NEUTROSOPHIC DUPLET STRUCTURES

 

Abstract.

The Neutrosophic Duplets and the Neutrosophic Duplet Algebraic Structures were introduced

by Florentin Smarandache in 2016.

 

Let U be a universe of discourse, and a set D included in U, endowed with a well-defined law #.

1. Definiton of the Neutrosophic Duplet (ND).

We say that <a, neut(a)>, where a, and its neutral neut(a) belong to D, is a neutrosophic duplet if:

1) neut(a) is different from the unitary element of D with respect to the law # (if any);

2) a#neut(a) = neut(a)#a = a;

3) there is no opposite anti(a) belonging to D for which a#anti(a) = anti(a)#a = neut(a).

2. Example of Neutrosophic Duplets.

In (Z8, #), the set of integers with respect to the regular multiplication

modulo 8, one has the following neutrosophic duplets:

<2, 5 >, <4, 3>, <4, 5>, <4, 7>, and <6, 5>.

 Proof:

Let Z8 = {0, 1, 2, 3, 4, 5, 6, 7}, having the unitary element 1 with respect to

the multiplication # modulo 8.

2 # 5 = 5 # 2 = 10 = 2 (mod 8),

so neut(2) = 5 ≠ 1.

There is no anti(2) Z₈, because:

2 # anti(2) = 5 (mod 8),

or 2y = 5 (mod 8) by denoting anti(2) = y, is equivalent to:

2y - 5 = M8 {multiple of 8}, or 2y - 5 = 8k, where k is an integer, or

2(y - 4k) = 5, where both y and k are integers, or:

even number = odd number, which is impossible.

Therefore, we proved that <2, 5> is a neutrosophic duplet.

Similarly for <4, 5>, <4, 3>, <4, 7>, and <6, 5>.

A counter-example: <0, 0> is not a neutrosophic duplet, because it

is a neutrosophic triplet: <0, 0, 0>, where there exists an anti(0) = 0.

 

3. Definiton of the Neutrosophic Extended Duplet (NED).

Let U be a universe of discourse, and a set D included in U, endowed with a well-defined law #.

We say that <a, eneut(a)>, where a, and its extended neutral eneut(a) belong to D, such that:

1) eneut(a) may be equal or different from the unitary element of D with respect to the law # (if any);

2) a#eneut(a) = eneut(a)#a = a;

3) there is no extended opposite eanti(a) belonging to D for which a#eanti(a) = eanti(a)#a = eneut(a).

 

4. Definition of Neutrosophic Duplet Strong Set (NDSS).

A neutrosophic Duplet Strong Set is a set D, such that for any x D there is a neut(x)D and no anti(x) D.

 

5. Definition of Neutrosophic Duplet Weak Set (NDWS).

A neutrosophic Duplet Weak Set is a set D, such that for any x D there is a neutrosophic duplet <y, neut(y>

included in D, such that x = y or x = neut(y).

 

6. Definition of Neutrosophic Extended Duplet Strong Set (NEDSS).

A Neutrosophic Extended Duplet Strong Set is a set D, such that for any x D there is an eneut(x)D and no eanti(x) D.

 

7. Definition of Neutrosophic Extended Duplet Weak Set (NEDWS).

A Neutrosophic Extended Duplet Weak Set is a set D, such that for any x D there is a neutrosophic duplet <y, eneut(y>

included in D, such that x = y or x = eneut(y).

8. Definition of Neutrosophic Duplet Strong Structures (NDSStr).

Neutrosophic Duplet Strong Structures are structures defined on the

neutrosophic duplet strong sets.

 

9. Definition of Neutrosophic Duplet Weak Structures (NDWStr).

Neutrosophic Duplet Weak Structures are structures defined on the

neutrosophic duplet weak sets.

 

10. Definition of Neutrosophic Extended Duplet Strong Structures (NEDSStr).

Neutrosophic Extended Duplet Strong Structures are structures defined on the

neutrosophic extended duplet strong sets.

 

11. Definition of Neutrosophic Extended Duplet Weak Structures (NEDWStr).

Neutrosophic Extended Duplet Weak Structures are structures defined on the

neutrosophic extended duplet weak sets.

 

References

[1] F. Smarandache, Neutrosophic Theory and Applications, Le Quy

Don Technical University, Faculty of Information technology,

Hanoi, Vietnam, 17th May 2016.

[2] Florentin Smarandache, Neutrosophic Duplet Structures, Joint

Fall 2017 Meeting of the Texas Section of the APS, Texas Section

of the AAPT, and Zone 13 of the Society of Physics Students, 

The University of Texas at Dallas, Richardson, TX, USA,

October 20-21, 2017, 

http://meetings.aps.org/Meeting/TSF17/Session/F1.32

[3] F. Smarandache, Neutrosophic Perspectives: Triplets, Duplets, Multisets,

Hybrid Operators, Modal Logic, Hedge Algebras. And Applications. Pons

Editions, Bruxelles, 323 p., 2017;

CHAPTER IX: 127-134

Neutrosophic Duplets 127

Definition of Neutrosophic Duplet. 127

Example of Neutrosophic Duplets. 127

Neutrosophic Duplet Set and Neutrosophic Duplet Structures 130

Definition of Neutrosophic Duplet Strong Set 130

Definition of Neutrosophic Duplet Weak Set 130

Proposition. 131

Theorem 131

Example of Neutrosophic Duplet Strong Set 132-133