# Neutrosophic Triplet Non-Associative Semihypergroups with Application

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**Let $\mathcal{H}$ be a non void set and $\circ :\mathcal{H}\ast \mathcal{H}$$\u27f6{P}^{\u2022}\left(\mathcal{H}\right)$ be a hyperoperation, where ${P}^{\u2022}\left(\mathcal{H}\right)$ is the family non-void subset of $\mathcal{H}$. The pair $\left(\mathcal{H},\ast \right)$ is called hypergroupoid.

**Definition**

**2**

**Definition**

**3**

**.**An element e of an LA-semihypergroup $\mathcal{H}$ is called left identity (resp., pure left identity) if for all ${w}_{1}\in \mathcal{H},$ ${w}_{1}\in e\ast {w}_{1}$ (resp., ${w}_{1}=e\ast {w}_{1}$). An element e of an LA-semihypergroup $\mathcal{H}$ is called right identity (resp., pure right identity) if for all ${w}_{1}\in \mathcal{H},$ ${w}_{1}\in {w}_{1}\ast e$ (resp., ${w}_{1}=e\ast {w}_{1}$). An element e of an LA-semihypergroup $\mathcal{H}$ is called identity (resp., pure right identity) if for all ${w}_{1}\in \mathcal{H},$ ${w}_{1}\in {w}_{1}\ast e\cap e\ast {w}_{1}$ (resp., ${w}_{1}={w}_{1}\ast e\cap e\ast {w}_{1}$).

**Definition**

**4**

**.**An $\mathcal{LA}$-smihypergroup with pure left identity satisfies the following property

**Definition**

**5**

**.**Let N be a non-void set with a binary operation ∗ and ${w}_{1}\in N$. Then ${w}_{1}$ is said to be neutrosophic triplet if there exist an element $neut\left({w}_{1}\right)\in N$ such that

**Example**

**1**

**.**Consider ${Z}_{6}$ under multiplication modulo $6.$ Then 2 is a neutrosophic triplet, because $neut\left(2\right)=4$, as $2\times 4=8$. Similarly $anti\left(2\right)=2$ because $2\times 2=4$. Thus 2 is a neutrosophic triplet, which is denoted by $\left(2,4,2\right)$. Similarly 4 is a neutrosophic triplet because $neut\left(a\right)=anti\left(4\right)=4$. So 4 is represented by as $(4,4,4)$. 3 is not a neutrosophic triplet as $neut\left(3\right)=5$ but $anti\left(3\right)$ does not exist in ${Z}_{6}$ and 0 is a trivial neutrosophic triplet as $neut\left(0\right)=anti\left(0\right)=0$. This is denoted by $(0,0,0).$

## 3. Neutrosophic Triplet $\mathcal{LA}$-Semihypergroups

**Definition**

**6.**

- left neutrosophic triplet set if for every ${w}_{1}\in \mathcal{H}$, there exist $neut\left({w}_{1}\right)$ and $anti\left({w}_{1}\right)$ such that$$\begin{array}{cc}\hfill {w}_{1}& \in neut\left({w}_{1}\right)\ast {w}_{1},\hfill \\ \hfill neut\left({w}_{1}\right)& \in anti\left({w}_{1}\right)\ast {w}_{1}.\hfill \end{array}$$
- right neutrosophic triplet set if for every ${w}_{1}\in \mathcal{H}$, there exist $neut\left({w}_{1}\right)$ and $anti\left({w}_{1}\right)$ such that$$\begin{array}{cc}\hfill {w}_{1}& \in {w}_{1}\ast neut\left({w}_{1}\right),\hfill \\ \hfill neut\left({w}_{1}\right)& \in {w}_{1}\ast anti\left({w}_{1}\right).\hfill \end{array}$$
- neutrosophic triplet set if for every ${w}_{1}\in \mathcal{H}$, there exist $neut\left({w}_{1}\right)$ and $anti\left({w}_{1}\right)$ such that$$\begin{array}{cc}\hfill {w}_{1}& \in (neut\left({w}_{1}\right)\ast {w}_{1})\cap ({w}_{1}\ast neut\left({w}_{1}\right)),\hfill \\ \hfill neut\left({w}_{1}\right)& \in (anti\left({w}_{1}\right)\ast {w}_{1})\cap ({w}_{1}\ast anti\left({w}_{1}\right)).\hfill \end{array}$$

