# Neutrosophic Components Semigroups and Multiset Neutrosophic Components Semigroups

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Concepts

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

- 1.
- $\{S,+\}$ is a commutative semigroup with 0 as its additive identity.
- 2.
- $\{S,\times \}$ is a semigroup.
- 3.
- $a\times (b+c)$ = $a\times b+a\times c$ for all $a,b,c,\in S$ follows distribution law.

**Definition**

**4.**

**Definition**

**5.**

**Example**

**1.**

## 3. Neutrosophic Components (NC) Semigroups under Usual Product and Sum

**Example**

**2.**

**Definition**

**6.**

**Theorem**

**1.**

- 1.
- $\{{S}_{1},\times \}$ is an infinite order commutative semigroup which is not a monoid and has no zero divisors.
- 2.
- Every $a=(x,y,z)$ in ${S}_{1}$ will generate an infinite cyclic subsemigroup under product of ${S}_{1}$ denoted by $(P,\times )$.
- 3.
- The elements of P forms a totally ordered set, (for if $a=(x,y,z)\in P$ we see ${a}^{2}=a\times a<a$).
- 4.
- $\{{S}_{1},\times \}$ has no idempotents and $\{{S}_{1},\times \}$ is a torsion free semigroup.

**Proof.**

**Definition**

**7.**

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

**Definition**

**8.**

**Theorem**

**5.**

**Proof.**

**Definition**

**9.**

**Theorem**

**6.**

**Proof.**

## 4. Multiset NC Semigroups

**Definition**

**10.**

**Definition**

**11.**

**Example**

**3.**

**Theorem**

**7.**

- 1.
- $\{M\left({S}_{1}\right),\times \}$ has no trivial or non-trivial special type of zero divisors and no trivial or non-trivial idempotents.
- 2.
- $\{M\left({S}_{2}\right),\times \}$ has infinite number of special type of zero divisors and no non-trivial idempotents.
- 3.
- $\{M\left({S}_{3}\right),\times \}$ has no trivial or non-trivial special zero divisors but has (1, 1, 1) as identity and has no non trivial idempotents.
- 4.
- $\{M\left({S}_{4}\right),\times \}$ has non-trivial special type of zero divisors and has (1, 1, 1) as its identity and has idempotents of the form $\left\{\right(0,1,0),(1,1,0),(0,0,1),(1,0,1)$ and so on }.

**Proof.**

- Follows from the fact that ${S}_{1}$ has no zero divisors and idempotents as it is built on the interval (0, 1).
- Evident from the fact ${S}_{2}$ is built on [0, 1) so has special type of zero divisors by definition but no idempotent.
- True from the fact ${S}_{3}$ is built on (0, 1], so (1, 1, 1) $\in M\left({S}_{3}\right)$.
- ${S}_{4}$ which is built on [0, 1] has infinite special type of zero divisors as (0, 0, 0) $\in {S}_{4}$ by Definition 11 and (1, 1, 1) $\in M\left({S}_{4}\right)$ and has idempotents of the form $\left\{\right(0,1,0),(1,1,0),(0,0,1),(1,0,1)$ and so on }.

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

## 5. n-Multiplicity Neutrosophic Set Semigroups Using ${S}_{1},{S}_{2},{S}_{3}$ and ${S}_{4}$

**Example**

**4.**

**Definition**

**12.**

**Theorem**

**11.**

- 1.
- n-$M\left({S}_{i}\right)$ is not closed under the binary operation ‘+’ under usual addition, for $i=$ 1, 3 and 4.
- 2.
- n-$M\left({S}_{i}\right)$ is a (n-multiplicity neutrosophic multiset) semigroup under the usual product for i = 1, 2, 3 and 4.
- 3.
- {n-$M\left({S}_{i}\right),\times \}$ is a monoid for i = 3 and 4.
- 4.
- {n-$M\left({S}_{i}\right),\times \}$ has no special zero divisors if ${S}_{i}={S}_{1}$ and ${S}_{3}$ but they have no non trivial idempotents. ${S}_{2}$ and special zero divisors and no non trivial idempotents, but ${S}_{4}$ has both non trivial special zero divisors and non trivial idempotents.

