# On Neutrosophic Extended Triplet LA-hypergroups and Strong Pure LA-semihypergroups

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction and Preliminaries

**Definition**

**1.**

^{∗}(H)

^{∗}(H)= P(H)/$\varphi $.

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

- (a)
- T ⊆ H, T ≠$\varphi $;
- (b)
- m ◦ n ⊆ T for all m, n ∈ T;
- (c)
- (H, ◦) is an LA-semihypergroup.

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

- (a)
- a left identity, if a ∈ e ◦ a for each a ∈ H;
- (b)
- a right identity, if a ∈ a ◦ e for each a ∈ H;
- (c)
- an identity, if a ∈ (e ◦ a) ∩ (a ◦ e) for each a ∈ H;
- (d)
- a pure left identity, if a = e ◦ a for each a ∈ H;
- (e)
- a pure right identity, if a = a ◦ e for each a ∈ H;
- (f)
- a pure identity, if a = (e ◦ a) ∩ (a ◦ e) for each a ∈ H;
- (g)
- a scalar identity, if a = e ◦ a = a ◦ e for each a ∈ H.

**Definition**

**8.**

- (a)
- (H, ◦) is an LA-hypergroup;
- (b)
- There exists e ∈ H such that e is identity of (H, ◦) ;
- (c)
- Every element a ∈ H has at least one inverse.

**Definition**

**9.**

**Definition**

**10.**

## 2. Neutrosophic Extended Triplet LA-Semihypergroups and Neutrosophic Extended Triplet LA-Hypergroups

**Definition**

**11.**

- (a)
- a left neutrosophic extended triplet LA-semihypergroup (LNET-LA-semihypergroup) if to any givena ∈ L, there are p ∈ L and q ∈ L, in such a way that

- (b)
- a right neutrosophic extended triplet LA-semihypergroup (RNET-LA-semihypergroup), if to any given a ∈ L, there are s ∈ L and t ∈ L, in such a way that

- (c)
- a neutrosophic extended triplet LA-semihypergroup (NET-LA-semihypergroup), if to any given a ∈ L, there are m ∈ L and n ∈ L, in such a way that

**Example**

**1.**

Python program 1 Verification of LA-semihypergroup 1 |

1: T = [ [[0],[0],[0]], [[0], [1], [0]], [[0], [0], [0,2]] ] |

2: count = 0 |

3: for x in range(3): |

4: for y in range(3): |

5: for z in range(3): |

6: T1 = T[x][y] |

7: T2 = set() |

8: k1 = len(T1)S3 = set(T[neut_t][t]) |

9: for m in range(k1) |

10: T2 = set(T[T1[m]][z]).union(T2) |

11: T3 = T[z][y] |

12: T4 = set() |

13: k2 = len(T3) |

14: for n in range(k2): |

15: T4 = set(T[T3[n]][x]).union(T4) |

16: if T2 = = T4:Ifif |

17: count += 1 18: while count = = 3**3: |

19: print(‘{} is an LA-semihypergroup’.format(T)) 20: break |

Python program 2 Verification of NET-LA-semihypergroup 1 |

1: T = [ [[0],[0],[0]], [[0], [1], [0]], [[0], [0], [0,2]] ] |

2: test = [[] |

3: for t in range(3): |

4: for neut_t in range(3): |

5: for anti_t in range(3): |

6: S1 = set(T[t][neut_t]) |

7: S2 = set(T[t][anti_t]) |

8: S3 = set(T[neut_t][t]) |

9: S4 = set(T[anti_t][t]) |

10: S5 = set(list([t])) |

11: S6 = set(list([neut_t])) |

12: if S5.issubset(S1 & S3) and S6.issubset(S2 & S4): |

13: test.append([t, neut_t, anti_t]) |

14: test2 = test |

15: test1 = set([test2[i][0] for i in range(len(test2))]) |

16: if test1 == set([x for x in range(3)]): |

17: print('{0} is an Net-LA-semihypergroup and hyper neutrosophic-triplet are {1}'.format(T, test2)) |

**Example**

**2.**

**Example**

**3.**

**Remark**

**1.**

**Example**

**4.**

Python program 3 Verification of LA-semihypergroup 2 |

1: T = [ [[0],[0],[0],[0,1,2,3]], [[0], [0], [0],[0,1,2,3]], [[0], [0], [0,1],[2,3]], [[1,2,3],[0,1,2,3],[2,3],[0,3]] ] |

