# On Some NeutroHyperstructures

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{v}-Semigroups and study their properties by providing several illustrative examples.

_{v}-semigroup; NeutroHyperideal; NeutroStrongIsomorphism

## 1. Introduction

## 2. Algebraic Hyperstructures

**Definition**

**1**

**.**Let H be a non-empty set and ${\mathcal{P}}^{\ast}\left(H\right)$ be the family of all non-empty subsets of H. Then, a mapping $\circ :H\times H\to {\mathcal{P}}^{\ast}\left(H\right)$ is called a binary hyperoperation on H. The couple $(H,\circ )$ is called a hypergroupoid.

**Example**

**1.**

**Example**

**2.**

+ | e | b | c |

e | e | {e,b} | {e,c} |

b | e | {e,b} | {e,c} |

c | e | {e,b} | {e,c} |

**Definition**

**2**

**Example**

**3.**

+ | 0 | 1 | 2 | 3 |

0 | 0 | 1 | {0,2} | 3 |

1 | 1 | 2 | 3 | 0 |

2 | 2 | 3 | 0 | 1 |

3 | 3 | 0 | 1 | 2 |

**Remark**

**1.**

**Definition**

**3**

**.**Let $(H,\circ )$ be a semihypergroup (${H}_{v}$-semigroup) and $M\ne \mathsf{\varnothing}\subseteq H$. Then M is a

- 1.
- subsemihypergroup (${H}_{v}$-subsemigroup) of H if $(M,\circ )$ is a semihypergroup (${H}_{v}$-semigroup).
- 2.
- left hyperideal of H if M is a subsemihypergroup (${H}_{v}$-subsemigroup) of H and $h\circ a\subseteq M$ for all $h\in H$.
- 3.
- right hyperideal of H if M is a subsemihypergroup (${H}_{v}$-subsemigroup) of H and $a\circ h\subseteq M$ for all $h\in H$.
- 4.
- hyperideal of H if M is both: a left hyperideal of H and a right hyperideal of H.

**Remark**

**2.**

## 3. NeutroHyperstructures

**Definition**

**4.**

- 1.
- There exist $x,y\in A$ with $x\xb7y\subseteq A$. (This condition is called degree of truth, “T”).
- 2.
- There exist $x,y\in A$ with $x\xb7y\u2288A$. (This condition is called degree of falsity, “F”).
- 3.
- There exist $x,y\in A$ with $x\xb7y$ is indeterminate in A. (This condition is called degree of indeterminacy, “I”).

**Definition**

**5.**

**Definition**

**6.**

- 1.
- $x\xb7(y\xb7z)=(x\xb7y)\xb7z$; (This condition is called degree of truth, “T”).
- 2.
- $a\xb7(b\xb7c)\ne (a\xb7b)\xb7c$; (This condition is called degree of falsity, “F”).
- 3.
- $e\xb7(f\xb7g)$ is indeterminate or $(e\xb7f)\xb7g$ is indeterminate or we cannot find if $e\xb7(f\xb7g)$ and $(e\xb7f)\xb7g$ are equal. (This condition is called degree of indeterminacy, “I”).

**Definition**

**7.**

**Definition**

**8.**

- 1.
- $[x\xb7(y\xb7z\left)\right]\cap \left[\right(x\xb7y)\xb7z]\ne \mathsf{\varnothing}$; (This condition is called degree of truth, “T”).
- 2.
- $[a\xb7(b\xb7c\left)\right]\cap \left[\right(a\xb7b)\xb7c]=\mathsf{\varnothing}$; (This condition is called degree of falsity, “F”).
- 3.
- $e\xb7(f\xb7g)$ is indeterminate or $(e\xb7f)\xb7g$ is indeterminate or we cannot find if $e\xb7(f\xb7g)$ and $(e\xb7f)\xb7g$ have common elements. (This condition is called degree of indeterminacy, “I”).

**Definition**

**9.**

- 1.
- NeutroHypergroupoid if “·” is a NeutroHyperoperation.
- 2.
- NeutroSemihypergroup if “·” is NeutroAssociative but not an AntiHyperoperation.
- 3.
- NeutroH${}_{v}$-Semigroup if “·” is NeutroWeakAssociative but not an AntiHyperoperation.

