# On Neutrosophic Triplet Groups: Basic Properties, NT-Subgroups, and Some Notes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

^{−}

^{1}of x is also relative to the unit element e, and the inverse element is unique in the classical group. In [14,15], starting from the basic idea of neutrosophic set, a new algebraic structure, neutrosophic triplet group (briefly, NTG), is proposed. In NTG, the unit element is generalized as a neutral element, which is relative and local; that is, each element has its own neutral element; and the original inverse element concept is generalized as an anti (opposite) element, and it is relative to own neutral element, and it cannot be unique. In this way, NTG can express more general symmetry and has important theoretical and applied value.

**Definition**

**1.**

**Remark**

**1.**

**Definition**

**2.**

- (1)
- The operation * is closed, i.e.,a *b∈N,∀a,b∈N;
- (2)
- The operation* is associative, i.e., (a *b) *c=a * (b *c),∀a,b,c∈N

## 3. Some Counterexamples and Misunderstandings on Neutrosophic Triplet Groups

**Example**

**1.**

**Example**

**2.**

## 4. Some New and Important Properties of Neutrosophic Triplet Groups

**Theorem**

**1.**

- (1)
- a ∈ N, neut(a) is unique.
- (2)
- a ∈ N, neut(a) * neut(a) = neut(a).

**Proof.**

**Remark**

**2.**

**Theorem**

**2.**

- (1)
- neut(a)*p = q*neut(a), for any p, q ∈ {anti(a)};.
- (2)
- neut(neut(a)) = neut(a);
- (3)
- anti(neut(a))*anti(a) ∈ {anti(a)};
- (4)
- neut(a*a)*a = a*neut(a*a) = a; neut(a*a)*neut(a) = neut(a)*neut(a*a) = neut(a);
- (5)
- neut(anti(a))*a = a*neut(anti(a)) = a; neut(anti(a))*neut(a) = neut(a)*neut(anti(a)) = neut(a);
- (6)
- anti(neut(a))*a = a*anti(neut(a)) = a, for any anti(neut(a)) ∈ {anti(neut(a))};
- (7)
- a ∈ {anti(neut(a)*anti(a))};
- (8)
- neut(a)*anti(a) ∈ {anti(a)}; anti(a)*neut(a) ∈ {anti(a)};
- (9)
- a ∈ {anti(anti(a))}, that is, there exists p ∈ {anti(a)} such that a ∈ {anti(p)};
- (10)
- neut(a)*anti(anti(a)) = a.

**Proof.**

- (1)
- For any p, q ∈ {anti(a)}, according the definition of neutral and opposite element, applying Theorem 1 (1), we have:p*a = a*p = neut(a), q*a = a*q = neut(a).

neut(a)*p = (q*a)*p = q*(a*p) = q*neut(a). - (2)
- For any anti(a) ∈ {anti(a)} and anti(neut(a)) ∈ {anti(neut(a))},[anti(neut(a))*anti(a)]*a = anti(neut(a))*[anti(a)*a] = anti(neut(a))*neut(a) = neut(neut(a)).{[anti(neut(a))*anti(a)]*a}*neut(a) = [anti(neut(a))*anti(a)]*[a*neut(a)] =

[anti(neut(a))*anti(a)]*a = neut(neut(a)).neut(neut(a))*neut(a) = {[anti(neut(a))*anti(a)]*a}*neut(a) = neut(neut(a)). - (3)
- For any anti(a) ∈ {anti(a)} and anti(neut(a)) ∈ {anti(neut(a))}, applying (2), we have:[anti(neut(a))*anti(a)]*a = anti(neut(a))*[anti(a)*a] = anti(neut(a))*neut(a) = neut(neut(a)) = neut(a);

a*[anti(neut(a))*anti(a)] = [a*neut(a)]*[anti(neut(a))*anti(a)] = a*[neut(a)*anti(neut(a))]*anti(a) =

a*neut(neut(a))*anti(a) = a*neut(a)*anti(a) = a*anti(a) = neut(a). - (4)
- According to the definition of neutral element, using the associative law, we get:(a*a)*neut(a*a) = (a*a),

anti(a)*[(a*a)*neut(a*a)] = anti(a)*(a*a),

[anti(a)*a]*[a*neut(a*a)] = [anti(a)*a]*a,

neut(a)*[a*neut(a*a)] = neut(a)*a,

[neut(a)*a]*neut(a*a) = neut(a)*a,

a*neut(a*a) = a.neut(a)*neut(a*a) = [anti(a)*a]*neut(a*a) = anti(a)*[a*neut(a*a)] = anti(a)*a = neut(a).