**Definition**

**7.**

- pure left neutrosophic triplet set if for every ${w}_{1}\in \mathcal{H}$, there exist $neut\left({w}_{1}\right)$ and $anti\left({w}_{1}\right)$ such that$$\begin{array}{cc}\hfill {w}_{1}& =neut\left({w}_{1}\right)\ast {w}_{1},\hfill \\ \hfill neut\left({w}_{1}\right)& =anti\left({w}_{1}\right)\ast {w}_{1}.\hfill \end{array}$$
- pure right neutrosophic triplet set if for every ${w}_{1}\in \mathcal{H}$, there exist $neut\left({w}_{1}\right)$ and $anti\left({w}_{1}\right)$ such that$$\begin{array}{cc}\hfill {w}_{1}& ={w}_{1}\ast neut\left({w}_{1}\right),\hfill \\ \hfill neut\left({w}_{1}\right)& ={w}_{1}\ast anti\left({w}_{1}\right).\hfill \end{array}$$
- pure neutrosophic triplet set if for every ${w}_{1}\in \mathcal{H}$, there exist $neut\left({w}_{1}\right)$ and $anti\left({w}_{1}\right)$ such that$$\begin{array}{cc}\hfill {w}_{1}& =(neut\left({w}_{1}\right)\ast {w}_{1})\cap ({w}_{1}\ast neut\left({w}_{1}\right)),\hfill \\ \hfill neut\left({w}_{1}\right)& =\left(anti\left({w}_{1}\right)\ast {w}_{1}\right)\cap \left({w}_{1}\ast anti\left({w}_{1}\right)\right).\hfill \end{array}$$

**Example**

**2.**

**Definition**

**8.**

- $\left(\mathcal{H},\ast \right)$ is well defined.
- $\left(\mathcal{H},\ast \right)$ satisfies the left invertive law.

**Example**

**3.**

**Definition**

**9.**

- $\left(\mathcal{H},\ast \right)$ is a well defined.
- $\left(\mathcal{H},\ast \right)$ satisfies the left invertive law.

**Example**

**4.**

**Remark**

**1.**

**Remark**

**2.**

**Definition**

**10.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

- $neut\left({w}_{1}\right)\ast neut\left({w}_{2}\right)=neut({w}_{1}\ast {w}_{2})$ for all ${w}_{1},{w}_{2}\in \mathcal{H}.$
- $ant\left({w}_{1}\right)\ast anti\left({w}_{2}\right)=anti\left({w}_{1}\ast {w}_{2}\right)$ for all ${w}_{1},{w}_{2}\in \mathcal{H}.$

**Proof.**

**Example**

**5.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Definition**

**11.**

**Example**

**6.**

**Lemma**

**1.**

- K is a neutrosophic triplet $\mathcal{LA}$-semihypergroup.
- For all ${w}_{1},{w}_{2}\in K,$${w}_{1}\ast {w}_{2}\in K.$

**Proof.**

**Definition**

**12.**

- $f\left({w}_{1}{\ast}_{1}{w}_{2}\right)=f\left({w}_{2}\right){\ast}_{2}f\left({w}_{2}\right),$
- $f\left(net\left({w}_{1}\right)\right)=neut\left(f\left({w}_{1}\right)\right),$
- $f\left(anti\left({w}_{1}\right)\right)=anti\left(f\left({w}_{1}\right)\right).$

**Theorem**

**6.**

- The image of f is a neutrosophic triplet $\mathcal{LA}$-subsemihypergroup of ${\mathcal{H}}_{2}.$
- The inverse image of f is a neutrosophic $\mathcal{LA}$-subsemihypergroup of ${\mathcal{H}}_{1}$.

**Proof.**

**Remark**

**3.**

- Every neutrosophic triplet $\mathcal{LA}$-semihypergroup is an $\mathcal{LA}$-semihypergroup, but the reverse may or may not true.
- In neutrosophic triplet $\mathcal{LA}$-semihypergroup, every element must have a left $neut\left(.\right),$ but in an $\mathcal{LA}$-semihypergroup the left $neut\left(.\right)$ of an element may or may not exist.
- In neutrosophic $\mathcal{LA}$-semihypergroup, every element must have left $anti\left(.\right),$ but in an $\mathcal{LA}$-semihypergroup the element may or may not have semihypergroup.
- In neutrosophic $\mathcal{LA}$-semihypergroup pure left $neut\left(.\right)$ is not equal to pure left Identity.

