# Introduction to Dependence Relations and Their Links to Algebraic Hyperstructures

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Motivation

## 3. Properties of Dependence Relations

#### 3.1. Definition of a Dependence Relation

**Definition**

**1.**

#### 3.2. Influence and Impact in a Dependence Relation

- $Infl\left(x\right)$ as the set of all influential elements of x,
- $Imp\left(x\right)$ as the set of all elements influenced by x.

**Remark**

**1.**

#### 3.3. Simple and Composed Dependencies

#### 3.4. Length and Level of a Dependence Relation

**Example**

**1.**

- 1.
- the isolated elements, e.g., ${B}_{k,1}$, ${B}_{k,2}$, ${B}_{k,3}$, ${C}_{k,5}$, $C{A}_{k,2}$, $C{A}_{k,3}$, etc. and
- 2.
- the non-influential elements: ${S}_{k,1}$, ${S}_{k,6}$, ${S}_{k,7}$, ${S}_{k,8}$.

## 4. Preliminaries and Notations in Hyperstructure Theory

- $(a\circ b)\circ c=a\circ (b\circ c)$ for all $a,b,c\in H$ (associativity),
- $a\circ H=H\circ a=H$ for all $a\in H$ (reproductivity),

**Definition**

**2.**

**Definition**

**3.**

## 5. Hypergroupoids Associated with Abstract Dependencies

- $t\in Infl\left(u\right)$, where $u\in Imp\left(x\right)$;
- $t\in Imp\left(u\right)$, where $u\in Infl\left(x\right)$;
- $t\in Imp\left(u\right)$, where $u\in Imp\left(x\right)$;
- $t\in Infl\left(u\right)$, where $u\in Infl\left(x\right)$;

- $t\in Infl\left(u\right)$, where $u\in Imp\left(x\right)$$\Rightarrow u\in t{\ast}_{1}x$;
- $t\in Imp\left(u\right)$, where $u\in Infl\left(x\right)$$\Rightarrow u\in t{\ast}_{2}x$.

