# Neutrosophic Triplet Group (revisited).

1 Introduction

Neutrosophy is a new branch of philosophy that studies the nature, origin and scope of neutralities as well as their interaction with ideational spectra. Florentin Smarandache [8] in 1995, first introduced the concept of neutrosophic logic and neutrosophic set where each proposition in neutrosophic logic is approximated to have the percentage of truth in a subset T, the percentage of indeterminacy in a subset I, and the percentage of falsity in a subset F so that this neutrosophic logic is called an extension of fuzzy logic especially of the intuitionistic fuzzy logic. In fact neutrosophic set is the generalization of classical sets [9], fuzzy set [12], intuitionistic fuzzy set [1,9], and interval valued fuzzy set[9] etc. This mathematical tool is used to handle problems consisting uncertainty, imprecision, indeterminacy, inconsistency, incompleteness and falsity. By utilizing the idea of neutrosophic theory, Vasantha Kandasamy and Florentin Smarandache studied neutrosophic algebraic structures in [4,5,6] by inserting an indeterminate element "I" in the algebraic structure and then combine "I" with each element of the structure with respect to corresponding binary operation *. They call it neutrosophic number {a + bI, with a, b real numbers, and I = literal indeterminacy, [I.sup.2] = I} and the generated algebraic structure is then termed as neutrosophic algebraic structure. They further study several neutrosophic algebraic structures such as neutrosophic fields, neutrosophic vector spaces, neutrosophic groups, neutrosophic bigroups, neutrosophic N-groups, neutrosophic semigroups, neutrosophic bisemigroups, neutrosophic N-semigroup, neutrosophic loops, neutrosophic biloops, neutrosophic N-loop, neutrosophic groupoids, and neutrosophic bigroupoids and so on.

Groups [2,3,11] are very important in algebraic structures because they play the role of a backbone in almost all algebraic structures theory. Groups are thought as old algebra due to its rich structure than any other notion. In many algebraic structures, groups provide concrete foundation such as, rings, fields, vector spaces, etc. Groups are also important in many other areas like physics, chemistry, combinatorics, biology etc. to study the symmetries and other behavior among their elements. The most important aspect of a group is group action. There are many types of groups, such as: permutation groups, matrix groups, transformation groups, Lie-groups etc. that are highly used as a practical perspective in our daily life. Generalized groups [7] are important in this aspect.

In this paper, for the first time, we introduced the idea of neutrosophic triplet. The newly born neutrosophic triplets are highly dependable on the proposed binary operation. These neutrosophic triplets have been discussed by Smarandache and Ali in Physics [10]. Moreover, we used these neutrosophic triplets to introduce neutrosophic triplet group, which is different from the classical group both in structural and foundational properties from all aspects. Furthermore, we gave some interesting and fundamental properties and notions with illustrative examples. We also introduced a new type of homomorphism called as neutrohomomorphism, which is in fact a generalization of the classical homomorphism under some conditions. We also study neutro-homomorphism for neutrosophic triplet groups. The rest of the paper is organized as follows. After the literature review in section 1, we introduced neutrosophic triplets in section 2. Section 3 is dedicated to the introduction of neutrosophic triplet groups with some of its interesting properties. In section 4, we developed neutro-homomorphism and in section 5, we gave distinction and comparison of neutrosophic triplet group with the Molaei's generalized group. We also draw a brief sketch of the possible applications of neutrosophic triplet group in other research areas. Conclusion is given in section 6.

2 Neutrosophic Triplet

Remark 2.1. All below theorems and propositions in a Neutrosophic Triplet Set (NTS) and Neutrosophic Triplet Group (NTG) are true when the multipliers are non-zero and cancellable multipliers.

An element a [member of] (S, *), where * is a binary law, is cancellable to the left if:

[for all] b, [member of] S, from a*b = a*c one gets only b = c.

The element a is cancellable to the right if:

[for all] b, [member of] S, from b*a = c*a one gets only b = c.

And, the element a is cancellable (in general) if the element a is both cancelable to the left and to the right. Definition 2.1.1. Let N be a set together with a binary operation *. Then N is called a neutrosophic triplet set if for any a [member of] N, there is a neutral of "a" called neut(a), different from the classical algebraic unitary element, and an opposite of "a" called anti(a), with neut(a) and anti(a) belonging to N, such that:

a * neut(a) = neut(a) *a = a

and

a * anti(a) = anti(a) * a =

The elements a, neut(a), and anti(a) are collectively called as neutrosophic triplet and we denote it by (a, neut(a), anti(a)). By neut(a), we mean neutral of a and apparently, a is just the first coordinate of a neutrosophic triplet and not a neutrosophic triplet.