2: count = 0 |

3: for x in range(4): |

4: for y in range(4): |

5: for z in range(4): |

6: T1 = T[x][y] |

7: T2 = set() |

8: k1 = len(T1)S3 = set(T[neut_t][t]) |

9: for m in range(k1) |

10: T2 = set(T[T1[m]][z]).union(T2) |

11: T3 = T[z][y] |

12: T4 = set() |

13: k2 = len(T3) |

14: for n in range(k2): |

15: T4 = set(T[T3[n]][x]).union(T4) |

16: if T2 = = T4:Ifif |

17: count += 1 18: while count = = 4**3: |

19: print(‘( T,∗) is an LA-semihypergroup.’) 20: break |

Python program 4 Verification of NET-LA-semihypergroup 2 |

1: T = [ [[0],[0],[0],[0,1,2,3]], [[0], [0], [0],[0,1,2,3]], [[0], [0], [0,1],[2,3]], [[1,2,3],[0,1,2,3],[2,3],[0,3]] ] |

2: test = [[] |

3: for t in range(4): |

4: for neut_t in range(4): |

5: for anti_t in range(4): |

6: S1 = set(T[t][neut_t]) |

7: S2 = set(T[t][anti_t]) |

8: S3 = set(T[neut_t][t]) |

9: S4 = set(T[anti_t][t]) |

10: S5 = set(list([t])) |

11: S6 = set(list([neut_t])) |

12: if S5.issubset(S1 & S3) and S6.issubset(S2 & S4): |

13: test.append([t, neut_t, anti_t]) |

14: test2 = test |

15: test1 = set([test2[i][0] for i in range(len(test2))]) |

16: if test1 == set([x for x in range(3)]): |

17: print('(T,∗) is an NET-LA-semihypergroup and hyper neutrosophic-triplet are {}'.format(test2). |

**Remark**

**2.**

**Definition**

**12.**

**Proposition**

**1.**

**Proof.**

**Remark**

**3.**

**Proposition**

**2.**

**Proof.**

**Remark**

**4.**

**Example**

**7.**

**Proposition**

**3.**

**Proposition**

**4.**

**Proof.**

**Remark**

**5.**

## 3. Strong Pure Neutrosophic Extended Triplet LA-Semihypergroups (SPNET-LA-Semihypergroups)

**Definition**

**13.**

- (a)
- a pure left neutrosophic extended triplet LA-semihypergroup (PLNET-LA-semihypergroup), if to any given a ∈ L, there are p ∈ L and q ∈ L, in such a way thata = p ∗ a and p = q ∗ a
- (b)
- a pure right neutrosophic extended triplet LA-semihypergroup (PRNET-LA-semihypergroup), if to any given a ∈ L, there are s ∈ L and t ∈ L, in such a way thata = a ∗ s and s = a ∗ t
- (c)
- a pure neutrosophic extended triplet LA-semihypergroup (PNET-LA-semihypergroup), if to any given a ∈ L, there are m ∈ L and n ∈ L, in such a way thata = (m ∗ a) ∩ (a ∗ m) and m = (n ∗ a) ∩ (a ∗ n)
- (d)
- a strong pure neutrosophic extended triplet LA-semihypergroup (SPNET-LA-semihypergroup), if to any given a ∈ L, there are m ∈ L and n ∈ L, in such a way thata = m ∗ a = a ∗ m and m = n ∗ a = a ∗ n

**Proposition**

**5.**

**Remark**

**6.**

**Proposition**

**6.**

**Proposition**

**7.**

- (1)
- if s ∈${\{\}}_{neut(a)}$, then s is unique and s∗ s = s;
- (2)
- if s = neut(a), then neut(s) = s and s∈${\{\}}_{anti{(s)}_{s}}$;
- (3)
- if s = neut(a), t ∈${\{\}}_{anti{(a)}_{s}}$, r ∈${\{\}}_{anti{(s)}_{s}}$, then r∗ t ⊆${\{\}}_{lanti{(a)}_{s}}$;
- (4)
- if s = neut(a), t ∈${\{\}}_{anti{(a)}_{s}}$, then s∗ t ⊆${\{\}}_{lanti{(a)}_{s}}$;
- (5)
- if p = neut(a), s = neut(b), q ∈${\{\}}_{anti{(a)}_{p}}$,t ∈${\{\}}_{anti{(b)}_{s}}$and |a∗ b|=| p∗ s| = 1, then$$\mathit{neut}(a\ast b)=p\ast s\mathit{and}q\ast t\subseteq {\{\}}_{anti{(a\ast b)}_{p\ast s}}$$
- (6)
- if s = neut(a) = neut(b), q ∈${\{\}}_{anti{(a)}_{s}}$, t ∈${\{\}}_{anti{(b)}_{s}}$and |a∗ b| = 1, then$$\mathit{neut}(a\ast b)=s\mathit{and}q\ast t\subseteq {\{\}}_{anti{(a\ast b)}_{s}}$$
- (7)
- if neut(a) = neut(b), then a ∗ b = b ∗ a;
- (8)
- then s ∗ b = s ∗ c if b ∗ a = c ∗ a, where s = neut(a);
- (9)
- if s = neut(a), q, t ∈${\{\}}_{anti{(a)}_{s}}$, then s∗ q = s∗ t.

