef] - Hošková, Š. Discrete transformation hypergroups. In Proceedings of the 4th International Conference Aplimat, Bratislava, Slovakia, 1–4 February 2005; pp. 275–279. [Google Scholar]
- Hošková, Š.; Chvalina, J. A survey of investigations of the Brno research group in the hyperstructure theory since the last AHA Congress. In Proceedings of the AHA 2008: 10th International Congress-Algebraic Hyperstructures And Applications, Brno, Czech Republic, 3–9 September 2008. [Google Scholar]
- Hošková, Š.; Chvalina, J.; Račková, P. Hypergroups of integral operators in connections with transformation structures. AiMT
**2006**, 1, 105–117. [Google Scholar]

**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 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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