On Homomorphism Theorem for Perfect Neutrosophic Extended Triplet Groups

Abstract

Some homomorphism theorems of neutrosophic extended triplet group (NETG) are proved in the paper [Fundamental homomorphism theorems for neutrosophic extended triplet groups, Symmetry 2018, 10(8), 321; doi:10.3390/sym10080321]. These results are revised in this paper. First, several counterexamples are given to show that some results in the above paper are not true. Second, two new notions of normal NT-subgroup and complete normal NT-subgroup in neutrosophic extended triplet groups are introduced, and their properties are investigated. Third, a new concept of perfect neutrosophic extended triplet group is proposed, and the basic homomorphism theorem of perfect neutrosophic extended triplet groups is established.

1. Introduction

As an extension of fuzzy sets and intuitionistic fuzzy sets, F. Smarandache proposed the new concept of neutrosophic sets [1]. Because the existence of intermediate states (neutral) is allowed, neutrosophic sets have more flexibility in expressing uncertainty, which has attracted much research interest. At present, neutrosophic sets have been applied to many fields, for examples, logical algebraic systems, decision making, medical diagnosis and data analysis [2,3,4,5,6,7,8,9,10,11], and more in-depth theoretical studies have also made new progress [12,13,14].

As an application of the basic idea of neutrosophic sets (more general, neutosophy), the new notion of neutrosophic triplet group (NTG) is proposed by F. Smarandache and M. Ali in [15,16]. As a new algebraic structure, NTG is a generalization of classical group, but it has different properties from the classical group. For NTG, the neutral element is relative and local, that is, for a neutrosophic triplet group (N, *), every element a in N has its own neutral element (denote by neut(a)) satisfying condition a*neut(a) = neut(a)*a = a, and there exists at least one opposite element (denote by anti(a)) in N relative to neut(a) such condition a*anti(a) = anti(a)*a = neut(a). In the original definition of NTG by the authors of [16], neut(a) is different from the traditional unit element. Later, the concept of neutrosophic extended triplet group (NETG) was introduced (see [15]), in which the neutral element may be a traditional unit element, it is just a special case.

It should be noted that from the point of view of Neutrosophy, the neutrosophic set and neutrosophic extended triplet group are related: for a neutrosophic set, the membership of each element x is divided into three independent parts, T(x), I(x), F(x); for a neutrosophic extended triplet group, every element a and its neutral element neut(a), opposition element anti(a) constitute a triple (a, neut(a), anti(a)). In other words, the concepts of the neutrosophic set and the neutrosophic extended triplet group reflect the thought “Trinity”. Of course, neutrosophic set and neutrosophic extended triplet group are two different mathematical concepts, one is expressed by function, the other is expressed by algebraic structure, and their deeper connection needs further study.

For the nature and structure of NTG, recently, some new results have been published: cancellable NTGs are discussed in [17]; several homomorphism theorems of commutative NTGs are proved in [18]; some new properties of NTGs are obtained in [19]; the relationships between generalized NTGs and logical algebraic systems are investigated in [20]. In these papers, the name “neutrosophic triplet group” essentially refers to “neutrosophic extended triplet group”, which is illustrated by the authors of [17,18,19,20].

As we know, for an algebraic system, homomorphism basic theorems are important, similar to the classical group. In [21], the authors studied the homomorphism basic theorems of NETGs, and obtained some useful results. Unfortunately, we found that some of the results need to be corrected. In this paper, we first give some counter examples to show that several theorems in [21] are wrong, and then we prove a quotient structure theorem of weak commutative NETGs. Moreover, we introduce a new concept of perfect NETG and establish basic homomorphism theorem of perfect NETGs, which will play a positive role in the further study of neutrosophic extended triplet groups.

2. Preliminaries

Definition 1

([15,16]). Let N be a non-empty set together with a binary operation*. Then, N is called a neutrosophic extended triplet set if for any a∈N, there exist a neutral of “a” (denote by neut(a)), and an opposite of “a” (denote by anti(a)), such that neut(a)∈N, anti(a)∈N and:

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

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

The triplet (a, neut(a), anti(a)) is called a neutrosophic extended triplet.

Note that, for a neutrosophic triplet set (N, *), a∈N, neut(a) and anti(a) may not be unique. In order not to cause ambiguity, we use the following notations to distinguish:

• neut(a): denote any certain one of neutral of a;
• {neut(a)}: denote the set of all neutral of a.
• anti(a): denote any certain one of opposite of a;
• {anti(a)}: denote the set of all opposite of a.

Definition 2

([15,16]). Let (N, *) be a neutrosophic extended triplet set. Then, N is called a neutrosophic extended triplet group (NETG), if the following conditions are satisfied:

(1)

(N, *) is well-defined, i.e., for any a, b∈N, one has a *b∈N.

