# Generalized Neutrosophic Extended Triplet Group

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

**Definition**

**1.**

**Definition**

**2.**

**Proposition**

**1.**

- (1)
- $neut(a)$ is unique for any $a\in N$.
- (2)
- $neut(a)\ast neut(a)=neut(a)$ for any $a\in N$.
- (3)
- $neut(neut(a))=neut(a)$ for any $a\in N$.

**Definition**

**3.**

**Definition**

**4.**

- (1)
- $(a\ast b)\ast c=a\ast (b\ast c),\forall \phantom{\rule{4pt}{0ex}}a,b,c\in G$.
- (2)
- For each $a\in G$, there exists a unique $e(a)\in G$ such that $a\ast e(a)=e(a)\ast a=a$.
- (3)
- For each $a\in G$, there exists ${a}^{-1}\in G$ such that $a\ast {a}^{-1}={a}^{-1}\ast a=e(a)$.

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Proposition**

**2.**

- (1)
- S is completely regular;
- (2)
- every element of S lies in a subgroup of S;
- (3)
- every H-class in S is a group.

**Definition**

**8.**

**Proposition**

**3.**

- (1)
- S is Clifford semigroup;
- (2)
- S is a semilattice of groups;
- (3)
- S is regular, and the idempotents of S are central.

**Theorem**

**2.**

**Theorem**

**3.**

## 3. The Relations of $({Z}_{n},\otimes )$ and NETG

**Lemma**

**1.**

**Lemma**

**2.**

**Proposition**

**4.**

**Proof.**

**Remark:**

**Theorem**

**4.**

**Proof.**

**Example**

**1.**

**Proposition**

**5.**

**Proof.**

**Remark:**

**Example**

**2.**

- (1)
- $[2]$ has not the neutral element and opposite element being $\mathrm{gcd}(\mathrm{gcd}(2,36),36/\mathrm{gcd}(2,36))=2\ne 1$. In fact, $[2],[3],[6],[10],[12],[14],[15],$$[18],[21],[22],[24],[26],[30],[33]$ and $[34]$ do not have the neutral element and opposite element by Theorem 4 for the same reason.
- (2)
- $[4]$ and $[8]$ have the same neutral element being $\mathrm{gcd}(4,36)=\mathrm{gcd}(8,36)=4$. In fact, $[4],[8],[16],[20],[28]$ and $[32]$ have the same neutral element, which is $[28]$.
- (3)
- $[9]$ and $[27]$ have the same neutral element $[9]$ being $\mathrm{gcd}(9,36)=\mathrm{gcd}(27,36)=9$.
- (4)
- $[1],[5],[7],[11],[13],[17],[19],[23],[25],[29],[31]$ and $[35]$ have the same neutral element $[1]$.
- (5)
- $[0]$ has the neutral element $[0]$.
- (6)
- We can see that the neutral element of each element is unique and explain that Proposition 5 is correct. Moreover, $([0],[0],[1]),([0],[0],[0]),([1],[1],[1]),([9],[9],[1]),([9],[9],[9]),([28],[28],[1])$ and $([28],[28],[28])$ are some neutrosophic extended triplets in $({Z}_{36},\otimes )$, which verify the above Remark.

**Theorem**

**5.**

**Proof.**

**Remark:**

**Example**

**3.**

- (1)
- $[1],[7],[11],[13],[17],[19],[23]$ and $[29]$ have the same neutral element, which is $[1]$.
- (2)
- $[2],[4],[8],[14],[16],[22],[26]$ and $[28]$ have the same neutral element, which is $[16]$.
- (3)
- $[3],[9],[21]$ and $[27]$ have the same neutral element, which is $[21]$.
- (4)
- $[5]$ and $[25]$ have the same neutral element, which is $[25]$.
- (5)
- $[6],[12],[18]$ and $[24]$ have the same neutral element, which is $[6]$.
- (6)
- $[10]$ and $[20]$ have the same neutral element, which is $[10]$.
- (7)
- $[15]$ has neutral element $[15]$.
- (8)
- $[0]$ has neutral element $[0]$.

## 4. GNETG and Quasi-Completely Regular Semigroup

**Definition**

**9.**

**Definition**

**10.**

- (1)
- $(N,\ast )$ is well-defined, i.e., for any $a,b\in N$, $a\ast b\in N$.
- (2)
- $(N,\ast )$ is associative, i.e., $(a\ast b)\ast c=a\ast (b\ast c)$ for all $a,b,c\in N$.

**Remark:**

**Example**

**4.**

**Example**

**5.**

- (1)
- $[0],[1],[3],[4],[5],[7],[8],[9]$ and $[11]$ exist the neutral element and opposite element.
- (2)
- $[2]$ does not exist the neutral element and opposite element, but we can see that ${[2]}^{2}=[4]$ exists the neutral element and opposite element.
- (3)
- $[6]$ does not exist the neutral element and opposite element, but we can see that ${[6]}^{2}=[0]$ exists the neutral element and opposite element.
- (4)
- $[10]$ does not exist the neutral element and opposite element, but we can see that ${[10]}^{2}=[4]$ exists the neutral element and opposite element.