## 4. Application

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Smarandache, F. A Unifying Field in Logics. Neutrosophy, Neutrosophic Probability, Set and Logic; Rehoboth American Research Press: Rehoboth, NM, USA, 1999. [Google Scholar]
- Kandasamy, W.B.V.; Smarandache, F. Some Neutrosophic Algebraic Structures and Neutrosophic N-Algebraic Structures; Hexis: Phoenix, AZ, USA, 2006; p. 219. [Google Scholar]
- Kandasamy, W.B.V.; Smarandache, F. N-Algebraic Structures And S-N-Algebraic Structures; Hexis: Phoenix, AZ, USA, 2006; p. 209. [Google Scholar]
- Kandasamy, W.B.V.; Smarandache, F. Basic Neutrosophic Algebraic Structures and Their Applications to Fuzzy and Neutrosophic Models; Hexis: Phoenix, AZ, USA, 2004; p. 149. [Google Scholar]
- Al-Quran, A.; Hassan, N. The complex neutrosophic soft expert set and its application in decision making. J. Intell. Fuzzy Syst.
**2018**, 34, 569–582. [Google Scholar] [CrossRef] - Al-Quran, A.; Hassan, N. The complex neutrosophic soft expert relation and its multiple attribute decision-making method. Entropy
**2018**, 20, 101. [Google Scholar] [CrossRef] - Abu Qamar, M.; Hassan, N. Q-neutrosophic soft relation and its application in decision making. Entropy
**2018**, 20, 172. [Google Scholar] - Abu Qamar, M.; Hassan, N. Entropy, measures of distance and similarity of Q-neutrosophic soft sets and some applications. Entropy
**2018**, 20, 672. [Google Scholar] - Uluçay, V.; Sahin, M.; Hassan, N. Generalized neutrosophic soft expert set for multiple-criteria decision-making. Symmetry
**2018**, 10, 437. [Google Scholar] [CrossRef] - Smarandache, F. Neutrosophic Perspectives: Triplets, Duplets, Multisets, Hybrid Operators, Modal Logic, Hedge Algebras. And Applications; Pons Publishing House: Brussels, Belgium, 2017. [Google Scholar]
- Zhang, X.H.; Smarandache, F.; Liang, X.L. Neutrosophic duplet semi-group and cancellable neutrosophic triplet groups. Symmetry
**2017**, 9, 275. [Google Scholar] [CrossRef] - Bal, M.; Shalla, M.M.; Olgun, N. Neutrosophic triplet cosets and quotient groups. Symmetry
**2018**, 10, 126. [Google Scholar] [CrossRef] - Jaiyeola, T.G.; Smarandache, F. Some results on neutrosophic triplet group and their applications. Symmetry
**2018**, 10, 202. [Google Scholar] [CrossRef] - Smarandache, F.; Ali, M. Neutrosophic triplet group. Neural Comput. Appl.
**2018**, 29, 595–601. [Google Scholar] [CrossRef] - Marty, F. Sur une generalization de la notion de groupe. In Proceedings of the 8th Congres des Mathematicians Scandinaves, Stockholm, Sweden; 1934; pp. 45–49. [Google Scholar]
- Corsini, P. Prolegomena of Hypergroup Theory; Aviani Editore: Udine, Italy, 1993. [Google Scholar]
- Vougiouklis, T. Hyperstructures and Their Representations; Hadronic Press: Palm Harbor, FL, USA, 1994. [Google Scholar]
- Corsini, P.; Leoreanu, V. Applications of Hyperstructure Theory; Kluwer Academic: Dordrecht, The Netherlands, 2003. [Google Scholar]
- Vougiouklis, T. A new class of hyperstructures. J. Comb. Inf. Syst. Sci.
**1995**, 20, 229–235. [Google Scholar] - Vougiouklis, T. ∂-operations and Hv-fields. Acta Math. Sin. (Engl. Ser.)
**2008**, 24, 1067–1078. [Google Scholar] [CrossRef] - Vougiouklis, T. The h/v-Structures, Algebraic Hyperstructures and Applications; Taru Publications: New Delhi, India, 2004; pp. 115–123. [Google Scholar]
- Spartalis, S. On Hv-semigroups. Ital. J. Pure Appl. Math.
