NEUTROSOPHIC TRIPLET STRUCTURES and NEUTROSOPHIC EXTENDED TRIPLET STRUCTURES

 

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

binary law *.

 

1. Definition of Neutrosophic Triplet (NT).

A neutrosophic triplet is an object of the form <x, neut(x), anti(x)>, for x  N, where

neut(x)  N is the neutral of x, different from the classical algebraic unitary element if any, such that:

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

and anti(x)N is the opposite of x such that:

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

In general, an element x may have more neut's and more anti's.

The neutrosophic triplets and their algebraic structures were first introduced

by Florentin Smarandache and Mumtaz Ali [1, 2, 3, 5, 6] in 2014 - 2016.

 

2. Definition of Neutrosophic Extended Triplet (NET).

A neutrosophic extended triplet is a neutrosophic triplet, defined as above, but where the neutral

of x {denoted by eneut(x) and called "extended neutral"} is allowed to also be equal to the classical

algebraic unitary element (if any). Therefore, the restriction "different from the classical algebraic

unitary element if any" is released.

As a consequence, the "extended opposite" of x, denoted by eanti(x), is also allowed to be equal

to the classical inverse element from a classical group.

Thus, a neutrosophic extended triplet is an object of the form <x, eneut(x), eanti(x)>, for xN, where

eneut(x)N is the extended neutral of x, which can be equal or different from the classical algebraic

unitary element if any, such that:

                                 x*eneut(x) = eneut(x)*x = x

and anti(x)N is the extended opposite of x such that:

                               x*eanti(x) = eanti(x)*x = eneut(x).

In general, for each x∈N there are many exist eneut's and eanti's.

The neutrosophic extended triplets were introduced by Smarandache [4, 7] in 2016.

 

3. Definition of Neutrosophic Triplet (Strong) Set (NTSS).

The set N is called a neutrosophic triplet (strong) set if for any x  N there exist neut(x)  N

and anti(x)  N.

 

4. Definition of Neutrosophic Extended Triplet (Strong) Set (NETSS).

The set N is called a neutrosophic extended triplet (strong) set if for any x  N there exist

eneut(x)  N and eanti(x)  N.

 

5. Definition of Neutrosophic Triplet Weak Set (NTWS).

The set N is called a neutrosophic triplet weak set if for any x  N there exist a neutrosophic

triplet <y, neut(y), anti(y)> included in N, such that x = y or x = neut(y) or x = anti(y).

 

6. Definition of Neutrosophic Extended Triplet Weak Set (NETWS).

The set N is called a neutrosophic extended triplet weak set if for any xN there exist a

neutrosophic extended triplet <y, eneut(y), eanti(y)> included in N, such that x = y or x = eneut(y)

or x = eanti(y).

 

7. Theorem 1.

a) A neutrosophic triplet strong set is also a neutrosophic triplet weak set, but not conversely.

b) A neutrosophic extended triplet strong set is also a neutrosophic extended triplet weak set,

but not conversely.

 

8. Definition of Neutrosophic (Strong) Triplet Group (NTG)

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

 

9. Definition of Neutrosophic Extended (Strong) Triplet Group (NETG)

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

(strong) 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.

NETG, 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 extended neutrals replace the classical unitary element,

and the neutrosophic extended opposites replace the classical inverse elements.

In the case when NETG includes a classical group, then NETG enriches the structure of

a classical group, since there may be elements with more extended neutrals and more

extended opposites.

 

10. Definition of Neutrosophic Triplet Ring (NTR)

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

such that:

a) (N, *) is a commutative neutrosophic triplet (strong) group;

which means that:
- N is a set of neutrosophic (strong) triplets with respect to the law *

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

also belong to N); we use the notations neut*(.) and respectively anti*(.) to mean

with respect to the law *;

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

(as in the classical sense);

b) (N, #) is a set such that the law # on N 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).

 

11. Definition of Neutrosophic Extended Triplet Ring (NETR)

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

such that:

a) (N, *) is a commutative neutrosophic extended triplet group;

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

(i.e. if x belongs to N, then eneut*(x) and eanti*(x), defined with respect to the law *,

also belong to N);

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

(as in the classical sense);

b) (N, #) is a set such that the law # on N 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).

 

12. 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 N into the NTR;

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

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

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

 

13. Remarks on Neutrosophic Extended Triplet Ring:

1) Similarly, The Neutrosophic Exteded 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 extended to

eneut*(x) with respect to the law * for each x in N into the NETR;

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

with respect to the law *, is extended to eanti*(x) with respect to the law * in N.