For the same element a in N, there may be more neutrals to it neut(a) and more opposites of it anti(a).

Remark 2.2

If a well-defined binary law * on the set N has a classical algebraic unitary element e in N, then no other triplet of the form (e, b, c) can be formed, except the (e, e, e), i.e. when b = c = e, which is not accepted as neutrosophic triplet.

Consequently, the set (N, *) with a classical unitary element cannot be a neutrosophic triplet set.

Remark 2.2.

It is important that there are at least two different neutral elements with respect to all set elements into a neutrosophic triplet set.

Definition 2.1.3. A Zero Neutrosophic Triplet on the neutrosophic triplet set N, is a neutrosophic triplet of the form (0, 0, a), where 0, a e N {of course, the triplet (0, 0, a) must satisfy the axioms of the neutrosophic triplet}.

Example 2.1.3.1. Let N=be a set with respect to multiplication x modulo 10 in a [member of] N, 6 x a = a x 6 = a(mod 10).

It should be remarked, to this example, that 6 is a classical algebraic unitary element on N, with respect to the multiplication x modulo 10, because for any a [member of] N, 6 x a = a x 6 = a (mod 10).

But 6 cannot be a neutral for the element 0 [member of] N, because (0, 6, ?) cannot form a neutrosophic triplet since there is no anti(0) such that:

0 x anti(0) = anti(0) x 0 = 6.

Therefore, the neutrosophic triplets of 0 [called Zero Neutrosophic Triplets] are (0, 0, 0), (0, 0, 2), (0, 0, 4), (0, 0, 6), (0, 0, 8).

N is not a neutrosophic triplet set since, except element 0, the other elements 2, 4, 6, and 8 do not have neutral elements different from the classical unitary element 6.

Theorem 2.1. Let N be a set endowed with the binary law *, which is well-defined and has the classical algebraic unitary element e [member of] N,

[for all] x [member of] N, e * x = x * e = x.

If (e, b, c) is a neutrosophic triplet, with b, c [member of] N, then b = c = e.

{In other words, if a set TV has a classical algebraic unitary element e, with respect to the binary well-defined law *, then the only neutrosophic triplet of e is (e, e, e), which is mutually called trivial neutrosophic triplet, the only triplet that makes exception from the definition of neutrosophic triplets.}

Proof.

Let (e, b, c) be a neutrosophic triplet. Since neut(e) = b, one has:

e *neut(e) = neut(e) *e = e,

but e*b = b and b*e = b too (since e is the classical algebraic unitary element on the set N),

whence b = e.

And, because anti(e) = c, one has:

e*c = c*e = e,

but e*c = c and c*e = c too (since e is the classical algebraic unitary element on the set N),

whence c = e.

Therefore, the only triplet of the classical algebraic unitary (identity) element is (e, e, e), but it cannot be considered a neutrosophic triplet.

Definition 2.2: The element b in (N, *) is the second component, denoted as neut(*) of a neutrosophic triplet, if there exist other elements a and c in N such that a * b = b * a = a and a * c = c * a = b. The formed neutrosophic triplet is (a, b, c).

Definition 2.3: The element c in (N,*) is the third component, denoted as neut(*), of a neutrosophic triplet, if there exist other elements a and c in N such that a * b = b * a = a and a * c = c * a = b. The formed neutrosophic triplet is (a, b, c).

Example 2.2. Consider Z under multiplication modulo 6 , where

[Z.sub.6] = {0,1,2,3,4,5}

The classical unitary element is e = 1.

Then 2 gives rise to a neutrosophic triplet because neut(2) = 4, as 2 x 4 = 8. Also anti(2) = 2 because 2 x 4 = 4. Thus (2,4,2) is a neutrosophic triplet. Similarly 4 gives rise to a neutrosophic triplet because neut(4) = anti(4) = 4. So (4,4,4) is a neutrosophic triplet. 3 has two neutrals, neut(3) = {3,5}, and forms one neutrosophic triplet (3, 3, 3), but 3 does not give rise to a neutrosophic triplet for neut(3) = 5 since anti(3) does not exist in [Z.sub.6] for this neutral,

5 has no neut(5) so no neutrosophic triplet related to 5, and last but not the least 0 gives rise to a trivial neutrosophic triplet as neut(0) = anti(0) = 0. The zero neutrosophic triplets are denoted by(0,0,0), (0, 0, 1), (0, 0, 2), (0, 0, 3), (0, 0, 4), (0, 0, 5).

[Z.sub.6] is not a neutrosophic set, since 1 and 5 have no corresponding neutrosophic triplets, but [M.sub.6] = {0,2,3,4} [subset] [Z.sub.6] is a commutative neutrosophic group [whose definition will be provided below].