**Theorem**

**6.**

**Proof.**

**Definition**

**11.**

**Proposition**

**6.**

**Proof.**

**Example**

**6.**

**Proposition**

**7.**

- (1)
- $neut({a}^{n})$ is unique.
- (1)
- $neut({a}^{n})\ast neut({a}^{n})=neut({a}^{n})$.

**Proof.**

**Proposition**

**8.**

- (1)
- ${a}^{n}\ast x\ast neut({a}^{n})=x\ast neut({a}^{n})\ast {a}^{n}=neut({a}^{n})$, for any $x\in \{anti({a}^{n})\}$.
- (2)
- ${a}^{n}\ast neut({a}^{n})\ast x=neut({a}^{n})\ast x\ast {a}^{n}=neut({a}^{n})$ for any $x\in \{anti({a}^{n})\}$.
- (3)
- $neut((neut({a}^{n})))=neut(x\ast neut({a}^{n}))=neut({a}^{n}),x\in \{anti({a}^{n})\}\cup \{{a}^{n}\}$.
- (4)
- $neut({a}^{n})\ast x=y\ast neut({a}^{n})=z\ast neut({a}^{n})$, for any $x,y,z\in \{anti({a}^{n})\}$.
- (5)
- $anti(x\ast neut({a}^{n}))\ast neut(x\ast neut({a}^{n}))={a}^{n}$, for any $x\in \{anti({a}^{n})\}$.

**Proof.**

**Example**

**7.**

- (1)
- Being ${a}^{2}\ast a\ast neut({a}^{2})=a\ast neut({a}^{2})\ast {a}^{2}=neut({a}^{2})$, ${a}^{2}\ast e\ast neut({a}^{2})=e\ast neut({a}^{2})\ast {a}^{2}=neut({a}^{2})$, that is, for any $x\in \{anti({a}^{2})\}$, ${a}^{2}\ast x\ast neut({a}^{2})=x\ast neut({a}^{2})\ast {a}^{2}=neut({a}^{2})$.
- (2)
- Being ${b}^{2}\ast neut({b}^{2})\ast b=neut({b}^{2})\ast b\ast {b}^{2}=neut({b}^{2})$, ${b}^{2}\ast neut({b}^{2})\ast f=neut({b}^{2})\ast f\ast {b}^{2}=neut({b}^{2})$, that is, for any $x\in \{anti({b}^{2})\}$, ${b}^{2}\ast neut({b}^{2})\ast x=neut({b}^{2})\ast x\ast {b}^{2}=neut({b}^{2})$.
- (3)
- Being $neut((neut({a}^{2})))=neut(a\ast neut({a}^{2}))=neut({a}^{2})$, $neut((neut({a}^{2})))=neut(e\ast neut({a}^{2}))=neut({a}^{2})$, that is, for any $x\in \{anti({a}^{n})\}\cup \{{a}^{n}\}$, $neut((neut({a}^{n})))=neut(x\ast neut({a}^{n}))=neut({a}^{n})$.
- (4)
- For each element $x,y,z\in S$, $neut({c}^{2})\ast x=y\ast neut({c}^{2})=z\ast neut({c}^{2})=g$, that is, for any $x,y,z\in \{anti({c}^{2})\}$, $neut({c}^{2})\ast x=y\ast neut({c}^{2})=z\ast neut({c}^{2})$.
- (5)
- Being $b\ast f={b}^{2}=f,f\ast f={b}^{2}=f$, that is, for any $x\in \{anti({b}^{2})\}$, $anti(x\ast neut({b}^{2}))\ast neut(x\ast neut({b}^{2}))={b}^{2}$.

**Proposition**

**9.**

- (1)
- $neut({a}^{n})\ast neut({b}^{m})=neut({a}^{n}\ast {b}^{m})$.
- (2)
- $anti({a}^{n})\ast anti({b}^{m})\in \{anti({a}^{n}\ast {b}^{m})\}$.

**Proof.**

**Example**

**8.**

**Example**

**9.**

- (1)
- Being for each $2\le n\in {Z}^{+},{a}^{n}=e$, and for each $m\in {Z}^{+},{f}^{m}=f$, so $neut({a}^{n})\ast neut({f}^{m})=e\ast f=c$. On the other hand, $neut({a}^{n}\ast {f}^{m})=neut(e\ast f)=neut(c)$, but $neut(c)$ does not exist, so the (1) of Proposition 9 does not hold.
- (2)
- Being $\{anti({a}^{n})\}=\{a,e\}$, $\{anti({f}^{m})\}=\{f\}$, ${a}^{n}\ast {f}^{m}=c$, but $anti(c)$ does not exist, we can get that the (2) of Proposition 9 does not hold.

**Theorem**

**7.**

**Proof.**

**Example**

**10.**

**Definition**

**12.**

**Example**

**11.**

**Example**

**12.**

**Proposition**

**10.**

- (1)
- $neut({a}^{n})\ast neut({b}^{m})=neut({b}^{m})\ast neut({a}^{n})$, for all $a,b\in N$.
- (2)
- $neut({a}^{n})\ast neut({b}^{m})\ast {a}^{n}={a}^{n}\ast neut({b}^{m})$, for all $a,b\in N$.

