# The Symmetry of Lower and Upper Approximations, Determined by a Cyclic Hypergroup, Applicable in Control Theory

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Notation and Context

**Example**

**1.**

#### Basic Notions of the Theory of Algebraic Hypercompositional Structures

**Definition**

**1**

**.**A hypergroup $(H,\circ )$ is called cyclic if, for some $h\in H$, there is

## 3. Single-Power Cyclic Hypergroup of Matrices

**Example**

**2.**

**Remark**

**1.**

**Example**

**3.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Example**

**4.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Example**

**5.**

**Remark**

**2.**

## 4. Approximation Space Determined by the Cyclic Hypergroup

**Definition**

**2**

**.**Let H be a set and R be an equivalence relation on H. Let A be subset of H. A rough set is a pair of subsets $\left(\overline{R}\left(A\right),\underline{R}\left(A\right)\right)$ of H which approximates A as closer as possible from outside and inside, respectively:

**Example**

**6.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**1.**

**Notation**

**1.**

**Example**

**7.**

**Theorem**

**6.**

**Proof.**

**Remark**

**3.**

**Corollary**

**2.**

**Theorem**

**7.**

- (1)
- $\overline{{R}_{M}}({\mathit{E}}^{\ast}\ast {\mathit{E}}^{\ast})={\mathbb{M}}_{m,n}\left(\mathcal{E}\right)=\underline{{R}_{M}}({\mathit{E}}^{\ast}\ast {\mathit{E}}^{\ast})$
- (2)
- $\underline{{R}_{M}}\left({\mathit{A}}_{e}\right)\subseteq \underline{{R}_{M}}\left({\mathit{A}}_{e}^{2}\right)\subseteq \underline{{R}_{M}}\left({\mathit{A}}_{e}^{3}\right)={\mathbb{M}}_{m,n}\left(\mathcal{E}\right)$
- (3)
- $\overline{{R}_{M}}\left({\mathit{A}}_{e}\right)\subseteq \overline{{R}_{M}}\left({\mathit{A}}_{e}^{2}\right)\subseteq \overline{R}\left({\mathit{A}}_{e}^{3}\right)={\mathbb{M}}_{m,n}\left(\mathcal{E}\right)$