**Example**

**4.**

+ | 0 | 1 |

0 | {0,1} | 0 |

1 | 1 | 0 |

**Example**

**5.**

**Example**

**6.**

· | m | a | d |

m | m | m | m |

a | m | {m,a} | d |

d | m | d | d |

**Remark**

**3.**

**Proposition**

**1.**

**Proof.**

**Example**

**7.**

⋄ | m | a | d |

m | m | {a,d} | d |

a | {a,d} | d | m |

d | d | m | a |

**Remark**

**4.**

**Example**

**8.**

**Example**

**9.**

**Example**

**10.**

**Example**

**11.**

• | m | a | d |

m | a | a | d |

a | {m,a} | m | d |

d | d | d | m |

**Remark**

**5.**

**Example**

**12.**

·_{1} | s | a | m |

s | s | m | s |

a | m | a | m |

m | m | m | m |

**Theorem**

**1.**

**Proof.**

**Example**

**13.**

⊛ | m | a | d |

m | a | {m,a} | d |

a | a | m | d |

d | d | d | m |

**Definition**

**10.**

**Remark**

**6.**

**Definition**

**11.**

- (1)
- S is a NeutroLeftHyperideal of H if there exists $x\in S$ such that $r\circ x\subseteq S$ for all $r\in H$.
- (2)
- S is a NeutroRightHyperideal of S if there exists $x\in S$ such that $x\circ r\subseteq S$ for all $r\in H$.
- (3)
- S is a NeutroHyperideal of H if there exists $x\in S$ such that $r\circ x\subseteq S$ and $x\circ r\subseteq S$ for all $r\in H$.

**Example**

**14.**

**Example**

**15.**

**Example**

**16.**

**Remark**

**7.**

**Lemma**

**1.**

**Proof.**

- T: $x\circ (y\circ z)=(x\circ y)\circ z$;
- F: $a\circ (b\circ c)\ne (a\circ b)\circ c$;
- I: $e\circ (f\circ g)$ is indeterminate or $(e\circ f)\circ g$ is indeterminate or we cannot find if $e\circ (f\circ g)$ and $(e\circ f)\circ g$ are equal.

**Example**

**17.**

**Remark**

**8.**

**Lemma**

**2.**

**Proof.**

**Definition**

**12.**

- (1)
- ϕ is called NeutroHomomorphism if $\varphi (x\circ y)=\varphi \left(x\right)\u2605\varphi \left(y\right)$ for some $x,y\in A$.
- (2)
- ϕ is called NeutroIsomomorphism if ϕ is a bijective NeutroHomomorphism.
- (3)
- ϕ is called NeutroStrongHomomorphism if for all $x,y\in A$, $\varphi (x\circ y)=\varphi \left(x\right)\u2605\varphi \left(y\right)$ when $x\circ y\subseteq H$, $\varphi \left(x\right)\u2605\varphi \left(y\right)\u2288{H}^{\prime}$ when $x\circ y\u2288H$, and $\varphi \left(x\right)\u2605\varphi \left(y\right)$ is indeterminate when $x\circ y$ is indeterminate.
- (4)
- ϕ is called NeutroStrongIsomomorphism if ϕ is a bijective NeutroOrderedStrongHomomorphism. In this case we say that $(H,\circ ){\cong}_{SI}({H}^{\prime},\u2605)$.

**Example**

**18.**

**Theorem**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

- T: $x\circ (y\circ z)=(x\circ y)\circ z$;
- F: $a\circ (b\circ c)\ne (a\circ b)\circ c$;
- I: $e\circ (f\circ g)$ is indeterminate or $(e\circ f)\circ g$ is indeterminate or we cannot find if $e\circ (f\circ g)$ and $(e\circ f)\circ g$ are equal.

- T: $\varphi \left(x\right)\u2605\left(\varphi \right(y)\u2605\varphi (z\left)\right)=\left(\varphi \right(x)\u2605\varphi (y\left)\right)\u2605\varphi \left(z\right)$;
- F: $\varphi \left(a\right)\u2605\left(\varphi \right(b)\u2605\varphi (c\left)\right)\ne \left(\varphi \right(a)\u2605\varphi (b\left)\right)\u2605\varphi \left(c\right)$;
- I: $\varphi \left(e\right)\u2605\left(\varphi \right(f)\u2605\varphi (g\left)\right)$ is indeterminate or $\left(\varphi \right(e)\u2605\varphi (f\left)\right)\u2605\varphi \left(g\right)$ is indeterminate or we cannot find if $\varphi \left(e\right)\u2605\left(\varphi \right(f)\u2605\varphi (g\left)\right)$ and $\left(\varphi \right(e)\u2605\varphi (f\left)\right)\u2605\varphi \left(g\right)$ are equal.