**Lemma**

**1.**

**Proof.**

**Remark**

**2.**

**Theorem**

**1.**

**Proof.**

**Remark**

**3.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Remark**

**4.**

**Lemma**

**4.**

**Lemma**

**5.**

**Example**

**2.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Remark**

**5.**

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Armstrong, W.W. Dependency structures of database relationships. In Proceedings of the IFIP Congress, Stockholm, Sweden, 5–10 August 1974; pp. 580–583. [Google Scholar]
- Matúš, F. Abstract functional dependency structures. Theor. Comput. Sci.
**1991**, 81, 117–126. [Google Scholar] [CrossRef] [Green Version] - More, S.M.; Naumov, P. The Functional Dependence Relation on Hypergraphs of Secrets. In Computational Logic in Multi-Agent Systems. CLIMA 2011. Lecture Notes in Computer Science; Leite, J., Torroni, P., Agotnes, T., Boella, G., Van der Torre, L., Eds.; Springer: Berlin/Heidelberg, Germany, 2011; Volume 6814. [Google Scholar]
- Jančič, M.; Kocijan, J.; Grašič, B. Identification of atmospheric variable using Deep Gaussian processes. IFAC-PapersOnLine
**2018**, 51, 43–48. [Google Scholar] [CrossRef] - Kocijan, J.; Perne, M.; Mlakar, P.; Grašič, B.; Zlata Božnar, M. Hybrid model of the near ground temperature profile. Stoch. Environ. Res. Risk. Assess.
**2019**. submitted. [Google Scholar] - Marty, F. Sur une généralisation de la notion de groupe. In Proceedings of the VIII Congrès des Mathématiciens Scandinaves, Stockholm, Sweden, 1934; pp. 45–49. [Google Scholar]
- Jantosciak, J. Transposition hypergroups, Noncommutative Join Spaces. J. Algebra
**1997**, 187, 97–119. [Google Scholar] [CrossRef] - Prenowitz, W. A contemporary approach to Classical Geometry. Am. Math. Mon.
**1961**, 68 Pt 2, 1–67. [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, 4–12 September 1994; pp. 19–30. [Google Scholar]
- Novák, M.; Křehlík, Š. EL-hyperstructures revisited. Soft Comput.
**2018**, 22, 7269–7280. [Google Scholar] [CrossRef] - Massouros, C.G. On path hypercompositions in graphs and automata. MATEC Web Conf.
**2016**, 41, 05003. [Google Scholar] [CrossRef] - Massouros, C.G.; Massouros, G.G. On open and closed hypercompositions. 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 Publishing: Melville, NY, USA, 2018. [Google Scholar]
- Polat, N. On bipartite graphs whose interval space is a closed join space. J. Geom.
**2017**, 108, 719–741. [Google Scholar] [CrossRef] - Novák, M.; Křehlík, Š.; Cristea, I. Cyclicity in EL-hypergroups. Symmetry
**2018**, 10, 611. [Google Scholar] [CrossRef] - Vougiouklis, T. Cyclicity in a special class of hypergroups. Acta Univ. Carolinae Math. Phys.
**1981**, 22, 3–6. [Google Scholar] - Al Tahan, M.; Davvaz, B. On some properties of single power cyclic hypergroups and regular relations. J. Algebra Appl.
**2017**, 16, 1750214. [Google Scholar] [CrossRef] - De Salvo, M.; Freni, D. Semi-ipergruppi e ipergruppi ciclici. Atti Sem. Mat. Fis. Univ. Modena
**1981**, 30, 44–59. [Google Scholar] - Massouros, G.G.; Mittas, J.D. Languages, Automata and Hypercompositional Structures. In Proceedings of the Fourth International Congress on Algebraic Hyperstructures and Applications, Xanthi, Greece, 27–30 June 1990; World Scientific Publishing Co. Pte. Ltd.: Singapore, 1991; pp. 137–147. [Google Scholar]
- Massouros, G.G. Hypercompositional structures from the computer theory. Ratio Math.
**1999**, 13, 37–42. [Google Scholar] - Chvalina, J. Functional Graphs, Quasi-ordered Sets and Commutative Hypergroups; Masaryk University: Brno, Czech Republic, 1995. (In Czech) [Google Scholar]
- Heidari, D.; Cristea, I. Breakable semihypergroups. Symmetry
**2019**, 11, 100. [Google Scholar] [CrossRef] - Chvalina, J.; Chvalinová, L. State hypergroups of automata. Acta Math. Inform. Univ. Ostrav.
**1996**, 4, 105–120. [Google Scholar] - Cristea, I.; Hassani Sadrabadi, E.; Davvaz, B. A fuzzy application of the group ${\mathbb{Z}}_{n}$ to complete hypergroups. Soft Comput.
**2019**. [Google Scholar] [CrossRef] - Cristea, I. Regularity of intuitionistic fuzzy relations on hypergroupoids. An. Şt. Univ. Ovidius Constanţa
**2014**, 22, 105–119. [Google Scholar] [CrossRef] - 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. Log.
**2002**, 8, 645–657. [Google Scholar] - Leoreanu–Fotea, V.; Davvaz, B. n-hypergroups and binary relations. Eur. J. Combin.
**2008**, 29, 1207–1218. [Google Scholar] [CrossRef] [Green Version] - Hošková-Mayerová, Š.; Maturo, A. Decision-making process using hyperstructures and fuzzy structures in social sciences. Stud. Fuzz. Soft Comput.
**2018**, 357, 103–111. [Google Scholar] - Saeid, A.B.; Flaut, C.; Hoškova-Mayerova, Š.; Afshar, M.; Rafsanjani, M.K. Some connections between BCK-algebras and n-ary block codes. Soft Comput.
**2018**, 22, 41–46. [Google Scholar] [CrossRef] - Kalampakas, A.; Spartalis, S.; Tsigkas, A. The Path Hyperoperation. An. Şt. Univ. Ovidius Constanţa
**2014**, 22, 141–153. [Google Scholar] [CrossRef] [Green Version] - Chvalina, J.; Hošková-Mayerová, Š.; Nezhad, A.D. General actions of hyperstructures and some applications. An. Şt. Univ. Ovidius Constanţa
**2013**, 21, 59–82. [Google Scholar] [CrossRef] - Al Tahan, M.; Davvaz, B. On a special single-power cyclic hypergroup and its automorphisms. Discret. Math. Algorithm. Appl.
**2016**, 8, 1650059. [Google Scholar] [CrossRef] - Novák, M.; Cristea, I. Composition in EL–hyperstructures. Hacet. J. Math. Stat.
**2019**, 48, 45–58. [Google Scholar] [CrossRef]

Measurement Station | Variable | The Time of Sample | Notation |
---|---|---|---|

Stolp Krško | temperature at 2 m | $k-2$ | ${S}_{k-2,1}$ |

Stolp Krško | temperature at 2 m | $k-1$ | ${S}_{k-1,1}$ |

Stolp Krško | global solar radiation | $k-2$ | ${S}_{k-2,9}$ |

Cerklje | temperature at 2 m | $k-1$ | ${C}_{k-1,1}$ |

Stolp Krško | wind speed | $k-1$ | ${S}_{k-1,2}$ |

Stolp Krško | global solar radiation | $k-1$ | ${S}_{k-1,9}$ |

WRF model | temperature at 2 m | k | ${W}_{k,1}$ |

WRF model | global solar radiation | k | ${W}_{k,9}$ |

$\ast $ | ${\mathit{a}}_{1,1}$ | ${\mathit{a}}_{1,2}$ | ${\mathit{a}}_{2,2}$ | ${\mathit{a}}_{2,3}$ | ${\mathit{a}}_{3,1}$ | ${\mathit{a}}_{3,2}$ | ${\mathit{a}}_{3,4}$ | ${\mathit{a}}_{3,5}$ |
---|---|---|---|---|---|---|---|---|