(2)

(N, *) is associative, i.e., (a*b)*c = a * (b*c) for all a, b, c∈N.

N is called a commutative neutrosophic extended triplet group if for all a, b∈N, a*b = b*a.

Proposition 1

([19]). Let (N, *) be a NETG. Then

(1)

neut(a) is unique for any a in N.

(2)

neut(a)*neut(a) = neut(a) for any a in N.

Proposition 2

([19]). Let (N, *) be a NETG. Then ∀a∈N, ∀anti(a)∈{anti(a)},

(1)

neut(a)*p = q*neut(a), for any p, q∈{anti(a)};

(2)

neut(neut(a)) = neut(a);

(3)

anti(neut(a))*anti(a)∈{anti(a)};

(4)

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

(5)

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

(6)

anti(neut(a))*a = a*anti(neut(a)) = a, for any anti(neut(a))∈{anti(neut(a))};

(7)

a∈{anti(neut(a)*anti(a))};

(8)

neut(a)*anti(a)∈{anti(a)}; anti(a)*neut(a)∈{anti(a)};

(9)

a∈{anti(anti(a))}, that is, there exists p∈{anti(a)} such that a∈{anti(p)};

(10)

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

Definition 3

([19]). Let (N, *) be a NETG. Then N is called a weal commutative neutrosophic extended triplet group (briefly, WCNETG) if a*neut(b) = neut(b)*a for all a, b∈N.

Proposition 3

([19]). Let (N, *) be a NETG. Then (N, *) is weak commutative if and only if N satisfies the following conditions:

(1)

neut(a)*neut(b) = neut(b)*neut(a) for all a, b∈N.

(2)

neut(a)*neut(b)*a = a*neut(b) for all a, b∈N.

Definition 4

([19]). Let (N, *) be a NETG and H be a subset of N. Then H is called a NT-subgroup of N if

(1)

a*b∈H for all a, b∈H;

(2)

there exists anti(a)∈{anti(a)} such that anti(a)∈H for all a∈H, where {anti(a)} is the set of opposite element of a in (N, *).

Proposition 4

([19]). Let (N, *) be a weak commutative NETG. Then (∀a, b∈N)

(1)

neut(a)*neut(b) = neut(b*a);

(2)

anti(a)*anti(b)∈{anti(b*a)}.

3. Some Counterexamples

The following examples show that Theorem 12 in [21] is not true.

Example 1.

Let N = {1, 2, 3, 4}. Define operation * on N as following

Table 1. Commutative neutrosophic extended triplet group (NETG).

*	1	2	3	4
1	1	1	4	4
2	1	2	4	4
3	4	4	3	4
4	4	4	4	4

. Then, (N, *) is a commutative NETG.

Denote H₁ = {2}, H₂ = {3}, then H₁, H₂ are neutrosophic extended triplet subgroup of N (according to Definition 3 in [21]), and H₁H₂ = H₂H₁ = {4}; but H₁H₂ is not a neutrosophic extended triplet subgroup of N (according to Definition 3 in [21]), since 4∈H₁H₂, 1∈{anti(4)}, 1∉H₁H₂.

Remark 1.

According to Definition 3 in [21], H₁H₂ is not a neutrosophic extended triplet subgroup of N. However, according to Definition 4 in this paper, H₁H₂ is a NT-subgroup of N.

Example 2.

Let N = {1, 2, 3, 4}. Define operation * on N as following

Table 2. Non-commutative NETG.

*	1	2	3	4
1	1	1	3	4
2	1	2	3	4
3	1	4	3	4
4	1	4	3	4

. Then, (N, *) is a non-commutative NETG.

Denote H₁ = {2}, H₂ = {3}, then H₁, H₂, H₁H₂ = {3} are neutrosophic extended triplet subgroup of N (according to Definition 3 in [21]); but H₁H₂ = {3} ≠ {4} = H₂H₁.

The following example shows that Theorem 13 in [21] is not true.

Example 3.

Let N₁ = {a, b, c, d}, N₂ = {1, 2, 3, 4, 5, 6}. Define operations *₁ and *₂ on N₁ and N₂ as following

Table 3. The operation *₁ on N₁.

*1	a	b	c	d
a	c	d	a	b
b	a	b	c	d
c	a	b	c	d
d	c	d	a	b

and

Table 4. The operation *₂ on N₂.

*1	1	2	3	4	5	6
1	3	4	1	2	1	1
2	1	2	3	4	3	3
3	1	2	3	4	3	3
4	3	4	1	2	1	1
5	1	2	3	4	5	3
6	1	2	3	4	3	6

. Then, (N₁, *₁) and (N₂, *₂) are commutative NETGs.

