#
Neutrosophic Duplets of {Z_{pn},×} and {Z_{pq},×} and Their Properties

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## Abstract

**:**

## 1. Introduction

## 2. Results

**Definition**

**1.**

- 1.
- $neut\left(a\right)$ is not the same as the unitary element of D in relation with the law # (if any);
- 2.
- $a\#$$neut\left(a\right)$ = neut(a) # a = a; and
- 3.
- $anti\left(a\right)\notin $ D for which a # anti(a) = anti(a) # a = neut(a).

- The neutrals of all nontrivial neutrosophic duplets are units of $\{{Z}_{{p}^{n}},\times \}$, $\{{Z}_{pq},\times \}$ and $\{{Z}_{{p}_{1}{p}_{2}\dots {p}_{n}},\times \}$.
- If p is a prime in anyone of the semigroups ($\{{Z}_{{p}^{n}},\times \}$ or $\{{Z}_{pq},\times \}$ or $\{{Z}_{{p}_{1}{p}_{2}\dots {p}_{n}},\times \}$) as mentioned in 1, then $mp$ has only p number of neutrals, for the appropriate m.
- The neutrals of any $m{p}^{t}$ for a prime p; $(m,p)=1$ are obtained and they form a special collection.

## 3. Neutrosophic Duplets of $\{{Z}_{{p}^{n}},\times \}$ and its Properties

**Example**

**1.**

- 1.
- Every non-unit of ${Z}_{16}$ is a neutrosophic duplet.
- 2.
- Every non-unit divisible by 2, viz. $\{2,6,10,14\}$, has only $\{1,9\}$ as their neutrals.
- 3.
- Every non-unit divisible by 4 are 4 and 12, which has $\{1,5,9,13\}$ as neutrals.

**Theorem**

**1.**

- (i)
- The set of units of S are $A=\{1,3,5,\dots ,{2}^{n}-1\}$, forms a group under × and $\left|A\right|={2}^{n-1}$.
- (ii)
- The set of all neutrosophic duplets with ${2}^{n-1}$ is A; neutrals of ${2}^{n-1}$ are A.
- (iii)
- All elements of the form $2m\in {Z}_{{2}^{n}}$ (m an odd number) has only the elements $\{1,{2}^{n-1}+1\}$ to contribute to neutrosophic duplets (neutrals are $1,{2}^{n-1}+1$).
- (iv)
- All elements of the form $m{2}^{t}\in {Z}_{{2}^{n}};1<t<n-1$; m odd has its neutrals from $B=\{1,{2}^{n-t}+1$, ${2}^{n-t+1}+1$, ${2}^{n-t+2}+1$, $\dots ,{2}^{n-1}+1,{2}^{n-t}+{2}^{n-t+1}+1,\dots ,{2}^{n-t}+{2}^{n-1}+1,\dots ,$$1+{2}^{n-t}+{2}^{n-t+1}+\dots +{2}^{n-1}\}$.

**Proof.**

- (i)
- Given $S=\{{Z}_{{2}^{n}},\times \}$ where $n\ge 2$ and S is a semigroup under product modulo ${2}^{n}$. $A=\{1,3,5,7,\dots ,{2}^{n}-1\}$ is a group under product as every element is a unit in S and closure axiom is true by property of modulo integers and $\left|A\right|={2}^{n-1}$. Hence, Claim (i) is true.
- (ii)
- Now, consider the element ${2}^{n-1}$; the set of duplets for ${2}^{n-1}$ is A for ${2}^{n-1}\times 1={2}^{n-1}$; ${2}^{n-1}\times 3={2}^{n-1}[2+1]$ = ${2}^{n}+{2}^{n-1}={2}^{n-1},\dots ,{2}^{n-1}\left(m\right);$ (m is odd) will give only $m{2}^{n-1}$. Hence, this proves Claim (ii).
- (iii)
- Consider $2m\in {Z}_{{2}^{n}}$; we see $2m\times 1=2m$ and $2m({2}^{n-1}+1)=m{2}^{n}+2m=2m$. $(2m,{2}^{n-1}+1$) is a neutrosophic duplet pair; hence, the claim.
- (iv)
- Let $m{2}^{t}\in {Z}_{{2}^{n}}$; clearly, $m{2}^{t}\times x=m{2}^{t}$ for all $x\in B$.

