NEUTROSOPHIC DUPLET STRUCTURES

 

The Neutrosophic Duplets and their algebraic structures were introduced

by Florentin Smarandache in 2016.

 

Definiton of the Neutrosophic Duplet.

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

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

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

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

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

 

Example of Neutrosophic Duplets.

In (Z8, #), the set of integers modulo 8; with respect to the

regular multiplicationone 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.

 

Definition of Neutrosophic Duplet Structures.

Neutrosophic Duplet Structures are structures defined on the

sets of neutrosophic duplets.

 

References

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

Don Technical University, Faculty of Information technology,

Hanoi, Vietnam, 17th May 2016.

[2] F. Smarandache, Neutrosophic Duplet Structures, 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, Texas, 2017.
 

[3] F. Smarandache, Neutrosophic Duplets, University of New Mexico,

Gallup Campus, USA, http://fs.unm.edu/NeutrosophicDuplets.htm

 

                                                                                             [under press]