Define mapping ϕ: N₁ → N₂; a ↦ 1, b ↦ 2, c ↦ 3, d ↦ 4. Then ϕ is a neutro-homomorphism (according to Definition 13 in [21]), this can be verified by software MATLAB (the program is omitted). However, according to Definition 8 in [21], ker(ϕ) = {b, c}, and

a ker(ϕ) = d ker(ϕ) = {a, d}, b ker(ϕ) = c ker(ϕ) = {b, c} = ker(ϕ).

However, a ker(ϕ) = {a, d} ≠ {a} = ker(ϕ)a, N₁/ker(ϕ) is not isomorphic to im(ϕ) = {1, 2, 3, 4}.

Remark 2.

It should be pointed out that many of the results in [22] are cited in [21], but in fact, some results in [22] are not true, please see [19].

The following example shows that Theorem 14 (a) and (c) in [21] are not true.

Example 4.

Let N = {1, 2, 3, 4, 5}. Define operation * on N as following

Table 5. The operation * on N.

*	1	2	3	4	5
1	1	2	1	1	1
2	2	1	2	2	2
3	1	2	4	3	1
4	1	2	3	4	1
5	1	2	1	1	5

. Then, (N, *) is a commutative NETG.

Denote H = {5}, K = {1, 3, 4, 5}, then H, K is two neutrosophic extended triplet subgroup of N (according to Definition 3 in [21]), and H is a neutrosophic triplet normal subgroup of K (according to Definition 11 in [21]). However, HK = {1, 5} is not a neutrosophic triplet subgroup of N, since

1∈HK, 3, 4∈{anti(1)} and 3, 4∉HK.

Hence, Theorem 14 (a) and (c) in [21] are not true.

4. On Complete Normal NT-Subgroups of NETGs and Homomorphism Theorem of WCNETGs

For the omissions in the literature mentioned above, one of the main reasons is that there is no careful analysis of the various definitions of subgroups of neutrosophic extended triplet group (NETG). In this section, we propose new concepts of normal NT-subgroups and complete normal NT-subgroups of NETGs and discuss their basic properties. Moreover, based on complete normal NT-subgroups, we establish homomorphism theorem of weak commutative neutrosophic extended triplet groups (WCNETGs).

Definition 5.

Let (N, *) be a NETG and H be a NT-subgroup of N. Then H is called a normal NT-subgroup of N if for all a∈N and every anti(a)∈{anti(a)}, aH(anti(a)) ⊆ H.

Obviously, for any commutative NETG (N, *), a NT-subgroup H of N is normal if and only if for all a∈N, H(neut(a)) ⊆ H. The following examples show that there exists some NT-subgroups which are not normal, for some commutative NETGs.

Example 5.

Let (N, *) be the commutative NETG in Example 1. Denote H = {1}, then H is a NT-subgroup of N. But,

3H(anti(3)) = {4} ⊆ {1} = H.

Example 6.

Let (N, *) be the commutative NETG in Example 4. Denote H = {5}, then H is a NT-subgroup of N. But, 2H(anti(2)) = {2} ⊆ {5} = H.

Definition 6.

Let (N, *) be a NETG and H be a normal NT-subgroup of N. Then H is called to be complete normal if it satisfies:

(1)

for all a∈N, neut(a)∈H.

(2)

for all h∈H, anti(h)∈H.

The following examples show that a normal NT-subgroup may be not a complete normal.

Example 7.

Let (N, *) be the commutative NETG in Example 1. Denote H = {4}, then H is a normal NT-subgroup of N. But, H is not a complete NT-subgroup of N, since 1 = neut(1)∉H. Moreover, 1, 2, 3∈{anti(4)}, but 1, 2, 3∉H.

Example 8.

Let N = {1, 2, 3, 4, 5, 6}. Define operation * on N as following

Table 6. Non-commutative NETG.

*	1	2	3	4	5	6
1	3	1	1	3	1	3
2	4	2	2	4	2	4
3	1	3	3	1	3	1
4	2	4	4	2	4	2
5	1	2	3	4	5	6
6	3	4	1	2	6	5

. Then, (N, *) is a non-commutative NETG.

Denote H = {2, 3}, then H is a normal NT-subgroup of N. But, H is not a complete normal NT-subgroup of N, since neut(5) = 5∉H. Moreover, 2∈H, 5∈{anti(2)}, but 5∉H.

It is easy to verify that the following proposition is true (the proof is omitted).

Proposition 5.

Let (N, *) be a commutative NETG and H be a non-empty subset of N. Then H is complete normal NT-subgroup of N if and only if it satisfies:

(1)

for all a, b∈H, a*b∈H.

(2)

for all a∈N, neut(a)∈H.

(3)

for all h∈H, anti(h)∈H.

Proposition 6.