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

 

14. Definition of Hybrid Neutrosophic Triplet Ring (HNTR)

 The Hybrid Neutrosophic Triplet Ring is a set N endowed with two binary

laws (N, *, #), such that:

a) (N, *) is a commutative neutrosophic triplet (strong) group; which means that:

- N is a neutrosophic triplets strong set with respect to the law * (i.e. if x belongs to N,

then neut*(x) and anti*(x), defined with respect to the law *, also belong to N);

- the law * is well-defined, associative, and commutative on N (as in the classical sense);

b) (N, #) is a neutrosophic triplet strong set with respect to the law # (i.e. if x belongs to N,

then neut#(x) and anti#(x), defined with respect to the law #, also belong to N);

- the law # is well-defined and non-associative on N (as in the classical sense);

c) the law # is distributive with respect to the law * (as in the classical sense).

 

15. Definition of Hybrid Neutrosophic Extended Triplet Ring (HNETR)

 The Hybrid Neutrosophic Extended Triplet Ring is a set N endowed with two binary

laws (N, *, #), such that:

a) (N, *) is a commutative neutrosophic extended triplet (strong) group; which means that:

- N is a neutrosophic extended triplet strong set with respect to the law * (i.e. if x belongs to N,

then eneut*(x) and eanti*(x), defined with respect to the law *, also belong to N);

- the law * is well-defined, associative, and commutative on N (as in the classical sense);

b) (N, #) is a neutrosophic extended triplet strong set with respect to the law # (i.e. if x belongs

to N, then eneut#(x) and eanti#(x), defined with respect to the law #, also belong to N);

- the law # is well-defined and non-associative on N (as in the classical sense);

c) the law # is distributive with respect to the law * (as in the classical sense).

 

16. Remarks on Hybrid Neutrosophic Triplet Ring

a) A Hybrid Neutrosophic Triplet Ring is a field (N, *, #) from which there

has been removed the associativity of the second law #.

b) Or, Hybrid Neutrosophic Triplet Ring is a set (N, *, #), such that (N, *) is a

commutative neutrosophic triplet group, and (N, #) is a neutrosophic triplet loop,

and the law # is distributive with respect to the law * (as in the classical sense).

 

17. Remarks on Hybrid Neutrosophic ExtendedTriplet Ring

a) A Hybrid Neutrosophic Extended Triplet Ring is a field (N, *, #) from which there

has been removed the associativity of the second law #.

b) Or, Hybrid Neutrosophic Extended Triplet Ring is a set (N, *, #), such that (N, *) is a

commutative neutrosophic extended triplet group, and (N, #) is a neutrosophic extended

triplet loop, and the law # is distributive with respect to the law * (as in the classical sense).

 

18. Definition of Neutrosophic Triplet Field (NTF)

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

such that:

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

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

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

also both belong to N);

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

(as in the classical sense);

b) (N, #) 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 N, then neut#(x) and anti#(x), defined with respect to the law #,

also both belong to N);

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

(as in the classical sense);

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

(as in the classical sense).

 

19. Definition of Neutrosophic Extended Triplet Field (NETF)

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

such that:

a) (N, *) is a commutative neutrosophic extended triplet group;

which means that:
- N is a neutrosophic extended triplet set with respect to the law *

(i.e. if x belongs to N, then eneut*(x) and eanti*(x), defined with respect to the law *,

also both belong to N);

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

(as in the classical sense);

b) (N, #) is a neutrosophic extended triplet group;

which means that:

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

(i.e. if x belongs to N, then eneut#(x) and eanti#(x), defined with respect to the law #,

also both belong to N);

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

(as in the classical sense);

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

(as in the classical sense).

 

20. 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 extended to

eneut*(x) with respect to the first law * for each x in N into the NTF;

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

with respect to the first law *, is extended to eanti*(x) with respect to the first

law * in N;

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

eneut#(x) with respect to the second law # for each x in N into the NTF;

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

with respect to the second law #, is extended to eanti#(x) with respect to the

second law # in N;

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

 

21. Hybrid Neutrosophic Triplet Field of Type 1 (HNTF1).

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

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

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

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

 

22. Hybrid Neutrosophic Extended Triplet Field of Type 1 (HNETF1).

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

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

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

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

 

23. Hybrid Neutrosophic Triplet Field of Type 2 (HNTF2).

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

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

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

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

 

24. Hybrid Neutrosophic Triplet Field of Type 2 (HNETF2).

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

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

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

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

 

25. Applications of Neutrosophic Triplet Structures (NTS)

and Neutrosophic Extended Triplet Structures (NETS)

This new fields of Neutrosophic Triplet Structures and Neutrosophic Extended Triplet

Structures are very important, because they reflect our everyday life [they are not simple imagination!].