Theorem 2.3. If (a, neut(a), anti(0)) form a neutrosophic triplet, then

1. (anti(a),neut(a),a) also form a neutrosophic triplet, and similarly

2. (neut(a), neut(a), neut(a)) form a neutrosophic triplet.

Proof. We prove both 1 and 2.

1. Of course, anti(a) * a = neut(a).

We need to prove that:

anti(a) * neut(a) = anti(a)

Multiply by a to the left and we get:

a * anti(a) * neut(a) = a * anti(a) Or

[a * anti(a)] * neut(a) = neut(a) Or

neut(a) * neut(a) = neut(a)

Again multiply by a to the left and we get:

a * neut(a) * neut(a) = a * neut(a)

Or

[a * neut(a)] * neut(a) = a

Or

a * neut(a) = a

2. To show that (neut(a), neut(a), neut(a)) is a neutrosophic triplet, it results from the fact that neut(a) * neut(a) = neut(a).

3 Neutrosophic Triplet Group

Definition 3.1: Let (N,*) be a neutrosophic triplet set (which includes the trivial neutrosophic triplet too, if any). Then N is called a neutrosophic triplet group, if the following conditions are satisfied.

1) If (N,*) is well-defined, i.e. for any a, b [member of] N, one has a * b [member of] N.

2) If (N,*) is associative, i.e. (a*b)*c= a*(b*c) for all a.b.c [member of] N.

The neutrosophic triplet group, in general, is not a group in the classical algebraic way.

We consider, as the neutrosophic neutrals replacing the classical unitary element, and the neutrosophic opposites as replacing the classical inverse elements.

Example 3.2. Consider([Z.sub.10], #), where # is defined as a#b = 3ab

Let [M.sub.10] = {0, 2, 4, 5, 6, 8} [subset] [Z.sub.10]. Then ([M.sub.10],#) is a neutrosophic triplet group under the binary #. It is also associative, i.e.

(a#b)#c=a#(b#c).

Now take L. H. S to prove the R. H. S, so

a#(b#c)=3ab#c.

3(3ab)c=9abc,

3a(3bc)=3a(b#c),

a#(b#c).

The classical unitary element on [Z.sub.10] with respect to the law # is e = 7, since:

a # e = e # a = 3ae = 3a(7) = 21a = a (mod 10) for any a [member of] [Z.sub.10].

Therefore, we choose all triplets whose neutral elements are different from 7, and we get the following neutrosophic triplets:

(0, 0, 0), (0, 0, 2), (0, 0, 4), (0, 0, 5), (0, 0, 6), (0, 0, 8), (2, 2, 2), (4, 2, 6), (5, 5, 5), (6, 2, 4), and (8, 2, 8).

All above neutrals neut(.) = 0, 2, and 5 are different from the classical unitary element 7.

[Z.sub.10] is not a neutrosophic triplet group, nor even a neutrosophic triplet set.

But its subset [M.sub.10] = {0, 2, 4, 5, 6, 8} is a commutative neutrosophic triplet group, since the law # is well-defined, commutative, associative, and each element belonging to M has a corresponding neutrosophic triplet.

Definition 3.3: Let (N,*) be a neutrosophic triplet group. Then N is called a commutative neutrosophic triplet group if for all a, b [member of] N we have a * b = b * a.

Example 3.4. Consider (M, *), where M = {0, 1}, and the binary law * is defined as a*b = a + b - ab (mod 4) for all a, b [MEMBER OF] M.

Then (M, *) is a not a neutrosophic triplet group, not even a neutrosophic triplet set. Proof.

The law * has a classical algebraic unitary element e = 0, since:

For any a [member of] M, a*0 = 0*a = a + 0- ax0 = a (mod 4).

Therefore, (M, *) cannot be a neutrosophic triplet group.

Theorem 3.5. Every idempotent element gives rise to a neutrosophic triplet.

Proof. Let a be an idempotent element. Then by definition [a.sup.2] = a. Since [a.sup.2] = a, which clearly implies that neut(a) = a and anti(a) = a. Hence a gives rise to a neutrosophic triplet (a, a, a).

Theorem 3.6. There are no neutrosophic triplets in [Z.sub.n] with respect to multiplication modulo n if n is a prime, except the zero neutrosophic triplets (0, 0, 0), (0, 0, 1), ..., (0, 0, n-1).

Proof. It is obvious. The multiplication modulo n is well-defined, associative, and commutative.