${a}_{1,1}$ | $\left\{{a}_{1,1}\right\}$ | $\{{a}_{1,1},$ ${a}_{1,2},{a}_{3,2}\}$ | $\{{a}_{1,1},{a}_{2,2}\}$ | $\{{a}_{1,1}$, ${a}_{2,2},{a}_{2,3}\}$ | $\{{a}_{1,1}$, ${a}_{2,2},{a}_{3,1}\}$ | $\{{a}_{1,1},{a}_{3,2}\}$ | $\{{a}_{1,1},{a}_{3,4}\}$ | $\{{a}_{1,1},{a}_{3,5}\}$ |

${a}_{1,2}$ | $\{{a}_{1,1}$, ${a}_{1,2},{a}_{3,2}\}$ | $\{{a}_{1,1},{a}_{1,2}$, ${a}_{2,2},{a}_{3,2}\}$ | $\{{a}_{1,1},{a}_{1,2},$ ${a}_{2,2}$,${a}_{2,3}$, ${a}_{3,2}\}$ | $\{{a}_{1,1},{a}_{1,2}$, ${a}_{2,2},{a}_{3,2}$ ${a}_{3,1}\}$ | $\{{a}_{1,1}$, ${a}_{1,2},{a}_{3,2}\}$ | $\{{a}_{1,1},{a}_{1,2}$, ${a}_{3,2},{a}_{3,4}\}$ | $\{{a}_{1,1},{a}_{1,2}$, ${a}_{3,2},{a}_{3,5}\}$ | |

${a}_{2,2}$ | $\{{a}_{1,1},{a}_{2,2}\}$ | $\{{a}_{1,1}$, ${a}_{2,2},{a}_{2,3}\}$ | $\{{a}_{1,1}$, ${a}_{2,2},{a}_{3,1}\}$ | $\{{a}_{1,1}$, ${a}_{2,2},{a}_{3,2}\}$ | $\{{a}_{1,1}$, ${a}_{2,2},{a}_{3,4}\}$ | $\{{a}_{1,1}$, ${a}_{2,2},{a}_{3,5}\}$ | ||

${a}_{2,3}$ | $\{{a}_{1,1}$, ${a}_{2,2},{a}_{2,3}\}$ | $\{{a}_{1,1},{a}_{2,2}$, ${a}_{2,3},{a}_{3,1}\}$ | $\{{a}_{1,1},{a}_{2,2}$, ${a}_{2,3},{a}_{3,2}\}$ | $\{{a}_{1,1},{a}_{2,2}$, ${a}_{3,4},{a}_{2,3}\}$ | $\{{a}_{1,1},{a}_{2,2}$, ${a}_{2,3},{a}_{3,5}\}$ | |||

${a}_{3,1}$ | $\{{a}_{1,1}$, ${a}_{2,2},{a}_{3,1}\}$ | $\{{a}_{1,1},{a}_{2,2}$, ${a}_{3,1},{a}_{3,2}\}$ | $\{{a}_{1,1},{a}_{2,2}$, ${a}_{3,1},{a}_{3,4}\}$ | $\{{a}_{1,1},{a}_{2,2}$, ${a}_{3,1},{a}_{3,5}\}$ | ||||

${a}_{3,2}$ | $\{{a}_{1,1},{a}_{3,2}\}$ | $\{{a}_{1,1}$, ${a}_{3,2},{a}_{3,4}\}$ | $\{{a}_{1,1}$, ${a}_{3,2},{a}_{3,5}\}$ | |||||

${a}_{3,4}$ | $\left\{{a}_{3,4}\right\}$ | $\{{a}_{1,1}$, ${a}_{3,4},{a}_{3,5}\}$ | ||||||

${a}_{3,5}$ | $\{{a}_{1,1},{a}_{3,5}\}$ |

© 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**

Cristea, I.; Kocijan, J.; Novák, M.
Introduction to Dependence Relations and Their Links to Algebraic Hyperstructures. *Mathematics* **2019**, *7*, 885.
https://doi.org/10.3390/math7100885

**AMA Style**

Cristea I, Kocijan J, Novák M.
Introduction to Dependence Relations and Their Links to Algebraic Hyperstructures. *Mathematics*. 2019; 7(10):885.
https://doi.org/10.3390/math7100885

**Chicago/Turabian Style**

Cristea, Irina, Juš Kocijan, and Michal Novák.
2019. "Introduction to Dependence Relations and Their Links to Algebraic Hyperstructures" *Mathematics* 7, no. 10: 885.
https://doi.org/10.3390/math7100885