** **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
(Z**_{8}, #),
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
Z**_{8 }=_{ }{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 = M*_{8}
{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:

**
e***ven 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, 17^{th} 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]**

**
**