**Proof.**

**Example**

**5.**

**Theorem**

**12.**

**Proof.**

**Theorem**

**13.**

**Proof.**

## 6. Discussions

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SVNS | Single Valued Neutrosophic Set |

## References

- Herstein, I.N. Topics in Algebra; John Wiley & Sons: Hoboken, NJ, USA, 2006. [Google Scholar]
- Hall, M. The Theory of Groups; Courier Dover Publications: Mineola, NY, USA, 2018. [Google Scholar]
- Howie, J.M. Fundamentals of Semigroup Theory; Clarendon Oxford: Oxford, UK, 1995. [Google Scholar]
- Godin, T.; Klimann, I.; Picantin, M. On torsion-free semigroups generated by invertible reversible Mealy automata. In International Conference on Language and Automata Theory and Applications; Springer: Berlin, Germany, 2015; pp. 328–339. [Google Scholar]
- East, J.; Egri-Nagy, A.; Mitchell, J.D.; Peresse, Y. Computing finite semigroups. J. Symb. Comput.
**2019**, 92, 110–155. [Google Scholar] [CrossRef] [Green Version] - Smarandache, F. A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Probability, and Statistics; American Research Press: Rehoboth, DE, USA, 2000. [Google Scholar]
- Smarandache, F.; Ali, M. Neutrosophic triplet group. Neural Comput Applic
**2018**, 29, 595–601. [Google Scholar] [CrossRef] [Green Version] - Kandasamy W.B., V.; Kandasamy, I.; Smarandache, F. Semi-Idempotents in Neutrosophic Rings. Mathematics
**2019**, 7, 507. [Google Scholar] [CrossRef] [Green Version] - Kandasamy W. B., V.; Kandasamy, I.; Smarandache, F. Neutrosophic Triplets in Neutrosophic Rings. Mathematics
**2019**, 7, 563. [Google Scholar] [CrossRef] [Green Version] - Kandasamy, W.V.; Kandasamy, I.; Smarandache, F. Neutrosophic Quadruple Vector Spaces and Their Properties. Mathematics
**2019**, 7, 758. [Google Scholar] - Saha, A.; Broumi, S. New Operators on Interval Valued Neutrosophic Sets. Neutrosophic Sets Syst.
**2019**, 28, 10. [Google Scholar] - Sahin, R.; Karabacak, M. A novel similarity measure for single-valued neutrosophic sets and their applications in medical diagnosis, taxonomy, and clustering analysis. In Optimization Theory Based on Neutrosophic and Plithogenic Sets; Elsevier: Amsterdam, The Netherlands, 2020; pp. 315–341. [Google Scholar]
- Jain, A.; Nandi, B.P.; Gupta, C.; Tayal, D.K. Senti-NSetPSO: Large-sized document-level sentiment analysis using Neutrosophic Set and particle swarm optimization. Soft Comput.
**2020**, 24, 3–15. [Google Scholar] [CrossRef] - Wu, X.; Zhang, X. The Decomposition Theorems of AG-Neutrosophic Extended Triplet Loops and Strong AG-(l, l)-Loops. Mathematics
**2019**, 7, 268. [Google Scholar] [CrossRef] [Green Version] - Ma, Y.; Zhang, X.; Yang, X.; Zhou, X. Generalized Neutrosophic Extended Triplet Group. Symmetry
**2019**, 11, 327. [Google Scholar] [CrossRef] [Green Version] - Li, Q.; Ma, Y.; Zhang, X.; Zhang, J. Neutrosophic Extended Triplet Group Based on Neutrosophic Quadruple Numbers. Symmetry
**2019**, 11, 696. [Google Scholar] [CrossRef] [Green Version] - Ali, M.; Smarandache, F.; Khan, M. Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field. Mathematics
**2018**, 6, 46. [Google Scholar] [CrossRef] [Green Version] - Smarandache, F. Neutrosophic Perspectives: Triplets, Duplets, Multisets, Hybrid Operators, Modal Logic, Hedge Algebras. And Applications; EuropaNova: Bruxelles, Belgium, 2017. [Google Scholar]
- Kandasamy, W.V.; Ilanthenral, K. Smarandashe Special Elements in Multiset Semigroups; EuropaNova ASBL: Brussels, Belgium, 2018. [Google Scholar]
- Forsberg, L. Multisemigroups with multiplicities and complete ordered semi-rings. Beitr Algebra Geom
**2017**, 58, 405–426. [Google Scholar] [CrossRef] [Green Version] - Kandasamy, W.V. Smarandache Semirings, Semifields, And Semivector Spaces. Smarandache Notions J.
**2002**, 13, 88. [Google Scholar] - Blizard, W.D. The development of multiset theory. Mod. Log.
**1991**, 1, 319–352. [Google Scholar]

© 2020 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**

W. B., V.; Kandasamy, I.; Smarandache, F.
Neutrosophic Components Semigroups and Multiset Neutrosophic Components Semigroups. *Symmetry* **2020**, *12*, 818.
https://doi.org/10.3390/sym12050818

**AMA Style**

W. B. V, Kandasamy I, Smarandache F.
Neutrosophic Components Semigroups and Multiset Neutrosophic Components Semigroups. *Symmetry*. 2020; 12(5):818.
https://doi.org/10.3390/sym12050818

**Chicago/Turabian Style**

W. B., Vasantha, Ilanthenral Kandasamy, and Florentin Smarandache.
2020. "Neutrosophic Components Semigroups and Multiset Neutrosophic Components Semigroups" *Symmetry* 12, no. 5: 818.
https://doi.org/10.3390/sym12050818