Fundamental Homomorphism Theorems for Neutrosophic Extended Triplet Groups

In classical group theory, homomorphism and isomorphism are significant to study the relation between two algebraic systems. Through this article, we propose neutro-homomorphism and neutro-isomorphism for the neutrosophic extended triplet group (NETG) which plays a significant role in the theory of neutrosophic triplet algebraic structures. Then, we define neutro-monomorphism, neutro-epimorphism, and neutro-automorphism. We give and prove some theorems related to these structures. Furthermore, the Fundamental homomorphism theorem for the NETG is given and some special cases are discussed. First and second neutro-isomorphism theorems are stated. Finally, by applying homomorphism theorems to neutrosophic extended triplet algebraic structures, we have examined how closely different systems are related.


Introduction
Groups are finite or infinite set of elements which are vital to modern algebra equipped with an operation (such as multiplication, addition, or composition) that satisfies the four basic axioms of closure, associativity, the identity property, and the inverse property.Groups can be found in geometry studied by "Felix klein in 1872" [1], characterizing phenomenality like symmetry and certain types of transformations.Group theory, firstly introduced by "Galois" [2], with the study of polynomials has applications in physics, chemistry, and computer science, and also puzzles like the Rubik's cube as it may be expressed utilizing group theory.Homomorphism is both a monomorphism and an epimorphism maintaining a map between two algebraic structures of the same type (such as two groups, two rings, two fields, two vector spaces) and isomorphism is a bijective homomorphism defined as a morphism, which has an inverse that is also morphism.Accordingly, homomorphisms are effective in analyzing and calculating algebraic systems as they enable one to recognize how intently distinct systems are associated.Similar to the classical one, neuro-homomorphism is the transform between two neutrosophic triplet algebraic objects N and H.That is, if elements in N satisfy some algebraic equation involving binary operation "*", their images in H satisfy the same algebraic equation.A neutro-isomorphism identifies two algebraic objects with one another.The most common use of neutro-homomorphisms and neutro-isomorphisms in this study is to deal with homomorphism theorems which allow for the identification of some neutrosophic triplet quotient objects with certain other neutrosophic triplet subgroups, and so on.
The neutrosophic logic and a neutrosophic set, firstly made known by Florentin Smarandache [3] in 1995, has been widely applied to several scientific fields.This study leads to a new direction, exploration, path of thinking to mathematicians, engineers, computer scientists, and Symmetry 2018, 10, 321 2 of 14 many other researchers, so the area of study grew extremely and applications were found in many areas of neutrosophic logic and sets such as computational modelling [4], artificial intelligence [5], data mining [6], decision making problems [7], practical achievements [8], and so forth.Florentin Smarandache and Mumtazi Ali investigated the neutrosophic triplet group and neutrosophic triplet as expansion of matter plasma, nonmatter plasma, and antimatter plasma [9,10].By using the concept of neutrosophic theory Vasantha and Smarandache introduced neutrosophic algebraic systems and N-algebraic structures [11] and this was the first neutrosofication of algerbraic structures.The characterization of cancellable weak neutrosophic duplet semi-groups and cancellable NTG are investigated [12] in 2017.Florentin Smarandache and Mumtaz Ali examined the applications of the neutrosophic triplet field and neutrosophic triplet ring [13,14] in 2017.Şahin Mehmet and Abdullah Kargın developed the neutrosophic triplet normed space and neutrosophic triplet inner product [15,16].The neutrosophic triplet G-module and fixed point theorem for NT partial metric space are given in literature [17,18].Similarity measures of bipolar neutrosophic sets and single valued triangular neutrosophic numbers and their appliance to multi-attribute group decision making investigated in [19,20].By utilizing distance-based similarity measures, refined neutrosophic hierchical clustering methods are achieved in [21].Single valued neutrosophic sets to deal with pattern recognition problems are given with their application in [22].Neutrosophic soft lattices and neutrosophic soft expert sets are analyzed in [23,24].Centroid single valued neutrosophic numbers and their applications in MCDM is considered in [25].Bal Mikail, Moges Mekonnen Shalla, and Necati Olgun reviewed neutrosophic triplet cosets and quotient groups [26] by using the concept of NET in 2018.The concepts concerning neutrosophic sets and neutrosophic modules are described in [27,28], respectively.A method to handle MCDM problems under the SVNSs are introduced in [29].Bipolar neutrosophic soft expert set theory and its basic operations are defined in [30].
The other parts of a paper is coordinated thusly.Subsequently, through the literature analysis in the first section and preliminaries in the second section, we investigated neutro-monomorphism, neutro-epimorphism, neutro-isomorphism, and neutro-automorphism in Section 3 and a fundamental homomorphism theorem for NETG in Section 4. We give and prove the first neutro-isomorphism theorem for NETG in Section 5, and then the second neutro-isomorphism theorem for NETG is given in Section 6.Finally, results are given in Section 7.

