# Cyclicity in EL–Hypergroups

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## Abstract

**:**

## 1. Introduction, Preliminaries and Historical Context

#### 1.1. The Origins by Wall

**Definition**

**1.**

**Definition**

**2**

**.**If a hypergroup H is generated by a single element a of H, then H will be called a cyclic hypergroup.

#### 1.2. The Original Approach of De Salvo and Freni

**Definition**

**3**

**.**We call $P\ne \varnothing $, $P\subseteq H$, a cyclic part of a semihypergroup H if there exists an element $x\in P$ such that, for all $a\in P$, there exists $n\in \mathbb{N}$ such that $a\in {x}^{n}$. The element $x\in P$ is called generator of P. If H is a cyclic part, we call H a cyclic semihypergroup.

**Definition**

**4**

**.**Let H be a cyclic semihypergroup with generator h. We call cyclicity (Italian: ciclicità) of an element $a\in H$ the minimum $m\in \mathbb{N}\backslash \left\{1\right\}$ such that $a\in {h}^{m}$; we write $cicl\left(a\right)=m$. We call cyclicity of Hand denote $cicl\left(H\right)=max\left\{cicl\right(a)\mid a\in H\}$.

#### 1.3. The Approach of Vougiouklis

**Definition**

**5**

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

#### 1.4. An Alternative Approach of the Italian School

**Definition**

**6**

**.**Let $(H,\ast )$ be a (semi)hypergroup and $n>1$ a natural number. We define relation ${\beta}_{n}$ as follows:

**Theorem**

**1**

**.**A hypergroup H is strongly cyclic, generated by x, if and only if $H/{\beta}^{\ast}$ is a cyclic group generated by ${\phi}_{H}\left(x\right)$.

**Definition**

**7**

**.**A hypergroup H is called cyclic with a generator x if ${\phi}_{H}\left(H\right)$ is a cyclic group generated from ${\phi}_{H}\left(x\right)$.

#### 1.5. Corsini’s Book: A Synthesis of Approaches

**Definition**

**8**

**.**A semihypergroup H is called cyclic if there exists $h\in H$ such that, for all $x\in H$, there exists $n\in \mathbb{N}$ such that $x\in {h}^{n}$. We call h the s–generator of H. A hypergroup is calleds-cyclic if it is a cyclic semihypergroup.

**Theorem**

**2**

**.**Every hypergroup H, if it is s-cyclic, is cyclic, and is generated by its s–generator h.

#### 1.6. Current State of the Art

#### 1.7. A Remark Concluding the Introduction

- There exist two main approaches towards cyclicity in the hyperstructure theory: through the fact that we get a hypergroup by repeatedly applying the hyperoperation on a generator, or through the canonical projection. Both were introduced by the Italian school. The former was independently introduced by Vougiouklis and is related to the original Wall’s definition. The latter is related to the notion of completeness and the results of Koskas.
- Vougiouklis defined numerous cases of cyclicity and periods such as single power (one generator only) and finite/infinite period. The purpose of his definition was to study a specific class of cyclic hypergroups, those based on P–hyperoperations.
- The original definition of Wall, the definition of Vougiouklis and the definition using canonical projection were provided for hypergroups. The original definition of De Salvo and Freni was provided for semihypergroups.
- Cyclicity of a (semi)hypergroup (defined by the Italian schools) and period of a hypergroup (defined by Vougiouklis) are, in a general case, different.

**Notation.**

## 2. Examples

#### 2.1. Theoretical Background

**Definition**

**9.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3**

**.**Every single power cyclic hypergroup $(H,\ast )$ has a trivial fundamental group $H/\beta $.

**Lemma**

**4**

**.**Let $(S,\xb7,\le )$ be a partially ordered semigroup. Binary hyperoperation $\ast :S\times S\to {\mathcal{P}}^{\ast}\left(S\right)$ defined by

**Lemma**

**5**

**.**Let $(S,\xb7,\le )$ be a partially ordered semigroup. The following conditions are equivalent:

- For any pair $a,b\in S$, there exists a pair $c,{c}^{\prime}\in S$ such that $b\xb7c\le a$ and ${c}^{\prime}\xb7b\le a$.
- The semi-hypergroup $(S,\ast )$ defined by Formula (6) is a hypergroup.