For n = 2 (even prime), [Z.sub.2] = {0, 1} has the classical algebraic unitary element, with respect to multiplication modulo 2, e = 1, and [Z.sub.2] has the zero neutrosophic triplets (0, 0, 0), (0, 0, 1).

Whence [Z.sub.2] is a not a neutrosophic triplet group, not even a neutrosophic triplet set.

Let [Z.sub.n] = {0, 1, 2, ..., n-1}, for n odd prime. The classical algebraic unitary element of [Z.sub.n] is 1, and the zero neutrosophic triplets are (0, 0, 0), (0, 0, 1), ..., (0, 0, n-1).

Let's compute the neutral of 2 [less than or equal to] p [less than or equal to] p - 1, if any, let neut(p) = x. We need to find x. px = p (mod n), or px-p = 0 (mod n), or p(x-1) = 0 (mod n),

whence x-1 = 0 (mod n) since n is an odd prime, and n and p are relatively prime numbers, or x = 1 (mod n), therefore there is no neutral of the elements p [member of] {2, 3, ..., n-1}, since 1 is excluded as classical algebraic unitary element. Thus no neutrosophic triplets corresponding to the elements 2, 3, ..., n-1 [member of] [Z.sub.n].

Remark 3.6.1. Let (N,*) be a neutrosophic triplet group under * and let a [member of] N. Then

neut(a) is not the same for all elements in N (as in classical group), but neut(a) depends on the a and on the operation *.

(In example 3.8, neut(0) = 0, neut(4) = 4, neut(8) = 4 and neut(9) = 9, so we have three different neutral elements: 0, 4, and 9.)

Theorem number 3.6.2. Let (N,*) be a neutrosophic triplet group under * that satisfies the cancellation law for all its elements. Then, for any a in N, the neut(a) and anti(a) are unique and depend on a.

Proof: Suppose [neut.sup.(1)] (a) and [neut.sup.(2)] (a) be two neutrals of a.

Then since [neut.sup.(1)](a)*a= [neut.sup.(2)] (a)*a, and by cancellation law to the right-hand side, we have [neut.sup.(1)](a)= [neut.sup.(2)] (a).

Similarly, if a*[neutf.sup.(1)](a)= a*[neut.sup.(2)] (a), by cancellation law to the left-hand side, we have [neut.sup.(1)](a)= [neu.sup.(2)] (a).

In the same way, since [anti.sup.(1)](a)*a= [anti.sup.(2)] (a)*a, we get [anti.sup.(1)](a)= [anti.sup.(2)](a), by cancellation law to the right-hand side.

Again, since a*[anti.sup.(1)](a) = a*[anti.sup.(2)](a), we get [anti.sup.(1)](a)= [anti.sup.(2)](a), by cancellation law to the left-hand side.

Theorem 3.6.3. If the elements a of NTG do not satisfy the cancellation law, then still for each a in NTG the neut(a) is unique and depending on a, but the anti(a) may not be unique.

Proof.

Let's suppose that [neut.sup.(1)](a) and [neut.sup.(2)](d) are two neutrals of a, we have [neut.sup.(1)](a) =a *[anti.sup.(1)] (a) =([neut.sup.(2)](a) *a) * [anti.sup.(1)](a)

=[neut.sup.(2)](a) *(a* [anti.sup.(1)](a))

=[neut.sup.(2)](a) * [neut.sup.(1)](a)

=([anti.sup.(2)](a) *a) * [neut.sup.(1)](a)

=[anti.sup.(2)](a) *(a*[neut.sup.(1)](a))

=[anti.sup.(2)](a)*a

=[neut.sup.(1)](a).

Yet, anti(a) is not unique.

To prove this, let's take a look at the following example.

Example 3.8. Let N=(0,4,8) be a commutative neutrosophic triplet group under multiplication modulo 12 in ([Z.sub.12], N). N does not have a classical unitary element.

Then neut(4) = 4, neut(8) = 4 and anti(0) == {0,4,8,9}, neut(0) = 0 and anti(0) = {0, 4, 8, 9}. This shows that neut(a) is not the same for all elements as in classical group theory; further, each element has only one neutral. Also, the element 0 has four anti(0) 's.

The neutrosophic triplets are: (0, 0, 0), (0, 0, 4), (0, 0, 8), (0, 0, 9), (4, 4, 4), (8, 4, 8), (9, 9, 9).

Remark 3.9. Let (N,*) be a neutrosophic triplet group with respect to * and let a [member of] N. Then anti(a) is not the same for all elements in N and also anti(N) depends on the element a and on the operation *, and some elements may have many anti's unlike classical group and generalized group.

To prove the above remark, let's take a look to the following example.