neut(a*a)*neut(a) = neut(a*a)*[a*anti(a)] = [neut(a*a)*a]*anti(a) = a *anti(a) = neut(a). - (5)
- For any anti(a) ∈ {anti(a)}, we have:anti(a)*neut(anti(a)) = anti(a); neut(anti(a))*anti(a) = anti(a).

a*[anti(a)*neut(anti(a))] = a*anti(a); [neut(anti(a))*anti(a)]*a = anti(a)*a.

[a*anti(a)]*neut(anti(a)) = a*anti(a); neut(anti(a))*[anti(a)*a] = anti(a)*a.

neut(a)*neut(anti(a)) = neut(a); neut(anti(a))*neut(a) = neut(a).

a*[neut(a)*neut(anti(a))] = a*neut(a); [neut(anti(a))*neut(a)]*a = neut(a)*a.

[a*neut(a)]*neut(anti(a)) = a*neut(a); neut(anti(a))*[neut(a)*a] = neut(a)*a.

a*neut(anti(a)) = a; neut(anti(a))*a = a.neut(a)*neut(anti(a)) = [anti(a)*a]*neut(anti(a)) = anti(a)*[a*neut(anti(a))] = anti(a)*a = neut(a).

neut(anti(a))*neut(a) = neut(anti(a))*[a*anti(a)] = [neut(anti(a))*a]*anti(a) = a *anti(a) = neut(a). - (6)
- For any anti(neut(a)) ∈ {anti(neut(a))}, by the definition of opposite element, we have:neut(a)*anti(neut(a)) = anti(neut(a))*neut(a) = neut(neut(a)).neut(a)*anti(neut(a)) = anti(neut(a))*neut(a) = neut(a).a*[neut(a)*anti(neut(a))] = a*neut(a); [anti(neut(a))*neut(a)]*a = neut(a)*a.

[a*neut(a)]*anti(neut(a)) = a*neut(a); anti(neut(a))*[neut(a)*a] = neut(a)*a.

a*anti(neut(a)) = a; anti(neut(a))*a = a. - (7)
- For any anti(a) ∈ {anti(a)}, we have:a*anti(a) = anti(a)*a = neut(a).

[a*neut(a)]*anti(a) = anti(a)*[neut(a)*a] = neut(a).

a*[neut(a)*anti(a)] = [anti(a)*neut(a)]*a = neut(a).a*[neut(a)*anti(a)] = [neut(a)*anti(a)]*a = neut(a).a*[neut(a)*anti(a)] = [neut(a)*anti(a)]*a = neut(a)*neut(anti(a)).[neut(a)*anti(a)]*[neut(a)*neut(anti(a))] = neut(a)*neut(a)*[anti(a)*neut(anti(a))] = neut(a)*anti(a);

[neut(a)*neut(anti(a))]*[neut(a)*anti(a)] = neut(a)*[neut(anti(a))*anti(a)]*neut(a) = neut(a)*anti(a). - (8)
- Assume anti(a) ∈ {anti(a)}, then [neut(a)*anti(a)]*a = neut(a)*[anti(a)*a] = neut(a)*neut(a). By Theorem 1 (2), neut(a)*neut(a) = neut(a). Thus, [neut(a)*anti(a)]*a = neut(a). On the other hand,a*[neut(a)*anti(a)] = [a*neut(a)]*anti(a) = a*anti(a) = neut(a).[neut(a)*anti(a)]*a = a*[neut(a)*anti(a)] = neut(a).
- (9)
- For any anti(a) ∈ {anti(a)}, denote p = neut(a)*anti(a). Using (8) we have p ∈ {anti(a)}. Moreover, by Theorem 1 (2):neut(a)*p = neut(a)*[neut(a)*anti(a)] = [neut(a)*neut(a)]*anti(a) = neut(a)*anti(a) = p.
- (10)
- Assume anti(a) ∈ {anti(a)} and anti(anti(a)) ∈ {anti(anti(a))}, by the definition of opposite element, we have:anti(a)*anti(anti(a)) = neut(anti(a)).a*[anti(a)*anti(anti(a))] = a*neut(anti(a)).