**Proof.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Proposition**

**8.**

**Proof.**

**Proposition**

**9.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**3.**

**Example**

**13.**

**Definition**

**14.**

**Proposition**

**10.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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∗ | 0 | 1 | 2 |
---|---|---|---|

0 | 0 | 0 | 0 |

1 | 0 | 1 | 0 |

2 | 0 | 0 | {0, 2} |

∗ | 0 | 1 | 2 |
---|---|---|---|

0 | 0 | 0 | 0 |

1 | 0 | 0 | 0 |

2 | 0 | 0 | {0, 1} |

∗ | 0 | 1 | 2 | 3 |
---|---|---|---|---|

0 | 0 | 0 | 0 | {0,1,2,3} |

1 | 0 | 0 | 0 | {0,1,2,3} |

2 | 0 | 0 | {0,1} | {2,3} |

3 | {1,2,3} | {0,1,2,3} | {2,3} | {0,3} |

∗ | 0 | 1 | 2 |
---|---|---|---|

0 | 0 | 0 | 0 |

1 | 0 | 1 | {0, 1} |

2 | 0 | {0, 1} | {2} |

∗ | 0 | 1 | 2 |
---|---|---|---|

0 | 0 | 0 | 0 |

1 | 0 | 2 | 2 |

2 | 0 | {0,1,2} | {0,1,2} |

∗ | 0 | 1 | 2 |
---|---|---|---|

0 | 0 | 0 | 0 |

1 | 0 | 1 | 0 |

2 | 0 | 0 | {0 ,2} |

∗ | 0 | 1 | 2 |
---|---|---|---|

0 | 0 | {0,1,2} | {0,1,2} |

1 | 0 | {0 ,2} | {1 ,2} |

2 | {0,1,2} | {0 ,2} | {0,1,2} |

∗ | 0 | 1 | 2 |
---|---|---|---|

0 | {1,2} | {0,1,2} | {0,1,2} |

1 | {0,1,2} | {0,2} | {0,2} |

2 | {0,1} | {1,2} | {0,1} |

∗ | 0 | 1 | 2 |
---|---|---|---|

0 | 0 | 0 | 0 |

1 | 0 | 2 | 2 |

2 | 0 | {1,2} | {1,2} |

∗ | 0 | 1 | 2 |
---|---|---|---|

0 | 0 | {0, 1, 2} | {0, 1, 2} |

1 | 2 | 2 | { 1, 2} |

2 | {0, 1, 2} | {0, 2} | {0, 1, 2} |

∗ | 0 | 1 | 2 |
---|---|---|---|

0 | 0 | 1 | {0,1,2} |

1 | 1 | 0 | {0,1,2} |

2 | {0,1,2} | {0,1,2} | 2 |

∗ | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

0 | 0 | 1 | {0, 1, 2} | 0 | 4 |

1 | 1 | 0 | {0, 1, 2} | 1 | 4 |

2 | {0, 1, 2} | {0, 1, 2} | 2 | {0, 1, 2} | 4 |

3 | 0 | 1 | {0, 1, 2} | 3 | 4 |

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

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## Share and Cite

**MDPI and ACS Style**

Hu, M.; Smarandache, F.; Zhang, X.
On Neutrosophic Extended Triplet LA-hypergroups and Strong Pure LA-semihypergroups. *Symmetry* **2020**, *12*, 163.
https://doi.org/10.3390/sym12010163

**AMA Style**

Hu M, Smarandache F, Zhang X.
On Neutrosophic Extended Triplet LA-hypergroups and Strong Pure LA-semihypergroups. *Symmetry*. 2020; 12(1):163.
https://doi.org/10.3390/sym12010163

**Chicago/Turabian Style**

Hu, Minghao, Florentin Smarandache, and Xiaohong Zhang.
2020. "On Neutrosophic Extended Triplet LA-hypergroups and Strong Pure LA-semihypergroups" *Symmetry* 12, no. 1: 163.
https://doi.org/10.3390/sym12010163