**Remark**

**1.**

#### 2.2. Single Power Cyclic Hypergroups

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

#### 2.3. s–Cyclic Hypergroups Which Are Not Single Power Cyclic

**Example**

**4.**

* | e | ${p}_{1}$ | ${p}_{2}$ | ${p}_{3}$ | ${p}_{4}$ | ${p}_{5}$ |

e | + | − | − | − | + | + |

${p}_{1}$ | + | + | + | − | − | |

${p}_{2}$ | + | + | − | − | ||

${p}_{3}$ | + | − | − | |||

${p}_{4}$ | + | + | ||||

${p}_{5}$ | + |

#### 2.4. Cyclic Hypergroups Which Are Not s–Cyclic

**Example**

**5.**

* | 1 | 2 | 3 | 4 |

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

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

3 | $\{1,2,3\}$ | $\{1,2,3\}$ | $\{1,2,3\}$ | $\{3,4\}$ |

4 | $\{1,2,4\}$ | $\{1,2,4\}$ | $\{3,4\}$ | $\{1,2,4\}$ |

## 3. New Results

**Theorem**

**5.**

**Example**

**6.**

**Corollary**

**1.**

**Proof.**

**Example**

**7.**

**Lemma**

**6.**

- If $a\le b$, then $a{\beta}_{2}b$, i.e., $a\beta b$.
- If $a,b\in H$ are such that $lb\{a,b\}$ exists, then $a{\beta}_{2}b$, i.e., $a\beta b$.