[a*anti(a)]*anti(anti(a)) = a*neut(anti(a)).

neut(a)*anti(anti(a)) = a*neut(anti(a)).neut(a)*anti(anti(a)) = a.

**Example**

**3.**

_{6}= {[0], [1], [2], [3], [4], [5]}, * is classical mod multiplication, then (Z

_{6}, *) is a commutative neutrosophic triplet group, see Example 10 in [16].

- (1)
- [2]*[4] = [5]*[2], [2]*[5] = [4]*[2], that is, for any p, q ∈ {anti([2])}, neut([2])*p = q*neut([2]).
- (2)
- neut(neut([0])) = neut([0]) = [0],neut(neut([1])) = neut([1]) = [1],neut(neut([2])) = neut([2]) = [4],neut(neut([3])) = neut([3]) = [3],neut(neut([4])) = neut([4]) = [4],neut(neut([5])) = neut([5]) = [1].
- (3)
- Since neut([2]) = [4], {anti([4])} = {[1], [4]} and {anti([2])} = {[2], [5]}, so anti(neut([2])) = anti([4]) = {[1], [4]}, and [1]*[2] = [2] ∈ {anti([2])}, [1]*[5] = [5] ∈ {anti([5])}, [4]*[2] = [2] ∈ {anti([2])}, [4]*[5] = [2] ∈ {anti([2])}. This means that anti(neut([2]))*anti([2]) ∈ {anti([2])} for any anti([2]) ∈ {anti([2])} and any anti(neut([2])) ∈ {anti(neut([2]))}.
- (4)
- neut([0]*[0])*[0] = [0]*neut([0]*[0]) = [0],neut([0]*[0])* neut([0]) = neut([0])*neut([0]*[0]) = [0]; neut([1]*[1])*[1] = [1]*neut([1]*[1]) = [1],neut([1]*[1])* neut([1]) = neut([1])*neut([1]*[1]) = [1]; and so on. This means that (4) hold for all a ∈ Z
_{6}. - (5)
- Since {anti([2])} = {[2], [5]}, so neut(anti([2])) = [4] or [1]. From [4]*[2] = [2]*[4] = [2] and [1]*[2] = [2]*[1] = [2] we know that neut(anti([2]))*[2] = [2]*neut(anti([2])) = [2] for any anti([2]) ∈ {anti([2])} and any neut(anti([2])) ∈ {neut(anti([2]))}. Note that, since {neut(anti([2]))} = {[4], [1]}; when anti([2]) = [5], neut(anti([2])) = [1] ≠ neut([2]), this means that neut(anti(a)) = neut(a) is not true in general.
- (6)
- Since {anti(neut([2]))} = {[1], [4]}, from this and [1]*[2] = [2]*[1] = [2] and [4]*[2] = [2]*[4] = [2] we know that anti(neut([2]))*[2] = [2]*anti(neut([2])) = [2] for any anti(neut([2])) ∈ {anti(neut ([2]))}. Note that, since {anti(neut([2]))} = {[1], [4]}; when anti(neut([2])) = [1], anti(neut([2])) ≠ neut([2]), this means that anti(neut(a)) = neut(a) is not true in general.
- (7)
- Since {anti(neut([2]))} = {[1], [4]} and {anti([2])} = {[2], [5]}, so {anti(neut([2]))*anti([2])} = {[2], [5]}, that is, [2] ∈ {anti(neut([2]))*anti([2])}.
- (8)
- Since neut([2]) = [4] and {anti([2])} = {[2], [5]}, from [4]*[2] = [4]*[5] = [2] we know that neut([2])*anti([2]) ∈ {anti([2])}.
- (9)
- Since neut([2]) = [4] and {anti([2])} = {[2], [5]}, so {anti(anti([2]))} = {[2], [5]}. Thus, from [4]*[2] = [4]*[5] = [2] we know that neut([2])*anti(anti([2])) = [2] for any anti([2]) ∈ {anti([2])} and anti(anti([2])) ∈ {anti(anti([2]))}. Note that, since {anti(2)} = {[2], [5]}; when anti([2]) = [5], anti(anti([2])) = [5] ≠ [2], this means that anti(anti(a)) = a is not true in general.