The neutrosophic triplets and neutrosophic extended triplets are based on real triads:

(friend, neutral, enemy), (positive particle, neutral particle, negative particle), (yes, unclear, no),

(pro, neutral, against), (victory, tie game, defeat), (taking a decision, undecided, not taking a decision),

(accept, pending, reject),  and in general (<A>, <neutA>, <antiA>) as in neutrosophy,

which is a new branch of philosophy generalizing the dialectics.

 

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, Neutrosophic Extended Triplets, Arizona State University,

Tempe, AZ, Special Collections, 2016.

[5] 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

[6] 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

[7] Florentin Smarandache, Seminar on Physics (unmatter, absolute theory of relativity,

general theory distinction between clock and time, superluminal and instantaneous physics,

neutrosophic and paradoxist physics), Neutrosophic Theory of Evolution, Breaking Neutrosophic

Dynamic Systems, and Neutrosophic Triplet Algebraic Structures, Federal University of

Agriculture, Communication Technology Resource Centre, Abeokuta,

Ogun State, Nigeria, 19th May 2017.

[8] F. Smarandache, Hybrid Neutrosophic Triplet Ring in Physical Structures, Annual

Meeting of the APS Four Corners Section, Fort Collins, CO, USA,  & Bulletin of

the American Physical Society, October 2021, 2017;

http://meetings.aps.org/Meeting/4CF17/Session/G1.33

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

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

Editions, Bruxelles, second edition, 323 p., 2017;

http://fs.gallup.unm.edu/neutrosophicPerspectives-ed2.pdf

CHAPTER VII: 74-110

Neutrophic Triplets 74

Definition of Neutrosophic Triplet 74

Example of Neutrosophic Triplet 74

Definition of Neutrosophic Triplet Strong Set (or Neutrosophic Triplet Set) 75

Example of Neutrosophic Triplet Strong Set 75

Definition of Neutrosophic Triplet Weak Set 76

Theorem 76

Definition of Neutrosophic Triplet Strong Group (or Neutrosophic Triplet Group) 77

Example of Neutrosophic Triplet Strong Group 78 Florentin Smarandache Neutrosophic Perspectives

Proposition 79

Definition of Neutrosophic Perfect Triplet 79

Definition of Neutrosophic Imperfect Triplet 79

Examples of Neutrosophic Perfect and Imperfect Triplets 79

Definition of Neutrosophic Triplet Relationship of Equivalence 82

Example of Neutrosophic Triplet Relationship of Equivalence 83

Example of Neutrosophic Perfect and Imperfect Triplets 84

Example of Neutrosophic Perfect and Imperfect Triplets 84

Example of Non-Associative Law 85

Definition of Neutrosophic Enemy of Itself 87

Definition of Two Neutrosophic Friends 87

Definition of n≥2 Neutrosophic Friends 88

Proposition 89

Example of Neutrosophic Friends 89

Neutrosophic Triplet Function 90

Theorems on Neutrosophic Triplets 91

Definition 1 91

Definition 2 91

Definition 3 91

Definition 4 92

Theorem 1 92

Theorem 2 94

Counter-Example 1 96

Theorem 3 97

Definition of Neutro-Homomorphism. 98

Example 99

Definition 6 100

Proposition 1 100

Proposition 2 100

Theorem 4 102

Theorem 5 103

Theorem 6 104

Neutrosophic Triplet Group vs. Generalized Group 105

Neutrosophic Triplet Multiple Order 109

CHAPTER VIII: 111-126

Neutrosophic Triplet Ring 111

Definition of Neutrosophic Triplet Ring 111

Hybrid Neutrosophic Triplet Ring 113

Definition 113

Hybrid Neutrosophic Triplet Ring of Second Type 115

Definition 115

Neutrosophic Triplet Field 117

Definition 117

Example of Neutrosophic Triplet Ring which is not a Neutrosophic Triplet Field. 119

Hybrid Neutrosophic Triplet Field 121

Hybrid Neutrosophic Triplet Field of Type 1. 121

Hybrid Neutrosophic Triplet Field of Type 2. 121

Neutrosophic Triplet Loop 122

Neutrosophic Triplet Structures 124-125