Example 3.10. Let N be the commutative neutrosophic triplet group in the above Example 3.8. Then anti(0) = {0, 4, 8, 9}, anti(4)=4 , anti(8)=8 and anti(9)=9. Therefore, the element 0 has four anti's, and the anti's of the elements are different from each other: 4, 8, 9, and 0.

Proposition 3.11. Let (N, *) be a neutrosophic triplet group with respect to * and let A

a, b, c [member of] N. Then

1) a*b=a*c if and only if neut(a)*b=neut(a)*c.

2) b*a = c*a if and only if b*neut(a)=c*neut(a).

Proof. 1. Suppose that a*b=a*c. Since N is a neutrosophic triplet group, so anti(a) [member of] N. Multiply anti(a) to the left side of a*b=a*c.

anti(a)*a*b=anti(a)*a*c

[anti(a)*a]*b=[anti(a)*a]*c

neut(a)*b=neut(a)*c

Conversely suppose that neut(a)*b=neut(a)*c.

Multiply a to the left side, we get:

[a * neut(a)] * b = [a * neut(a)] * c.

a * b = a * c

2. The proof is similar to 1.

Proposition 3.12. Let (N,*) be a neutrosophic triplet group with respect to * and let a,b,c [member of] N.

1) If anti(a)*b=anti(b)*c, then neut(a)*b=neut(a)*c.

2) If b*anti(a), then c*anti(a), thenb*neut(a)=c*neut(a).

Proof. 1-. Suppose that anti(a)*b=anti(a)*c. Since N is a neutrosophic triplet group with respect to *, so a [member of] N. Multiply a to the left side of anti(a)*b=anti(a)*c, we get:

a*anti(a)*b=a*anti(a)*c

[a*anti(a)]*b=[a*anti(a)]*c

neut(a)*b=neut(a)*c.

2. The proof is the same as (1).

Theorem 3.13. Let (N,*) be a commutative neutrosophic triplet group with respect to * and a, b, n [member of] N. Then neut(a)*neut(b)=neut(a*b).

Proof. Consider left hand side,neut(a)*neut(b)=neut(a*b).

Now multiply to the left with a and to the right with b, we get:

a*neut(a*b)*b=[a*b]*[neut(a*b)], as * in asociative =a*b.

Now consider right hand side, we have neut(a*b).

Again multiply to the left with a and to the right with b, we get:

a*neut(a*b)*b, as is associative, =a*b.

This completes the proof.

Theorem 3.14. Let (N,*) be a commutative neutrosophic triplet group with respect to * and a, b [member of] N. Then anti(a)*anti(b)=anti(a*b).

Proof. Consider left hand side, anti(a)*anti(b).

Multiply to the left with a and to the right with b, we get:

a*anti(a) *anti(b) *b=[a*anti(a)] * [anti(b)*b]

=neut(a)*neut(b)

=neut(a*b), from the above theorem.

Now consider right hand side, which is anti(a*b).

Multiply to the left with a and to the right with b, we get:

a*anti(a*b), since * is associative. neut(a*b).

This shows that anti(a)*anti(b) is true for all a, b [member of] N.

Theorem 3.15. Let (N,*) be a commutative neutrosophic triplet group under * and a, b [member of] N. Then

1) neut(a)*neut(b)=neut(b)*neut(a).

2) anti(a)*anti(b)=anti(b)*anti(a).

Proof 1. Consider right hand side neut(b)*neut(a). By Theorem 3, we have

neut(b)*neut(a)=neut(b)*neut(a), as N is commutative, neut(a)*neut(b), again by Theorem 3.

Hence neut(a)*neut(b)=neut(b)*neut(a).

2) On similar lines, one can easily obtained the proof of (2).

{Actually, both proofs could also result straightforwardly from the commutative property of the neutrosophic triplet group.}

Definition 3.16. Let (N,*) be a neutrosophic triplet group under * and let H be a subset of N. Then H is called a neutrosophic triplet subgroup of N if H itself is a neutrosophic triplet group with respect to*.

Proposition 3.18. Let (N,*) be a neutrosophic triplet group and let a [member of] N be a subset of N. Then H is a neutrosophic triplet subgroup of N if and only if the following conditions hold.

1) a*b [member of] H for all a, b [member of] H.

2) neut(a) [member of] H for all a [member of] H.

3) anti(a) [member of] H for all a [member of] H.

Proof. The proof is straightforward.

Definition 3.19. Let N be a neutrosophic triplet group and let a [member of] N. A smallest positive integer n [greater than or equal to] 1 such that is called neutrosophic triplet order {with respect to a given neut(a), when the case when there are many neutrals of a}. It is denoted by nto(a).