**Theorem**

**3.**

- (1)
- neut(a) * neut(b) = neut(a*b).
- (2)
- anti(a) * anti(b)∈ {anti(a*b)}.

**Proof.**

= a*[neut(b)*b] = a*b.

= neut(a)*[anti(b)*b] = neut(a)*neut(b).

## 5. NT-subgroups and Weak Commutative Neutrosophic Tripet Groups

**Definition**

**3.**

- (1)
- a*b∈ H for all a, b∈ H;
- (2)
- there exists anti(a)∈ {anti(a)} such that anti(a)∈ H for all a∈ H, where{anti(a)} is the set of opposite element of a in (N,*).

**Proposition**

**1.**

**Proof.**

**Remark**

**3.**

_{6}in Example 3 and H = {[0], [2], [3], [4]}, then H is a NT-subgroup of (Z

_{6}, *), and (1) [2] ∈ H but {anti([2])} is not a subset of H; (2) {neut(a)|a ∈ N = Z

_{6}} = {[0], [1], [3], [4]} is not a subset of H.

**Definition**

**4.**

**Example**

**4.**

neut(5) = 1, anti(5) = 6; neut(6) = 1, anti(6) = 5; neut(7) = 7, {anti(7)} = {1, 2, 3, 4, 5, 6, 7}.

**Proposition**

**2.**

- (1)
- neut(a)*neut(b) = neut(b)*neut(a) for all a, b∈ N.
- (2)
- neut(a)*neut(b)*a = a*neut(b) for all a, b∈ N.

**Proof.**

**Proposition**

**3.**

- (1)
- neut(a)*neut(b) = neut(b*a);
- (2)
- anti(a)*anti(b) ∈ {anti(b*a)}.

**Proof.**

= [b*neut(a)]*a = b*[neut(a)*a] = b*a.

= anti(a)*[neut(b)*a] = anti(a)*[a*neut(b)] = [anti(a)*a]*neut(b) = neut(a)*neut(b).

**Definition**

**5.**

- (1)
- neut(a) ∈ H for all a ∈ N.
- (2)
- if there exists anti(a) ∈ {anti(a)} and p ∈ N such that anti(a)*b*neut(p) ∈ H, then there exists anti(b) ∈ {anti(b)} and q ∈ N such that a*anti(b)*neut(q) ∈ H; and the inverse is true.

**Example**

**5.**

_{1}= {1, 7}, H

_{2}= {1, 5, 6, 7}. Then H

_{1}and H

_{2}are two strong NT-subgroups of N.

**Proposition**

**4.**

**Proof.**

^{−1}for any a ∈ N.

^{−1}*b ∈ H. Denote h = a

^{−1}*b ∈ H. Then a = b*h

^{−1}. Since H is a normal subgroup of N, h

^{−1}∈ H and there exists h

_{1}∈ H such that b*h

^{−1}= h

_{1}*b. Thus, a = h

_{1}*b, a*b

^{−1}= h

_{1}∈ H. That is, there exists b

^{−1}= anti(b) ∈ {anti(b)} and a ∈ N such that a*anti(b)*neut(a) = a*b

^{−1}*e = a*b

^{−1}= h

_{1}∈ H. Similarly, we can prove the inverse is true.

**Theorem**

**4.**

_{H}on N as follows: ∀a, b ∈ N:

_{H}b if and only if there exists anti(a) ∈ {anti(a)} and p ∈ N such that anti(a)*b*neut(p) ∈ H.