**Example**

**19.**

**Lemma**

**4.**

**Proof.**

- T: $\varphi \left(x\right)\u2605\left(\varphi \right(y)\u2605\varphi (z\left)\right)=\left(\varphi \right(x)\u2605\varphi (y\left)\right)\u2605\varphi \left(z\right)$;
- F: $\varphi \left(a\right)\u2605\left(\varphi \right(b)\u2605\varphi (c\left)\right)\ne \left(\varphi \right(a)\u2605\varphi (b\left)\right)\u2605\varphi \left(c\right)$;
- I: $\varphi \left(e\right)\u2605\left(\varphi \right(f)\u2605\varphi (g\left)\right)$ is indeterminate or $\left(\varphi \right(e)\u2605\varphi (f\left)\right)\u2605\varphi \left(g\right)$ is indeterminate or we cannot find if $\varphi \left(e\right)\u2605\left(\varphi \right(f)\u2605\varphi (g\left)\right)$ and $\left(\varphi \right(e)\u2605\varphi (f\left)\right)\u2605\varphi \left(g\right)$ are equal.

- T: $x\circ (y\circ z)=(x\circ y)\circ z$;
- F: $a\circ (b\circ c)\ne (a\circ b)\circ c$;
- I: $e\circ (f\circ g)$ is indeterminate or $(e\circ f)\circ g$ is indeterminate or we cannot find if $e\circ (f\circ g)$ and $(e\circ f)\circ g$ are equal.

**Theorem**

**3.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Example**

**20.**

- 1.
- $(\mathbb{R}\times M,\circ )$ is a NeutroSemihypergroup,
- 2.
- $(M\times \mathbb{R},\circ )$ is a NeutroSemihypergroup, and
- 3.
- $(M\times M,\circ )$ is a NeutroSemihypergroup.

