# Introduction to Dependence Relations and Their Links to Algebraic Hyperstructures

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## 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

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

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

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

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

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

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

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

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

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

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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