Preliminaries
In this section, we provide basic definitions, notations and facts which are significant to develop the paper.

Neutrosophic Extended Triplet
Let U be a universe of discourse, and (N, * ) a set included in it, endowed with a well-defined binary law * .

Definition 1 ([3]
).The set N is called a neutrosophic extended triplet set if for any x ∈ N there exist e neut(x) ∈ N and e anti(x) ∈ N. Thus, a neutrosophic extended triplet is an object of the form (x, e neut(x) , e anti(x) ) where e neut(x) is extended neutral of x, which can be equal or different from the classical algebraic unitary element if any, such that x * e neut(x) = e neut(x) * x = x and e anti(x) ∈ N is the extended opposite of x such that x * e anti(x) = e anti(x) * x = e neut(x) In general, for each x ∈ N there are many existing e neut(x) s and e anti(x) s .We consider, that the extended neutral elements replace the classical unitary element as well the extended opposite elements replace the inverse element of classical group.Therefore, NETGs are not a group in classical way.In the case when NETG enriches the structure of a classical group, since there may be elements with more extended opposites.

Definition 3 ([26]
).Given a NETG (N, * ), a neutrosophic triplet subset H is called a neutrosophic extended triplet subgroup of N if it itself forms a neutrosophic extended triplet group under * .Explicity this means (1) The extended neutral element e neut(x) lies in H.
In general, we can show H ≤ N as x ∈ H and then e anti(x) ∈ H, i.e x * e anti(x) = e neut(x) ∈ H.
The neutrosophic triplet image of f is is the neutrosophic triplet inverse image of B under f.

Definition 8 ([26]
).Let f : The neutrosophic triplet kernel of f is a subset where neut(x) denotes the neutral element of N 2 .
Definition 9.The neutrosophic triplet kernel of φ is called the neutrosophic triplet center of NETG N and it is denoted by Z(N).Explicitly, Hence Z(N) is the neutrosophic triplet set of elements in N that commute with all elements in N. Note that obviously Z(N) is a neutrosophic triplet.We have Z(N) = N in the case that N is abelian.
Proof.It is obvious to show that f is one to one and onto.Now let's show that f is neutro-homomorphism.Definition 17.Let N be NETG.ϕ ∈ AutN is called a neutro-inner automorphism if there is a n ∈ N such that ϕ = ϕ n .

Proposition 1.
Let N be a NETG.For a ∈ N, f a : N → N such that x → ax(anti(a) is a neutro-automorphism (AutN).
On the other hand, assume that

Fundamental Theorem of Neutro-Homomorphism
The fundamental theorem of neutro-homomorphism relates the structure of two objects between which a neutrosophic kernel and image of the neutro-homomorphism is given.It is also significant to prove neutro-isomorphism theorems.In this section, we give and prove the fundamental theorem of neutro-homomorphism.Then, we discuss a few special cases.Finally, we give examples by using NETG.

H H H H =
On the other hand, assume that

Fundamental Theorem of Neutro-Homomorphism
The fundamental theorem of neutro-homomorphism relates the structure of two objects between which a neutrosophic kernel and image of the neutro-homomorphism is given.It is also significant to prove neutro-isomorphism theorems.In this section, we give and prove the fundamental theorem of neutro-homomorphism.Then, we discuss a few special cases.Finally, we give examples by using NETG.
Theorem 13.Let N1, N2 be NETG's and Proof.We will construct an explicit map ( ) ( ) and prove that it is a neutro-isomorphism and well defined.Since ker(ϕ) is neutrosophic triplet normal subgroup of N1.

Let ( ),
K ker φ = and recall that Thus, we need to check the following conditions. ( i aK a b i bK Therefore, it is well defined. Proof.We will construct an explicit map i : N 1 /ker(φ) → im(φ) and prove that it is a neutro-isomorphism and well defined.Since ker(ϕ) is neutrosophic triplet normal subgroup of N 1 .
Let K = ker(φ), and recall that Thus, we need to check the following conditions.