**Proof.**

**Proposition**

**11.**

- (1)
- $neut({a}^{n})\ast neut({b}^{m})=neut({b}^{m}\ast {a}^{n})$;
- (2)
- $anti({a}^{n})\ast anti({b}^{m})\in \{anti({b}^{m}\ast {a}^{n})\}$.

**Proof.**

**Example**

**13.**

- (1)
- Being $neut({b}^{2})\ast neut({e}^{1})=f\ast e=g$, and $neut({e}^{1}\ast {b}^{2})=neut(g)=g$, so, $neut({b}^{2})\ast neut({e}^{1})=neut({e}^{1}\ast {b}^{2})$.
- (2)
- Being $\{anti({b}^{2})\}=\{b,f\}$, $\{anti({e}^{1})\}=\{a,e\}$, $\{anti({b}^{2}\ast {e}^{1})\}=\{anti(g)\}=\{a,e,b,f,c,g\}$, thus, $anti({b}^{2})\ast anti({e}^{1})\in \{anti({b}^{2}\ast {e}^{1})\}$.

**Example**

**14.**

- (1)
- Being $neut({a}^{2})\ast neut({f}^{1})=e\ast f=e$, but $neut({f}^{1}\ast {a}^{2})=neut(f)=f$, so, $neut({a}^{2})\ast neut({f}^{1})\ne neut({f}^{1}\ast {a}^{2})$.
- (2)
- Being $\{anti({a}^{2})\}=\{a,e\}$, $\{anti({f}^{1})\}=\{f\}$, $\{anti({f}^{1}\ast {a}^{2})\}=\{anti(f)\}=\{f\}$, thus, $anti({a}^{2})\ast anti({f}^{1})\notin \{anti({f}^{1}\ast {a}^{2})\}$.

**Definition**

**13.**

**Theorem**

**8.**

**Proof.**

**Example**

**15.**

**Example**

**16.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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⊗ | $[0]$ | [1] | [2] | [3] | [4] | [5] | [6] | [7] |

$[0]$ | $[0]$ | [0] | [0] | [0] | [0] | [0] | [0] | [0] |

$[1]$ | $[0]$ | [1] | [2] | [3] | [4] | [5] | [6] | [7] |

$[2]$ | $[0]$ | [2] | [4] | [6] | [0] | [2] | [4] | [6] |

$[3]$ | $[0]$ | [3] | [6] | [1] | [4] | [7] | [2] | [5] |

$[4]$ | $[0]$ | [4] | [0] | [4] | [0] | [4] | [0] | [4] |

$[5]$ | $[0]$ | [5] | [2] | [7] | [4] | [1] | [6] | [3] |

$[6]$ | $[0]$ | [6] | [4] | [2] | [0] | [6] | [4] | [2] |

$[7]$ | $[0]$ | [7] | [6] | [5] | [4] | [3] | [2] | [1] |

$\ast $ | a | e | f | g |

a | e | e | g | e |

e | e | e | e | e |

f | f | f | f | f |

g | g | g | g | g |

$\ast $ | a | b | c |

a | a | b | c |

b | b | a | c |

c | c | c | c |

$\ast $ | a | e | b | f | c | g |

a | e | e | c | c | c | g |

e | e | e | c | c | c | g |

b | g | g | f | f | g | g |

f | g | g | f | f | g | g |

c | g | g | c | c | g | g |

g | g | g | g | g | g | g |

⊗ | $[1]$ | $[4]$ | $[7]$ |

$[1]$ | $[1]$ | $[4]$ | $[7]$ |

$[4]$ | $[4]$ | $[4]$ | $[4]$ |

$[7]$ | $[1]$ | $[4]$ | $[1]$ |

$\ast $ | a | b | c | d | f |

a | a | b | c | d | f |

b | b | a | c | d | f |

c | c | c | c | c | c |

d | c | c | c | c | c |

f | f | f | c | d | f |

$\ast $ | a | b | c | f |

a | a | b | c | f |

b | b | a | c | f |

c | c | c | c | c |

f | f | f | c | f |

$\ast $ | a | e | b | f | c | g |

a | e | e | c | c | g | g |

e | e | e | g | g | g | g |

b | c | g | f | f | c | g |

f | c | g | f | f | c | g |

c | g | g | c | c | g | g |

g | g | g | g | g | g | g |

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**MDPI and ACS Style**

Ma, Y.; Zhang, X.; Yang, X.; Zhou, X.
Generalized Neutrosophic Extended Triplet Group. *Symmetry* **2019**, *11*, 327.
https://doi.org/10.3390/sym11030327

**AMA Style**

Ma Y, Zhang X, Yang X, Zhou X.
Generalized Neutrosophic Extended Triplet Group. *Symmetry*. 2019; 11(3):327.
https://doi.org/10.3390/sym11030327

**Chicago/Turabian Style**

Ma, Yingcang, Xiaohong Zhang, Xiaofei Yang, and Xin Zhou.
2019. "Generalized Neutrosophic Extended Triplet Group" *Symmetry* 11, no. 3: 327.
https://doi.org/10.3390/sym11030327