NEUTROSOPHIC TRIPLET STRUCTURES

 

The neutrosophic triplets and their algebraic structures were introduced

by Florentin Smarandache and Mumtaz Ali in 2014 - 2016.

 

1. Definition of Neutrosophic Triplet Set

Let U be a universe of discourse, and (N,*) a set of it, endowed with a well-defined

binary law. Then N is called a neutrosophic triplet set if for any x  N there exist a neutral of x,

denoted neut(x)∈  N, different from the classical algebraic unitary element, and an

opposite of x, called anti(x)∈  N, such that:

                                 x*neut(x) = neut(x)*x = x

and

                               x*anti(x) = anti(x)*x = neut(x).

 

2. Definition of Neutrosophic Triplet

A neutrosophic triplet is an object of the form <x, neut(x), anti(x)>, defined as above.

 

3. Definition of Neutrosophic Triplet Group

Let (N, *) be a neutrosophic triplet set. Then (N, *) is called a neutrosophic triplet group,

if the following classical axioms are satisfied.

1)       (N, *) is well-defined, i.e. for any x, y   N one has x*y N.

2)       (N, *) is associative, i.e.  for any x, y, z N one has x*(y*z) = (x*y)*z.

NTG, in general, is not a group in the classical way, because it may not have a classical

unitary element, nor classical inverse elements.
We consider, that the neutrosophic neutrals replace the classical unitary element,

and the neutrosophic opposites replace the classical inverse elements.

Theorem.

If <a, neut(a), anti(a)> is a neutrosophic triplet into a NTG with no zero-divisors and

a 0, then <anti(a), neut(a), a> and <neut(a), neut(a), neut(a)> are also neutrosophic

triplets.

4. Definition of Neutrosophic Triplet Ring

1) Neutrosophic Triplet Ring (NTR) is a set endowed with two binary laws (M, *, #),

such that:

a) (M, *) is a commutative neutrosophic triplet group;

which means that:
- M is a set of neutrosophic triplets with respect to the law *

(i.e. if x belongs to M, then neut(x) and anti(x), defined with respect to the law *,

also belong to M);

- the law * is well-defined, associative, and commutative on M

(as in the classical sense);

b) (M, #) is a set such that the law # on M is well-defined and associative

(as in the classical sense);

c) the law # is distributive with respect to the law *

(as in the classical sense).

 

4.1. Remarks on Neutrosophic Triplet Ring:

1) The Neutrosophic Triplet Ring is defined on the steps of the classical ring,

the only two distinctions are that:

- the classical unit element with respect to the law * is replaced

by neut(x) with respect to the law * for each x in M into the NTR;

- in the same way, the classical inverse element of an element x in M,

with respect to the law *, is replaced by anti(x) with respect to the law * in M.

2) A Neutrosophic Triplet Ring, in general, is different from a classical ring.

 

5. Definition of Neutrosophic Triplet Field

2) Neutrosophic Triplet Field (NTF) is a set endowed with two binary laws (M, *, #),

such that:

a) (M, *) is a commutative neutrosophic triplet group;

which means that:
- M is a set of neutrosophic triplets with respect to the law *

(i.e. if x belongs to M, then neut(x) and anti(x), defined with respect to the law *,

also both belong to M);

- the law * is well-defined, associative, and commutative on M

(as in the classical sense);

b) (M, #) is a neutrosophic triplet group;

which means that:

- M is a set of neutrosophic triplets with respect to the law #

(i.e. if x belongs to M, then neut(x) and anti(x), defined with respect to the law #,

also both belong to M);

- the law # is well-defined and associative on M

(as in the classical sense);

c) the law # is distributive with respect to the law *

(as in the classical sense).

 

5.1. Remarks on Neutrosophic Triplet Field:

1) The Neutrosophic Triplet Field is defined on the steps of the classical field,

the only four distinctions are that:

- the classical unit element with respect to the first law * is replaced

by neut(x) with respect to the first law * for each x in M into the NTF;

- in the same way, the classical inverse element of an element x in M,

with respect to the first law *, is replaced by anti(x) with respect to the first

law * in M;

- and the classical unit element with respect to the second law # is replaced

by neut(x) with respect to the second law # for each x in M into the NTF;

- in the same way, the classical inverse element of an element x in M,

with respect to the second law #, is replaced by anti(x) with respect to the

second law # in M;

2) A Neutrosophic Triplet Field, in general, is different from a classical field.

 

6. Hybrid Neutrosophic Triplet Field of Type 1.

It is a set F endowed with two laws * and # such that:

1: (F, *) is a commutative neutrosophic triplet group;

2: (F, #) is a classical group;

3: The law # is distributive over the law *.

 

7. Hybrid Neutrosophic Triplet Field of Type 2.

It is a set F endowed with two laws * and # such that:

1: (F, *) is a classical commutative group;

2: (F, #) is a neutrosophic triplet group; 

3: The law # is distributive over the law *.

 

8. Applications

This new field of neutrosophic triplet structures is important, because

it reflects our everyday life [it is not simple imagination!].

The neutrosophic triplets are based on real triads: (friend, neutral, enemy),

(positive particle, neutral particle, negative particle),

(yes, undecided, no), (pro, neutral, against), and in general (<A>, <neutA>, <antiA>)

as in neutrosophy.

 

References:

 

[1] Florentin Smarandache and Mumtaz Ali, Neutrosophic Triplet Group, Neural

Computing and Applications, Springer, 1-7, 2016,

https://link.springer.com/article/10.1007/s00521-016-2535-x;

DOI: 10.1007/s00521-016-2535-x.

 

[2] F. Smarandache, M. Ali, Neutrosophic triplet as extension of matter plasma,

unmatter plasma, and antimatter plasma, 69th annual gaseous electronics

conference, Bochum, Germany, Veranstaltungszentrum & Audimax,

Ruhr-Universitat, 1014 Oct. 2016,

http://meetings.aps.org/Meeting/GEC16/Session/HT6.111

 

[3] Florentin Smarandache, Mumtaz Ali, The Neutrosophic Triplet Group and its

Application to Physics, presented by F. S. to Universidad Nacional de Quilmes,

Department of Science and Technology, Bernal, Buenos Aires,

Argentina, 02 June 2014.

 

[4] F. Smarandache, M. Ali, Neutrosophic Triplet Field Used in Physical Applications,

(Log Number: NWS17-2017-000061), 18th Annual Meeting of the APS Northwest Section,

Pacific University, Forest Grove, OR, USA, June 1-3, 2017;

http://meetings.aps.org/Meeting/NWS17/Session/D1.1

 

[5] F. Smarandache, M. Ali, Neutrosophic Triplet Ring and its Applications,

(Log Number: NWS17-2017-000062), 18th Annual Meeting of the APS Northwest Section,

Pacific University, Forest Grove, OR, USA, June 1-3, 2017.

http://meetings.aps.org/Meeting/NWS17/Session/D1.2



[6] F. Smarandache, Neutrosophic Triplets, University of New Mexico,

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

 

                                                                                             [under press]