**Example**

**2.**

**Theorem**

**2.**

**Proof.**

## 4. Neutrosophic Duplets of ${Z}_{pq}$ and ${Z}_{{p}_{1}{p}_{2}\dots {p}_{n}}$

**Example**

**3.**

**Theorem**

**3.**

**Proof.**

**Example**

**4.**

**Example**

**5.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

## 5. Discussions and Conclusions

- For ${Z}_{pq},$p and q odd primes, how many neutrosophic duplet pairs are there?
- For ${Z}_{{p}_{1}\dots {p}_{n}}$, what are the neutrals of ${p}_{i}{p}_{j},{p}_{i}{p}_{j}{p}_{k},\dots ,{p}_{1}{p}_{2}\dots {p}_{i-1}{p}_{i+1}\dots {p}_{n}$?
- The study of neutrosophic duplets of ${Z}_{{p}_{1}^{{t}_{1}}{p}_{2}^{{t}_{2}}\dots {p}_{n}^{{t}_{n}}}$; ${p}_{1},\dots ,{p}_{n}$ are distinct primes and ${t}_{i}\ge 1;1\le i\le n$ is left open.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SVNS | Single Valued Neutrosophic Sets |

DVNS | Double Valued Neutrosophic Sets |

TRINS | Triple Refined Indeterminate Neutrosophic Sets |

IFS | Intuitionistic Fuzzy Sets |

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Neutrals for duplets | Neutrals for duplets |
---|---|

5, 10, 15, 20, 25, 30 | 7, 14, 21, 28 |

1, 8, 15, 22, 24 | 1, 6, 11, 16, 21, 26, 31 |

Neutrals for duplets | Neutrals for duplets |
---|---|

3, 6, 9, 12, 18, 21, 24, 27, | 5, 10, 20, 25, 40, 50, |

33, 36, 39, 48, 51, 54, 57, 66, | 55, 65, 80, 85, 95, 100 |

69, 78, 81, 87, 93, 96, 99, 102 | |

1, 36, 71 | 1, 22, 43, 64, 84 |

Neutrals for duplets | Neutrals for duplets |

7, 14, 28, 49, 56, 77, 91, 98 | 15, 30, 45, 60, 75, 90 |

1, 16, 31, 46, 61, 76, 91 | 1, 8, 15, 22, 29, 36, 43, 50, |

57, 64, 71, 78, 85, 92, 99 | |

Neutrals for duplets | Neutrals for duplets |

21, 42, 63, 84 | 35, 70 |

1, 6, 11 16, 21, 26, 31, 36, | 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, |

41, 46, 51, 56, 61, 66, 71, | 31, 34, 37, 40, 43, 46, 49, 52, 55, |

76, 81, 86, 91, 96, 101 | 58, 61, 64, 67, 70, 73, 76, 79, |

82, 85, 88, 91, 94, 97, 100, 103 |

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**MDPI and ACS Style**

Kandasamy W.B., V.; Kandasamy, I.; Smarandache, F.
Neutrosophic Duplets of {*Z*_{pn},×} and {*Z*_{pq},×} and Their Properties. *Symmetry* **2018**, *10*, 345.
https://doi.org/10.3390/sym10080345

**AMA Style**

Kandasamy W.B. V, Kandasamy I, Smarandache F.
Neutrosophic Duplets of {*Z*_{pn},×} and {*Z*_{pq},×} and Their Properties. *Symmetry*. 2018; 10(8):345.
https://doi.org/10.3390/sym10080345

**Chicago/Turabian Style**

Kandasamy W.B., Vasantha, Ilanthenral Kandasamy, and Florentin Smarandache.
2018. "Neutrosophic Duplets of {*Z*_{pn},×} and {*Z*_{pq},×} and Their Properties" *Symmetry* 10, no. 8: 345.
https://doi.org/10.3390/sym10080345