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

#### Application in the Control Theory

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Pavlak, Z. Rough Set: Theorerical Aspects of Reasoning Above Data; Kluver Academic Publishers: Dordrecht, The Netherlands, 1991. [Google Scholar]
- Pavlak, Z. Rough set. Int. J. Comput. Inf. Sci.
**1982**, 11, 341–356. [Google Scholar] - Molodtsov, D. Soft set—First results. Comp. Math. Appl.
**1999**, 37, 19–31. [Google Scholar] [CrossRef] [Green Version] - Chvalina, J. Functional Graphs, Quasi-ordered Sets and Commutative Hypergroups; Masaryk University: Brno, Czech Republic, 1995. (In Czech) [Google Scholar]
- Hošková, Š.; Chvalina, J.; Račková, P. Transposition hypergroups of Fredholm integral operators and related hyperstructures. Part I. J. Basic Sci.
**2008**, 4, 43–54. [Google Scholar] - Novák, M. On EL-semihypergroups. Eur. J. Combin.
**2015**, 44 Pt B, 274–286. [Google Scholar] - 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] - Křehlík, Š.; Novák, M. From lattices to H
_{v}-matrices. An. Şt. Univ. Ovidius Constanţa**2016**, 24, 209–222. [Google Scholar] [CrossRef] [Green Version] - Leoreanu-Foneta, V. The lower and upper approximation in hypergroup. Inf. Sci.
**2008**, 178, 3605–3615. [Google Scholar] - Novák, M.; Křehlík, Š.; Staněk, D. n-ary Cartesian composition of automata. Soft Comput.
**2019**. [Google Scholar] [CrossRef] - Novák, M.; Ovaliadis, K.; Křehlík, Š. A hyperstructure of Underwater Wireless Sensor Network (UWSN) design. In AIP Conference Proceedings 1978, Proceedings of the International Conference on Numerical Analysis and Applied Mathematics (ICNAAM 2017), The MET Hotel, Thessaloniki, Greece, 25–30 September 2017; American Institute of Physics: Thessaloniki, Greece, 2018. [Google Scholar]
- Račková, P. Hypergroups of symmetric matrices. In Proceedings of the 10th International Congress of Algebraic Hyperstructures and Applications (AHA 2008), Brno, Czech Republic, 3–9 September 2008; University of Defence: Brno, Czech Republic, 2009; pp. 267–272. [Google Scholar]
- Novák, M.; Křehlík, Š.; Cristea, I. Cyclicity in EL-hypergroups. Symmetry
**2018**, 10, 611. [Google Scholar] [CrossRef] [Green Version] - Vougiouklis, T. Cyclicity in a Special Class of Hypergroups. Acta Universitatis Carolinae Mathematica Physica
**1981**, 22, 3–6. [Google Scholar] - Skowron, A.; Dutta, S. Rough sets: Past, present, and future. Nat. Comput.
**2018**, 17, 855–876. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Anvariyeh, S.M.; Mirvakili, S.; Davvaz, B. Pavlak’s approximations in Γ-semihypergroups. Comput. Math. Appl.
**2010**, 60, 45–53. [Google Scholar] [CrossRef] [Green Version] - Leoreanu-Foneta, V. Approximations in hypergroups and fuzzy hypergroups. Comput. Math. Appl.
**2011**, 61, 2734–2741. [Google Scholar] [CrossRef] [Green Version] - Corsini, P.; Leoreanu, V. Applications of Hyperstructure Theory; Kluwer Academic Publishers: Dodrecht, The Netherlands; Boston, MA, USA; London, UK, 2003. [Google Scholar]
- Davvaz, B.; Leoreanu–Fotea, V. Applications of Hyperring Theory; International Academic Press: Palm Harbor, FL, USA, 2007. [Google Scholar]
- Vougiouklis, T. Hyperstructures and Their Representations; Hadronic Press: Palm Harbor, FL, USA, 1994. [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] - Massouros, C.G. On the semi-sub-hypergroups of a hypergroup. Int. J. Math. Math. Sci.
**1991**, 14, 293–304. [Google Scholar] [CrossRef] [Green Version] - Kokovkina, V.A.; Antipov, V.A.; Kirnos, V.P.; Priorov, A.L. The Algorithm of EKF-SLAM Using Laser Scanning System and Fisheye Camera. In Proceedings of the 2019 Systems of Signal Synchronization, Generating and Processing in Telecommunications (SYNCHROINFO), Yaroslavl, Russia, 1–3 July 2019; pp. 1–6. [Google Scholar]
- Vu, T.-D.; Aycard, O.; Appenrodt, N. Online Localization and Mapping with Moving Object Tracking in Dynamic Outdoor Environments. In Proceedings of the 2007 IEEE Intelligent Vehicles Symposium, Istanbul, Turkey, 13–15 June 2007; pp. 190–195. [Google Scholar]
- Zhu, D.; Sun, X.; Wang, L.; Liu, B.; Ji, K. Mobile Robot SLAM Algorithm Based on Improved Firefly Particle Filter. In Proceedings of the 2019 International Conference on Robots & Intelligent System (ICRIS), Haikou, China, 15–16 June 2019; pp. 35–38. [Google Scholar]

**Figure 5.**The lower and upper approximation with relation ${R}_{D}$ for matrices $2\times 2$–$5\times 5$.

**Figure 7.**The lower and upper approximation with relation ${R}_{M}$ for matrices $2\times 2$ and $\mathcal{E}=\{0,1,2\}$.

© 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**

Křehlík, Š.; Vyroubalová, J.
The Symmetry of Lower and Upper Approximations, Determined by a Cyclic Hypergroup, Applicable in Control Theory. *Symmetry* **2020**, *12*, 54.
https://doi.org/10.3390/sym12010054

**AMA Style**

Křehlík Š, Vyroubalová J.
The Symmetry of Lower and Upper Approximations, Determined by a Cyclic Hypergroup, Applicable in Control Theory. *Symmetry*. 2020; 12(1):54.
https://doi.org/10.3390/sym12010054

**Chicago/Turabian Style**

Křehlík, Štěpán, and Jana Vyroubalová.
2020. "The Symmetry of Lower and Upper Approximations, Determined by a Cyclic Hypergroup, Applicable in Control Theory" *Symmetry* 12, no. 1: 54.
https://doi.org/10.3390/sym12010054