**Theorem**

**6.**

**Proof.**

**Example**

**21.**

⊚ | m | a | d | n |

m | m | m | m | {m,a,d,n} |

a | m | {m,a} | d | {m,a,d,n} |

d | m | d | d | {m,a,d,n} |

n | {m,a,d,n} | {m,a,d,n} | {m,a,d,n} | {m,a,d,n} |

**Theorem**

**7.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Corollary**

**3.**

**Proof.**

## 4. Conclusions

- Find all NeutroSemihypergroups (NeutroH${}_{v}$-Semigroups) of small order (up to NeutroStrongIsomorphism).
- Find bounds for the number of finite NeutroSemihypergroups (NeutroH${}_{v}$-Semigroups) of arbitrary order n (up to NeutroStrongIsomorphism).
- Classify simple NeutroSemihypergroups (NeutroH${}_{v}$-Semigroups) up to NeutroStrongIsomorphism.
- Define other NeutroHyperstructures such as NeutroPolygroup, NeutroHyperring, etc.
- Find applications of NeutroHyperstructures in some fields like Biology, Physics, Chemistry, etc.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Smarandache, F. NeutroAlgebra is a Generalization of Partial Algebra. Int. J. Neutrosophic Sci. IJNS
**2020**, 2, 8–17. Available online: http://fs.unm.edu/NeutroAlgebra.pdf (accessed on 1 January 2020). - Smarandache, F. Introduction to NeutroAlgebraic Structures and AntiAlgebraic Structures. In Advances of Standard and Nonstandard Neutrosophic Theories; Pons Publishing House: Brussels, Belgium, 2019; Chapter 6; pp. 240–265. Available online: http://fs.unm.edu/AdvancesOfStandardAndNonstandard.pdf (accessed on 1 January 2020).
- Smarandache, F. Introduction to NeutroAlgebraic Structures and AntiAlgebraic Structures (revisited). Neutrosophic Sets Syst.
**2020**, 31, 1–16. [Google Scholar] [CrossRef] - Agboola, A.A.A. Introduction to NeutroGroups. Int. J. Neutrosophic Sci. IJNS
**2020**, 6, 41–47. [Google Scholar] - Agboola, A.A.A. Introduction to NeutroRings. Int. J. Neutrosophic Sci. IJNS
**2020**, 7, 62–73. [Google Scholar] - Al-Tahan, M.; Smarandache, F.; Davvaz, B. NeutroOrderedAlgebra: Applications to Semigroups. Neutrosophic Sets Syst.
**2021**, 39, 133–147. [Google Scholar] - Al-Tahan, M.; Davvaz, B.; Smarandache, F.; Osman, O. On Some Properties of Productional NeutroOrderedSemigroups. 2021; submitted. [Google Scholar]
- Hamidi, M.; Smarandache, F. Neutro-BCK-Algebra. Int. J. Neutrosophic Sci. IJNS
**2020**, 8, 110–117. [Google Scholar] - Rezaei, A.; Smarandache, F. On Neutro-BE-algebras and Anti-BE-algebras. Int. J. Neutrosophic Sci. IJNS
**2020**, 4, 8–15. [Google Scholar] - Smarandache, F.; Rezaei, A.; Kim, H.S. A New Trend to Extensions of CI-algebras. Int. J. Neutrosophic Sci. IJNS
**2020**, 5, 8–15. [Google Scholar] [CrossRef] - Marty, F. Sur une generalization de la notion de groupe. In Proceedings of the 8th Congress on Mathmatics, Scandinaves, Stockholm, Sweden, 14–18 August 1934; pp. 45–49. [Google Scholar]
- Al-Tahan, M.; Davvaz, B. Chemical Hyperstructures for Astatine, Tellurium and for Bismuth. Bull. Comput. Appl. Math.
**2019**, 7, 9–25. [Google Scholar] - Al-Tahan, M.; Davvaz, B. On the Existence of Hyperrings Associated to Arithmetic Functions. J. Number Theory
**2017**, 174, 136–149. [Google Scholar] [CrossRef] - Al-Tahan, M.; Davvaz, B. Algebraic Hyperstructures Associated to Biological Inheritance. Math. Biosci.
**2017**, 285, 112–118. [Google Scholar] [CrossRef] - Davvaz, B.; Subiono; Al-Tahan, M. Calculus of Meet Plus Hyperalgebra (Tropical Semihyperrings). Commun. Algebra
**2020**, 48. [Google Scholar] [CrossRef] - Corsini, P. Prolegomena of Hypergroup Theory; Udine Aviani Editore: Tricesimo (Udine), Italy, 1993. [Google Scholar]
- Davvaz, B. Polygroup Theory and Related Systems; World Scientific Publishing Co., Pte. Ltd.: Hackensack, NJ, USA, 2013; viii+200p. [Google Scholar]
- Davvaz, B.; Leoreanu-Fotea, V. Hyperring Theory and Applications; International Academic Press: Cambridge, MA, USA, 2008. [Google Scholar]
- Vougiouklis, T. The Fundamental Relation in Hyperrings. The General Hyperfield. In Proceedings of the Fourth International Congress on Algebraic Hyperstructures and Applications (AHA 1990); World Scientific: Singapore, 1991; pp. 203–211. [Google Scholar]
- Vougiouklis, T. Hyperstructures and Their Representations; Hadronic Press, Inc.: Palm Harber, FL, USA, 1994. [Google Scholar]
- Vougiouklis, T. H
_{v}-groups Defined on the Same Set. Discret. Math.**1996**, 155, 259–265. [Google Scholar] [CrossRef] [Green Version] - Vougiouklis, T.; Spartalis, S.; Kessoglides, M. Weak Hyperstructures on Small Sets. Ratio Math.
**1997**, 12, 90–96. [Google Scholar] - Ibrahim, M.A.; Agboola, A.A.A. Introduction to NeutroHyperGroups. Neutrosophic Sets Syst.
**2020**, 38, 15–32. [Google Scholar] - Smarandache, F. Extension of HyperGraph to n-SuperHyperGraph and to Plithogenic n-SuperHyperGraph, and Extension of HyperAlgebra to n-ary (Classical-/Neutro-/Anti-)HyperAlgebra. Neutrosophic Sets Syst.
**2020**, 33, 290–296. [Google Scholar]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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**

Al-Tahan, M.; Davvaz, B.; Smarandache, F.; Anis, O.
On Some NeutroHyperstructures. *Symmetry* **2021**, *13*, 535.
https://doi.org/10.3390/sym13040535

**AMA Style**

Al-Tahan M, Davvaz B, Smarandache F, Anis O.
On Some NeutroHyperstructures. *Symmetry*. 2021; 13(4):535.
https://doi.org/10.3390/sym13040535

**Chicago/Turabian Style**

Al-Tahan, Madeleine, Bijan Davvaz, Florentin Smarandache, and Osman Anis.
2021. "On Some NeutroHyperstructures" *Symmetry* 13, no. 4: 535.
https://doi.org/10.3390/sym13040535