- (1)
- the binary relation≈
_{H}is an equivalent relation on N; - (2)
- a ≈
_{H}b implies c*a ≈_{H}c*b for all c ∈ N; - (3)
- a ≈
_{H}b implies a*c ≈_{H}b*c and c*a ≈_{H}c*b for all c ∈ N; - (4)
- denote the equivalent class contained a by [a]
_{H}, and denote N/H = {[a]_{H}|a∈ N}, define binary operation * on N/H as follows: [a]_{H}*[b]_{H}= [a*b]_{H},∀a, b∈ N. We can obtain a homomorphism from(N, *) to (N/H, *), that is, f: N→N/H; f(a) = [a]_{H}for all a∈ N.

**Proof.**

- (1)
- For any a ∈ N, applying Theorem 1 we have:anti(a)*a*neut(a) = [anti(a)*a]*neut(a) = neut(a)*neut(a) = neut(a) ∈ H.
_{H}a.- If a ≈
_{H}b, then there exists anti(a) ∈ {anti(a)} and p ∈ N such that anti(a)*b*neut(p) ∈ H. Denote h = anti(a)*b*neut(p), then h ∈ H and:a*h = a*[anti(a)*b*neut(p)],

a*h = neut(a)*b*neut(p),

a*h = b*neut(a)*neut(p), (by Definition 4)

anti(b)*(a*h) = anti(b)*[b*neut(a)*neut(p)],

[anti(b)*a]*h = neut(b)*neut(a)*neut(p),

{[anti(b)*a]*h}*anti(h) = [neut(b)*neut(a)*neut(p)]*anti(h),

anti(b)*a*neut(h) = [neut(b)*neut(a)*neut(p)]*anti(h)._{H}a. - If a≈
_{H}b and b ≈_{H}c, then there exists anti(a) ∈ {anti(a)}, anti(a) ∈ {anti(a)}, p ∈ N and q ∈ N such that anti(a)*b*neut(p) ∈ H, anti(b)*c*neut(q) ∈ H. Denote h_{1}= anti(a)*b*neut(p), h_{2}= anti(b)*c*neut(q), then h_{1}∈ H, h_{2}∈ H and:b*h_{2}= b*[anti(b)*c*neut(q)] = [b*anti(b)]*[c*neut(q)] = neut(b)*c*neut(q)._{1}= anti(a)*b*neut(p), using Definition 4 we get:h_{1}*h_{2}

= [anti(a)*b*neut(p)]*h_{2}

= [anti(a)*b]*[neut(p)*h_{2}]

= [anti(a)*b]*[h_{2}*neut(p)]

= anti(a)*(b*h_{2})*neut(p)

= anti(a)*[neut(b)*c*neut(q)]*neut(p)

= anti(a)*[neut(b)*c]*[neut(q)*neut(p)]

= anti(a)*[c*neut(b)]*[neut(q)*neut(p)]

= [anti(a)*c]*[neut(b)*neut(q)*neut(p)]_{1}*h_{2}∈ H; using Proposition 3 (1), neut(b)*neut(q)*neut(p) = neut(p*q*b). Hence:anti(a)*c*neut(p*q*b) = h_{1}*h_{2}∈ H._{H}c. Therefore, ≈_{H}is an equivalent relation on N.

- (2)
- Assume a ≈
_{H}b. Then there exists anti(a) ∈ {anti(a)} and p ∈ N such that anti(a)*b*neut(p) ∈ H. Denote:h = anti(a)*b*neut(p), then h ∈ H and:h*neut(c)

= [anti(a)*b*neut(p)]*neut(c)

= [anti(a)*b]*[neut(p)*neut(c)]

= [anti(a)*b]*[neut(c)*neut(p)]

= anti(a)*[b*neut(c)]*neut(p)

= anti(a)*[neut(c)*b]*neut(p)

= [anti(a)*neut(c)]*[b*neut(p)]

= [anti(a)*anti(c)*c]*[b*neut(p)]

= [anti(a)*anti(c)]*(c*b)*neut(p).anti(c*a)*(c*b)*neut(p) = h*neut(c) ∈ H._{H}(c*b).