Theorem 3.21. Let (N,*) be a neutrosophic triplet group with respect to * and let a [member of] N. Then

1) neut(a)*neut(a)=neut(a).

In general [(neut(a)).sup.2] = neut(a), where n is a non-zero positive integer.

2) neut(a)*anti(a)=anti(a)*neut(a)=anti(a).

(Theorem 3.21 (1) and (2) were proven in Theorem 2.3. (1) and (2), except [(neut(a)).sup.n] = mut(a).)

Proof. Consider neut(a)*neut(a)=neut(a).

Multiply a to the left side, we get;

a*neut(a)*neut(a)=a*neut(a)

[a* neut(a)]*neut(a)=[a*neut(a)]

a*neut(a)=a

a=a.

On the same lines, we can see that [(neut(a)).sup.2] = neut(a) for a non-zero positive integer n. 2) Consider neut(a)*anti(a)=anti(a).

Multiply to the left with a, we get;

a* neut(a)*anti(a)=a*anti(a)

a* anti(a)=neut(a)

neut(a)=neut(a)

a=a.

Similarly anti(a)*neut(a)=anti(a).

Definition 3.22. Let N be a NTG. If N = <a> for some a [member of] N, then N is called a neutro-cyclic triplet group" is better for the definition.

We say that a is a generator part of the neutrosophic triplet group.

Theorem 3.24. Let N be a neutro-cyclic triplet group and let a be a generator part of the neutrosophic triplet.

Then

1) <neut(a)> generates neutro-cyclic triplet subgroup of N.

2) <anti(a)> generates neutro-cyclic triplet subgroup of N.

Proof. Straightforward.

4 Neutro-Homomorphism

In this section, we introduce the neutro-homomorphism for the neutrosophic triplet groups. We also study some of their properties. Further, we defined neutro-isomorphisms.

Definition 4.1. Let ([N.sub.1],[*.sub.1]) and ([N.sub.2],[*.sub.2])be two neutrosophic triplet groups. Let

[Florin] or [Guilder]: [N.sub.1] [right arrow] [N.sub.2]

be a mapping. Then f is called neutro-homomorphism if for all a, b [member of] [N.sub.1]we have

1) f (a [*.sub.1] b)= f (a) [*.sub.1]] f (b),

2) f ([[neu[t.sub.2] (a)) = [[neu[t.sub.2] (f (a)), and

3) [mathematical expression not reproducible]

where [[k.sub.1] = 1, 2, ... is the order of the neutrosophic triplet (a, [neut.sub.*1](a), [anti.sub.*1](a)) in the case when the element a has more opposites, and one uses the notations:

[mathematical expression not reproducible]

and similar notations for the second law [*.sub.2]:

[mathematical expression not reproducible]

Theorem 4.1.

Axioms (1), (2), and (3) are equivalent to extending the one-variable (neutro-homomorphism) function f(x) to three-variable (neutro-homomorphism) function F(x, y, z), defined as follows:

[mathematical expression not reproducible]

if (a, b, c) [member of] [N.sub.[1.sup.3] is a neutrosophic triplet, then

[mathematical expression not reproducible]

is also a neutrosophic triplet.

Hence in general, for [k.sub.1], [k.sub.2] = 1, 2, ..., one has:

[mathematical expression not reproducible]

Proof. Almost straightforwardly.

We construct a well-defined law of neutrosophic [triplets.sub.#1] on [N.sub.[1.sup.3] as follows:

for any two neutrosophic triplets (a, b, c) and ([alpha], [beta], [gamma]) from [N.sub.[1.sup.3], one has:

(a, b, c) [.sub.#1] ([alpha], [beta], [gamma]) = (a [.sub.*1] [alpha], b [.sub.*1] [beta], c [.sub.*1] [gamma]),

and a well-defined law of neutrosophic triplets [.sub.#2] on [N.sub.2.sup.3] as follows:

for any two neutrosophic triplets (u, v, w) and ([delta], [epsilon], [zeta]) from [N.sub.2.sup.3], one has:

(u, v, w) [.sub.#2] ([delta], [epsilon], [zeta] ) = (u [*.sub.2] [delta], v [*.sub.2] [epsilon], w [*.sub.2] [zeta]).

Whence,

F ((a, b, c) #1 ([alpha], [beta], [gamma])) = F (a *1 [alpha], b *1 [beta], c *1 [gamma]) = (f(a *1 [alpha]), f(b *1 [beta]), f(c *1 [gamma]))

= (f([alpha]) *2 f([alpha]), f(b) [*.sub.2] f([beta]), f(c) [*.sub.2] f([gamma])).