**Proof.**

**Theorem**

**6.**

**Corollary**

**2.**

**Proof.**

**Example**

**8.**

**Remark**

**2.**

## 4. Conclusions and Future Work

- operationally equivalent if $x\circ a=y\circ a$ and $a\circ x=a\circ y$, for all $a\in H$,
- inseparable if for $a,b\in H$ we have $x\in a\circ b$ if and only if $y\in a\circ b$,
- essentially indistinguishable if they are operationally equivalent and inseparable.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Marty, F. Sur une généralisation de la notion de groupe. IV Congrès des Mathématiciens Scandinaves
**1934**, 45–49. [Google Scholar] - Wall, H.S. Hypergroups. Am. J. Math.
**1934**, 59, 77–98. [Google Scholar] [CrossRef] - De Salvo, M.; Freni, D. Sugli ipergruppi ciclici e completi. Matematiche (Catania)
**1980**, 35, 211–226. [Google Scholar] - Corsini, P. Sur les semi-hypergroupes complètes et les groupoides. Atti Soc. Pel. Sci. Fis. Mat. Nat.
**1980**, 26, 391–398. [Google Scholar] - Corsini, P.; Romeo, G. Hypergroupes complètes et T–groupoids; Atti Convegno su “Sistemi binari e loro applicazioni”: Taormina, Italy, 1978; pp. 129–146. [Google Scholar]
- De Salvo, M. Sugli ipergruppi completi finiti. Riv. Mat. Univ. Parma
**1982**, 8, 269–280, (submitted 1980). [Google Scholar] - Freni, D. Ipergruppi ciclici e torsione negli ipergruppi. Matematiche (Catania)
**1980**, 35, 270–286. [Google Scholar] - De Salvo, M.; Freni, D. Semi-ipergruppi e ipergruppi ciclici. Atti Sem. Mat. Fis. Univ. Modena
**1981**, 30, 44–59. [Google Scholar] - Freni, D. Una nota su gli ipergruppoidi ciclici. Ratio Mathematica
**1995**, 9, 101–111. [Google Scholar] - Vougiouklis, T. Cyclicity in a special class of hypergroups. Acta Univ. Carolinae Math. Phys.
**1981**, 22, 3–6. [Google Scholar] - Konguetsof, L.; Vougiouklis, T.; Kessoglides, M.; Spartalis, S. On cyclic hypergroups with period. Acta Univ. Carolinae Math. Phys.
**1987**, 28, 3–7. [Google Scholar] - Vougioulis, T. Isomorphisms on P–hypergroups and cyclicity. Ars Combinatoria
**1990**, 29A, 241–245. [Google Scholar] - De Salvo, M.; Freni, D. Ipergruppi finitamente generati. Riv. Mat. Univ. Parma
**1986**, 12, 177–186. [Google Scholar] - De Salvo, M. Su le potenze ad esponente intero in un ipergruppo e gli r–ipergruppi. Riv. Mat. Univ. Parma
**1985**, 4, 409–421. [Google Scholar] - Antampoufis, N.; Hošková-Mayerová, Š. A brief survey on the two different approaches of fundamental equivalence relations on hyperstructures. Ratio Math.
**2017**, 33, 47–60. [Google Scholar] - Koskas, M. Groupoids, demi-hypergroupes et hypergroupes. J. Math. Pure Appl.
**1970**, 48, 155–192. [Google Scholar] - Freni, D. A note on the core of a hypergroup and the transitive closure β
^{⋆}of β. Riv. Mat. Pura Appl.**1991**, 8, 153–156. [Google Scholar] - Corsini, P. Prolegomena of Hypergroup Theory; Aviani Editore: Tricesimo, Italy, 1993. [Google Scholar]
- Corsini, P.; Leoreanu, V. Applications of Hyperstructure Theory; Kluwer Academic Publishers: Dodrecht, The Netherlands; Boston, MA, USA; London, UK, 2003. [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] [Green Version] - Al Tahan, M.; Davvaz, B. On a special single-power cyclic hypergroup and its automorphisms. Discrete Math. Algorithm. Appl.
**2016**, 8, 1650059. [Google Scholar] [CrossRef] - Karimian, M.; Davvaz, B. On the γ-cyclic hypergroups. Commun. Algebra
**2006**, 34, 4579–4589. [Google Scholar] [CrossRef] - Freni, D. A new characterization of the derived hypergroup via strongly regular equivalencies. Commun. Algebra
**2002**, 30, 3977–3989. [Google Scholar] [CrossRef] - Chvalina, J. Commutative hypergroups in the sense of Marty and ordered sets. In General Algebra and Ordered Sets, Proceedings of the International Conference Olomouc; Verlag Johannes Heyn: Olomouc, Czech Republic, 1994; pp. 19–30. [Google Scholar]
- Chvalina, J. Functional Graphs, Quasi-Ordered Sets and Commutative Hypergroups; Masaryk University: Brno, Czech Republic, 1995. (In Czech) [Google Scholar]
- Massouros, C.G. On path hypercompositions in graphs and automata. MATEC Web Conf.
**2016**, 41, 05003. [Google Scholar] [CrossRef] [Green Version] - Novák, M. On EL–semihypergroups. Eur. J. Combin.
**2015**, 44 Pt B, 274–286. [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.
**2018**, in press. [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.; Křehlík, Š. EL–hyperstructures revisited. Soft Comput.
**2018**, 22, 7269–7280. [Google Scholar] [CrossRef] - 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] - Pickett, H.E. Homomorphisms and subalgebras of multialgebras. Pac. J. Math.
**1967**, 21, 327–342. [Google Scholar] [CrossRef] [Green Version] - Novák, M. EL–semihypergroups in which the quasi-ordering is not antisymmetric. In Mathematics, Information Technologies and Applied Sciences 2017: Post-Conference Proceedings of Extended Versions of Selected Papers; University of Defence: Brno, Czech Republic, 2017; pp. 183–192. [Google Scholar]
- Jantosciak, J. Reduced hypergroups. In Algebraic Hyperstructures and Applications, Proceedings of the Fourth International Congress, Xanthi, Greece, 1990; Vougiouklis, T., Ed.; World Scientific: Singapore, 1991; pp. 119–122. [Google Scholar]

**Figure 1.**Various notions of cyclicity in hypergroups and their relations; from the most special case (single-power cyclic hypergroups) to the most general one (cyclic hypergroups).

**Figure 2.**Related to Example 8 and Remark 2 when $H/\beta \cong \mathbb{Z}/5\mathbb{Z}$. Elements are in relation based on the imaginary part and within the same class based on the real part.

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Novák, M.; Křehlík, Š.; Cristea, I.
Cyclicity in *EL*–Hypergroups. *Symmetry* **2018**, *10*, 611.
https://doi.org/10.3390/sym10110611

**AMA Style**

Novák M, Křehlík Š, Cristea I.
Cyclicity in *EL*–Hypergroups. *Symmetry*. 2018; 10(11):611.
https://doi.org/10.3390/sym10110611

**Chicago/Turabian Style**

Novák, Michal, Štepán Křehlík, and Irina Cristea.
2018. "Cyclicity in *EL*–Hypergroups" *Symmetry* 10, no. 11: 611.
https://doi.org/10.3390/sym10110611