First Neutro-Isomorphism Theorem
The first neutro-isomorphism theorem relates two neutrosophic triplet quotient groups involving products and intersections of neutrosophic extended triplet subgroups.In this section, we give and prove the first neutro-isomorphism theorem.Finally, we give an example by using NETG.Theorem 14.Let N be NETG and H, K be two neutrosophic extended triplet subgroup of N and H is a neutrosophic triplet normal in K. Then . Now let's define a mapping ϕ: HK→ K D by φ(hk) = KD.
Since for every KD ∈ K/D, neut.k∈ HK under ϕ such that φ(neut.k)= KD.Hence, ϕ is onto.Now by Theorem 13, HK/Kerφ ∼ = K/D Now it is enough to prove that kerφ = H.Let h ∈ H, h(neut) ∈ HK.Thus Example 3. Let N be NETG.Neutro-isomorphism theorems are for instance useful in the calculation of NETG orders, since neutro-isomorphic groups have the same order.If H ≤ N and K N so that HK is finite, then Lagrange's theorem [26] in neutrosophic triplet with theorem 13 yield

Second Neutro-Isomorphism Theorem
The second neutro-isomorphism theorem is extremely useful in analyzing the neutrosophic extended normal subgroups of a neutrosophic triplet quotient group.In this section, we give and prove the second neutro-homomorphism theorem for NETG.Theorem 15.Let N be a NETG.Let H and K be neutrosophic triplet normal subgroup of N with K ⊆ H. Then H/K N/K and N/KH/K ∼ = N/H Proof.Consider the natural map Ψ:N→N/H.The neutrosophic triplet kernel, H contains K. Thus, by the universal property of N/K, it follows that there is a neutro-homomorphism N/K → N/H.This map is clearly surjective.In fact, it sends the neutrosophic triplet left coset nK to the neutrosophic triplet left coset nH.Now suppose that nK is in the neutrosophic triplet kernel.Then the neutrosophic triplet left coset nH is the neutral neutrosophic triplet coset, that is, nH = H, so that n ∈ H. Thus the neutrosophic triplet kernel consists of those neutrosophic triplet left cosets of the form nK, for n ∈ H, that is, H/K.

Conclusions
This paper is mainly focused on fundamental homomorphism theorems for neutrosophic extended triplet groups.We gave and proved the fundamental theorem of neutro-homomorphism, as well as first and second neutro-isomorphism theorems explained for NETG.Furthermore, we define neutro-monomorphism, neutro-epimorphism, neutro-automorphism, inner neutro-automorphism, and center for neutrosophic extended triplets.Finally, by applying them to neutrosophic algebraic structures, we have examined how closely different systems are related.By using the concept of a fundamental theorem of neutro-homomorphism and neutro-isomorphism theorems, the relation between neutrosophic algebraic structures (neutrosophic triplet ring, neutrosophic triplet field, neutrosophic triplet vector space, neutrosophic triplet normed space, neutrosophic modules, etc.) can be studied and the field of study in neutrosophic algebraic structures will be extended.

Definition 10 ([26]).
Let N be a NETG and H ⊆ N.∀x ∈ N, the set xh/h ∈ H is called neutrosophic triplet coset denoted by xH.Analogously, the right neutrosophic triplet coset of H in N containing x. | xH | and | Hx | are used to denote the number of elements in xH and Hx, respectively.2.5.Neutrosophic Triplet Normal Subgroup and Quotient Group Definition 11 ([26]).A neutrosophic extended triplet subgroup H of a NETG of N is called a neutrosophic triplet normal subgroup of N if aH(anti(a)) ⊆ H, ∀x ∈ N and we denote it as H N and H N i f H = N.
Hx = hx/h ∈ H and (xH)anti(x) = (xh)anti(x)/h ∈ H.When h ≤ N, xH is called the left neutrosophic triplet coset of H in N containing x, and Hx is called neutrosophic triplet quotient group N/H has elements xH : x ∈ N, the neutrosophic triplet cosets of H in N, and operation (xH)(yH) = (xy)H.
As to the neutrosophic triplet kernel, a ∈ kerϕ ⇔ ϕ(a) = H (since H is neutral in N/H) ⇔ aH = H (by definition of φ) ⇔ a ∈ H. Let N be NETG and H ⊆ N be a non-empty neutrosophic extended triplet subset.Then H N, if and only if there exists a neutro-homomorphism ϕ : N 1 → N 2 with H = kerϕ.Let (N 1 , * ) and (N 2 , •) be two NETGs.If a mapping f : N 1 → N 2 neutro-homomorphism is one to one and onto f is called neutro-isomorphism.Here, N 1 and N 2 are called neutro-isomorphic and denoted as N 1 ∼ = N 2 .