And further, for b = [neut.sub.*1](a), c = [anti.sub.*1.sup.(k11)](a), and respectively [beta] = [neut.sub.*1]([alpha]), [gamma] = [anti.sub.*2.sup.(k12)]([alpha]), one gets:

F((a, [neut.sub.*1]([alpha]), [anti.sub.*1.sup.(k11)](a)) [.sub.#1] ([alpha], [neut.sub.*2]([alpha]), [anti.sub.*2.sup.(k12)] ([alpha]))) =

F( (a [.sub.*1] [alpha]), ([neut.sub.*1](a) [.sub.*1] [neut.sub.*1]([alpha])), ([anti.sub.*1.sup.(k11)](a)) [.sub.*1] ([anti.sub.*1.sup.(k12)]([alpha]))) =

( f(a [.sub.*1] [alpha]), f([neut.sub.*1](a) [.sub.*1] [neut.sub.*1]([alpha])), f([anti.sub.*1.sup.(k11)](a) [.sub.*1] [anti.sub.*1.sup.(k12)]([alpha]))) =

([f(a)*.sub.2] f([alpha]), f[([neut.sub.*1](a)).sub.*2] f([neut.sub.*1]([alpha])), f[([anti.sub.*1.sup.(k11)](a)).sub.*2] f([anti.sub.*1.sup.(k12)]([alpha]))) =

([f(a)*.sub.2] f([alpha]), [neut.sub.*2] [(f(a)).sub.*2] [neut.sub.*2] (f([alpha])), [anti.sub.*2.sup.(k21)][(f(a)).sub.*2] [anti.sub.*2.sup.(k22)](f([alpha]))) =

F( a, [neut.sub.*1](a), [anti.sub.*1.sup.(k11)](a)) [.sub.#2] F([alpha], [neut.sub.*1] ([alpha]), [anti.sub.*1.sup.(k12)]([alpha])).

Therefore F(x, y z), for (x, y z) neutrosophic triplets in [N.sub.1.sup.3], is its self a neutro-homomorphism.

Proposition 4.3. Every neutro-homomorphism is a classical homomorphism by neglecting the classical unitary element in classical homomorphism.

Proof. First, we neglect the classical unitary element that classical homomorphism maps unitary element to the corresponding unitary element. Now suppose that f is a neutro-homomorphism from a neutrosophic triplet group [N.sub.1] to a neutrosophic triplet group [N.sub.2]. Then by condition (1), it follows that f is a classical homomorphism.

Definition 4.4. A neutro-homomorphism is called neutro-isomorphism if it is one-to-one and onto.

5 Distinctions and Comparison

The distinctions between Molaei's Generalized Group [7] and Neutrosophic Triplet Group are:

I. - in MGG for each element there exists a unique neutral element, which can be the classical group unitary element; while in NTG each element may have a unique neutral element but which is different form the classical element;

II. - in MGG there exists a unique inverse of an element, while in NTG there may be many inverses for the same given element;

III. - MGG has a weaker structure than NTG.

IV. - Smarandache (2016-2017) has generalized the NTG to Neutrosophic Extended Triplet Group (NETG), where the neut(x) is allowed to be equal to the classical unitary algebraic element of the group theory [14-16].

So far the applications of neutrosophic triplet sets are in Z, modulo n, n [greater than or equal to] 2.

But new applications can be found, for example in social science:

One person <A> that has an enemy <anti([A.sub.d1]>(enemy in a degree [d.sub.1] of enemy-city), and a neutral person <neut([A.sub.d1]> with respect to <anti([A.sub.d1]>. Then another enemy <anti([A.sub.d2]> in a different degree of enemycity, and a neutral <neut([A.sub.d2]>, and so on. Hence one has the neutrosophic triplets:

<A, (neut ([A.sub.d1])>, <anti ([A.sub.d1])>>, <A, (neut ([A.sub.d2])>, (anti ([A.sub.d2])>>, and so on.

Then we take another person B in the same way...

(A, <neut([B.sub.d1])>,(anti ([B.sub.d1])>>,

<A, (neut ([B.sub.d2])>, <anti ([B.sub.d2])>> etc.

More applications may be found, if we deeply think about cases where we have neutrosophic triplets <A, (neut (A)>, (anti (A)>> in technology and in science.

Ackno wled gement :

There is no conflict of interest in the manuscript. We are very thankful to Prof. Muhammad Zafarullah from USA for his valuable comments and suggestion that improved this paper.

We are also grateful to Dr. Yilmaz Ceven, from Suleyman Demirel University, Isparta, Turkey, for his valuable remarks about the paper.