- (1)
- Assume a ≈
_{H}b. Then there exists anti(a) ∈ {anti(a)} and p ∈ N such that anti(a)*b*neut(p) ∈ H. Applying Definition 5 (2), there exists anti(b) ∈ {anti(b)} and q ∈ N such that a*anti(b)*neut(q) ∈ H. Denote h = a*anti(b)*neut(q), then h ∈ H and:neut(c)*h

= neut(c)*[a*anti(b)*neut(q)]

= [neut(c)*a]*[anti(b)*neut(q)]

= [a*neut(c)]*[anti(b)*neut(q)]

= {a*[c*anti(c)]}*[anti(b)*neut(q)]

= (a*c)*[anti(c)*anti(b)]*neut(q).(a*c)*anti(b*c)*neut(q) = neut(c)*h ∈ H.anti(a*c)*(b*c)*neut(r) ∈ H._{H}(b*c). - (2)
- Using (1)–(3) we can obtain (4). ☐

**Example**

**6.**

neut(3) = 1, anti(3) = 3; neut(4) = 1, anti(4) = 4;

neut(5) = 1, anti(5) = 6; neut(6) = 1, anti(6) = 5; neut(7) = 7, anti(7) = 7.

- (1)
- The relation ≈
_{H}is an equivalent relation on N and N/H = {{1, 5, 6, 7}, {2, 3, 4}}. - (2)
- 1 ≈
_{H}5 implies 2*1 = 2 ≈_{H}4 = 2*5, and so on. - (3)
- 1 ≈
_{H}5 implies 1*2 = 2 ≈_{H}3 = 5*2, and so on. - (4)
- (N/H, *) = {[1]
_{H}, [2]_{H}}, (N, *)$\stackrel{f}{\simeq}$(N/H, *), where f(1) = f(5) = f(6) = f(7) = [1]_{H}, and f(2) = f(3) = f(4) = [2]_{H}.

**Remark**

**4.**

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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* | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1 | 1 | 1 | 1 | 4 | 5 |

2 | 1 | 3 | 2 | 4 | 5 |

3 | 1 | 2 | 3 | 4 | 5 |

4 | 4 | 4 | 4 | 4 | 5 |

5 | 5 | 5 | 5 | 5 | 4 |

* | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 1 |

2 | 2 | 2 | 2 | 2 | 2 |

3 | 4 | 4 | 3 | 4 | 5 |

4 | 4 | 4 | 4 | 4 | 4 |

5 | 4 | 4 | 5 | 4 | 3 |

* | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

2 | 2 | 1 | 6 | 5 | 4 | 3 | 7 |

3 | 3 | 5 | 1 | 6 | 2 | 4 | 7 |

4 | 4 | 6 | 5 | 1 | 3 | 2 | 7 |

5 | 5 | 3 | 4 | 2 | 6 | 1 | 7 |

6 | 6 | 4 | 2 | 3 | 1 | 5 | 7 |

7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |

**Table 4.**Weak commutative neutrosophic triplet group and its strong neutrosophic triplet (NT)-subgroup.

* | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

1 | 1 | 2 | 3 | 4 | 5 | 6 | 1 |

2 | 2 | 1 | 6 | 5 | 4 | 3 | 2 |

3 | 3 | 5 | 1 | 6 | 2 | 4 | 3 |

4 | 4 | 6 | 5 | 1 | 3 | 2 | 4 |

5 | 5 | 3 | 4 | 2 | 6 | 1 | 5 |

6 | 6 | 4 | 2 | 3 | 1 | 5 | 6 |

7 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, X.; Hu, Q.; Smarandache, F.; An, X.
On Neutrosophic Triplet Groups: Basic Properties, NT-Subgroups, and Some Notes. *Symmetry* **2018**, *10*, 289.
https://doi.org/10.3390/sym10070289

**AMA Style**

Zhang X, Hu Q, Smarandache F, An X.
On Neutrosophic Triplet Groups: Basic Properties, NT-Subgroups, and Some Notes. *Symmetry*. 2018; 10(7):289.
https://doi.org/10.3390/sym10070289

**Chicago/Turabian Style**

Zhang, Xiaohong, Qingqing Hu, Florentin Smarandache, and Xiaogang An.
2018. "On Neutrosophic Triplet Groups: Basic Properties, NT-Subgroups, and Some Notes" *Symmetry* 10, no. 7: 289.
https://doi.org/10.3390/sym10070289