**2002**, 11, 165–174. [Google Scholar] - Spartalis, S. On reversible Hv-group. In Proceedings of the 5th International Congress on Algebraic Hyperstructures and Applications, Jasi, Romania, 4–10 July 1993; pp. 163–170. [Google Scholar]
- Vougiouklis, T. The fundamental relation in hyperrings: The general hyperfield. In Proceedings of the Algebraic Hyperstructures and Applications, Xanthi, Greece, April 1991; pp. 203–211. [Google Scholar]
- Spartalis, S. Quoitients of P-Hv-rings. In New Frontiers in Hyperstructures; Hadronic Press: Palm Harbor, FL, USA, 1996; pp. 167–176. [Google Scholar]
- Spartalis, S.; Vougiouklis, T. The fundamental relations on Hv-rings. Riv. Mat. Pura Appl.
**1994**, 7, 7–20. [Google Scholar] - Spartalis, S. On the number of Hv-rings with P-hyperoperations. Discret. Math.
**1996**, 155, 225–231. [Google Scholar] [CrossRef] - Davvaz, B.; Fotea, V.L. Hyperring Theory and Applications; International Academic Press: New York, NY, USA, 2007. [Google Scholar]
- Kazim, M.A.; Naseerudin, N. On almost semigroups. Aligarh Bull. Math.
**1972**, 2, 41–47. [Google Scholar] - Kamran, M.S. Conditions for LA-Semigroups to Resemble Associative Structures. Ph.D. Thesis, Quaid-i-Azam University, Islamabad, Pakistan, 1993. [Google Scholar]
- Protic, P.V.; Stevanovic, N. AG-test and some general properties of Abel-Grassmann’s groupoids. Pure Math. Appl.
**1995**, 6, 371–383. [Google Scholar] - Protic, P.V.; Stevanovic, N. The structural theorem for AG
^{*}-groupoids. Ser. Math. Inform.**1995**, 10, 25–33. [Google Scholar] - Yusuf, S.M. On Left Almost Ring. In Proceedings of the 7th International Pure Mathematics Conference, Islamabad, Pakistan, 5–7 August 2006. [Google Scholar]
- Hila, K.; Dine, J. On hyperideals in left almost semihypergroups. ISRN Algebra
**2011**, 2011, 953124. [Google Scholar] [CrossRef] - Yaqoob, N.; Corsini, P.; Yousafzai, F. On intra-regular left almost semihypergroups with pure left identity. J. Math.
**2013**, 2013, 510790. [Google Scholar] [CrossRef] - Yousafzai, F.; Hila, K.; Corsini, P.; Zeb, A. Existence of non-associative algebraic hyperstructures and related problems. Afr. Mat.
**2015**, 26, 981–995. [Google Scholar] [CrossRef] - Amjad, V.; Hila, K.; Yousafzai, F. Generalized hyperideals in locally associative left almost semihypergroups. N. Y. J. Math.
**2014**, 20, 1063–1076. [Google Scholar] - Gulistan, M.; Yaqoob, N.; Shahzad, M. A Note On Hv-LA-semigroup. UPB Sci. Bull. Ser. A
**2015**, 77, 93–106. [Google Scholar] - Yaqoob, N.; Gulistan, M. Partially ordered left almost semihypergroups. J. Egypt. Math. Soc.
**2015**, 23, 231–235. [Google Scholar] [CrossRef] - Rehman, I.; Yaqoob, N.; Nawaz, S. Hyperideals and hypersystems in LA-hyperrings. Songklanakarin J. Sci. Technol.
**2017**, 39, 651–657. [Google Scholar] - Nawaz, S.; Rehman, I.; Gulistan, M. On left almost semihyperrings. Int. J. Anal. Appl.
**2018**, 16, 528–541. [Google Scholar] - Yaqoob, N.; Cristea, I.; Gulistan, M.; Nawaz, S. Left almost polygroups. Ital. J. Pure Appl. Math.
**2018**, 39, 465–474. [Google Scholar]

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Gulistan, M.; Nawaz, S.; Hassan, N.
Neutrosophic Triplet Non-Associative Semihypergroups with Application. *Symmetry* **2018**, *10*, 613.
https://doi.org/10.3390/sym10110613

**AMA Style**

Gulistan M, Nawaz S, Hassan N.
Neutrosophic Triplet Non-Associative Semihypergroups with Application. *Symmetry*. 2018; 10(11):613.
https://doi.org/10.3390/sym10110613

**Chicago/Turabian Style**

Gulistan, Muhammad, Shah Nawaz, and Nasruddin Hassan.
2018. "Neutrosophic Triplet Non-Associative Semihypergroups with Application" *Symmetry* 10, no. 11: 613.
https://doi.org/10.3390/sym10110613