Conclusion

Inspired on the Neutrosophic philosophy, we defined for the first time the neutrosophic triplet. Basically, a neutrosophic triplet is a triad of certain elements, which satisfy certain axioms, which highly depend upon the proposed binary operation. The main theme of this paper is first to introduce the neutrosophic triplets, which are completely new notions, and then apply these neutrosophic triplets to introduce the neutrosophic triplet groups. This neutrosophic triplet group has several extra-ordinary properties as compared to the classical group. We also studied some interesting properties of this newly born structure. We further defined neutrohomomorphisms for neutrosophic triplet groups. A neutro-homomorphism is the generalization of the classical homomorphism with two extra conditions. As a further generalization, we gave rise to a new field or research called Neutrosophic Triplet Structures (such as neutrosophic triplet ring, neutrosophic triplet field, neutrosophic triplet vector space, etc.). In the end, we offered main distinctions and comparison of neutrosophic triplet group with the Molaei's generalized group as well as the possible application areas for the neutrosophic triplet groups.

References

[1] Atanassov TK, Intuitionistic Fuzzy Sets. Fuzzy Sets and Systems. 20, (1986) 87-96.

[2] Dummit DS, Foote R M, Abstract Algebra, 3rd Ed., John Viley & Sons Inc (2004).

[3] Herstein IN, Topics in algebra, Xerox College Publishing, Lexington, Mass., 1975.

[4] Kandasamy WB V, and Smarandache F, Some Neutrosophic Algebraic Structures and Neutrosophic N-Algebraic Structures, 219 p., Hexis, 2006.

[5] Kandasamy WBV, and Smarandache F, N-Algebraic Structures and S-N-Algebraic Structures, 209 pp., Hexis, Phoenix, 2006.

[6] Kandasamy WBV, and Smarandache F, Basic Neutrosophic Algebraic Structures and their Applications to Fuzzy and Neutrosophic Models, Hexis, 149 pp., 2004.

[7] Molaei, MR, Generalized groups, Bulet. Inst. Politehn. Ia, si Sect,. I 45(49), (1999), 21-24.

[8] Smarandache F, Neutrosophy. Neutrosophic Probability, Set and Logic, ProQuest Information & Learning, Ann Arbor, Michigan, USA, 105 p. (1998).

[9] Smarandache F, Neutrosophic set, a generalization of the intuitionistic fuzzy set, 2006 IEEE International Conference on Granular Computing, 10-12 May 2006, pages 38-42. DOI:10.1109/GRC.2006.1635754

[10] Smarandache F, Ali M, Neutrosophic Triplet as extension of Matter Plasma, Unmatter Plasma, and Antimatter Plasma, 69th Annual Gaseous Electronics Conference, Bochum, Germany, Veranstaltungszentrum & Audimax, Ruhr-Universitat, October 10-14, 2016, http://meetings.aps.org/Meeting/GEC16/Session/HT6.112

[11] Surowski DB, The Uniqueness Aspect of the Fundamental Theorem of Finite Abelian Groups. Amer. Math. Monthly, 102 (1995), 162-163.

[12] Zadeh A L, "Fuzzy sets," Inform. Control, vol. 8, (1965) 338-353.

[13] Smarandache F., Ali, M., 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.

[14] Smarandache, F. Neutrosophic Perspectives: Triplets, Duplets, Multisets, Hybrid Operators, Modal Logic, Hedge Algebras. And Applications. Pons Editions, Bruxelles, first edition 324 p. June 2017; second edition 346 p., September 2017.

[15] Smarandache, F. Neutrosophic Extended Triplets, mss., Arizona State University,

[16] Tempe, AZ, Special Collections, 2016, http://fs.unm.edu/NeutrosophicTriplets.htm

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

[18] 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.

[19] Florentin Smarandache, Mumtaz Ali, The Neutrosophic Triplet Group and its Application to Physics, presented by Florentin Smarandache to Universidad Nacional de Quilmes, Department of Science and Technology, Bernal, Buenos Aires, Argentina, 02 June 2014.

[20] F. Smarandache, Neutrosophic Extended Triplets, Arizona State University,

[21] Tempe, AZ, Special Collections, 2016.

[22] F. Smarandache, Neutrosophic Theory and Applications, Le Quy Don Technical University, Faculty of Information technology, Hanoi, Vietnam, 17th May 2016.

Accepted: May 8, 2019

Florentin Smarandache (1), Mumtaz Ali (2)

(1) University of New Mexico, 705 Gurley Ave., Gallup, New Mexico 87301, USA, E-mail: fsmarandache@gmail.com

(2) Department of Mathematics, Quaid-i-Azam University, Islamabad, 44000, Pakistan, E-mail: mumtazali7288@gmail.com
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