** **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
(Z**_{8}, #),
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
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*.

**
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, **_{e}neut(a)>,
where
*a, *and* * its extended neutral* *
_{e}*neut(a) *
belong
to
D,
such that:

**
1)
**_{e}neut(a) may
be equal or
different from the unitary element of
*D*
with respect to the law
#
(if any);

**
2)
a#**_{e}neut(a) =
_{e}*neut(a)#a = a;
*

**
3) there is no
extended opposite
**_{e}*anti(a)* belonging to
*
D* for
which
*a#*_{e}*anti(a) = *
_{e}*anti(a)#a = *
_{e}*neut(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_{
e}neut(x) ∈
*D *and *no *_{e}anti(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, *_{e}neut(y>

**
included in ***D*,
such that *x = y *or* x = *_{e}neut(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, 17^{th} 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 **

**
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**