Let (N, *) be a weak commutative NETG and H be a complete normal NT-subgroup of N, a, b∈N. Then the following conditions are equivalent:

(1)

there exists anti(a)∈{anti(a)} and p N such that anti(a)*b*neut(p)∈H;

(2)

for any anti(b)∈{anti(b)}, there exists p∈N such that anti(b)*a*neut(p)∈H;

(3)

for any anti(a)∈{anti(a)}, there exists p∈N such that anti(a)*b*neut(p)∈H.

Proof. 

(1) ⇒ (2): Assume that anti(a)*b*neut(p)∈H, p∈N. By Defition 6 (2), anti(anti(a)*b*neut(p))∈H, for any anti(anti(a)*b*neut(p))∈{anti(anti(a)*b*neut(p))}. On the other hand, using Proposition 4 (2),

anti(neut(p))*anti(b)*anti(anti(a))∈{anti(anti(a)*b*neut(p))}.

It follows that anti(neut(p))*anti(b)*anti(anti(a))∈H, for any anti(b)∈{anti(b)}. Then, by Definition 4 (1), Definition 6 (1), Definition 3, Proposition 2 (10) and (2) we get

(anti(neut(p))*anti(b)*anti(anti(a)))*neut(a)∈H,

(anti(neut(p))*anti(b))*(neut(a)*anti(anti(a)))∈H,

anti(neut(p))*anti(b)*a∈H,

neut(p)*(anti(neut(p))*anti(b)*a)∈H,

(neut(p)* anti(neut(p)))*anti(b)*a∈H,

neut(neut(p))*anti(b)*a∈H,

neut(p)*anti(b)*a∈H,

anti(b)*a*neut(p)∈H.

(2) ⇒ (3): Assume that anti(b)*a*neut(p)∈H, p∈N, for any anti(b)∈{anti(b)}. Using the proof process similar to the previous one, we can get that anti(a)*b*neut(p)∈H, p∈N, for any anti(a)∈{anti(a)}.

(3) ⇒ (1): Obviously. □

Theorem 1.

Let (N, *) be a weak commutative neutrosophic triplet group and H be a complete normal NT-subgroup of N. Define binary relation ≈_H on N as follows: ∀a, b∈N,

A ≈ _Hb if and only if there exists p∈N such that anti(a)*b*neut(p)∈H, where anti(a)∈{anti(a)}.

Then

(1)

≈_H is an equivalent relation on N;

(2)

∀ c∈N, a ≈ _Hb ⇒ c*a ≈_H c*b;

(3)

∀ c∈N, a ≈ _Hb ⇒ a*c ≈_H b*c;

(4)

define binary operation * on N/H={[a]_H|a∈N} as follows: [a]_H *[b]_H = [a*b]_H, ∀a, b∈N. We can obtained a homomorphism from (N, *) to (N/H, *), that is, f: N→N/H; f(a)= [a]_H for all a∈N.

Proof. 

(1) Suppose a∈N, then anti(a)*a*neut(a) = neut(a)*neut(a) = neut(a)∈H, applying Proposition 2 (2) and Definition 6 (1). Hence, a ≈ _Ha.

Assume a ≈ _Hb, then there exists p∈N such that anti(a)*b*neut(p)∈H, where anti(a)∈{anti(a)}. By Proposition 6 (2) and (3), anti(b)*a*neut(p)∈H, ∀ anti(b)∈{anti(b)}. Thus, b ≈_Ha.

If a ≈ _Hb and b ≈ _Hc, then there exists p∈N and q∈N such that anti(a)*b*neut(p)∈H, anti(b)*c*neut(q)∈H, where anti(a)∈{anti(a)}, anti(b)∈{anti(b)}. Using Definition 4 (1), Definition 3 and Proposition 4 (1) we have

(anti(a)*b*neut(p))*(anti(b)*c*neut(q))∈H,

anti(a)*(b*anti(b))*c*(neut(p)*neut(q))∈H,

anti(a)*neut(b)*c*(neut(p)*neut(q))∈H,

anti(a)*c*(neut(b)*neut(p)*neut(q))∈H,

anti(a)*c*(neut(q*p*b)∈H.

It follows that a ≈ _Hc.

Combining the results above, ≈_H is an equivalent relation on N.

(2) Suppose a ≈ _Hb. Then there exists p∈N such that anti(a)*b*neut(p)∈H, where anti(a)∈{anti(a)}. By Definition 3, Definition 6 (1) and Definition 4 (1),

(anti(a)*anti(c))*(c*b)*neut(p)  

= anti(a)*(anti(c)*c)*b*neut(p)  

= anti(a)*neut(c)*b*neut(p)   

= anti(a)*b*neut(c)       

= (anti(a)*b*neut(p))*neut(c)∈H. 

By Proposition 4 (2), anti(a)*anti(c)∈{anti(c*a)}. Hence, there exists anti(c*a)∈{anti(c*a)} and p∈N such that anti(c*a)*(c*b)*neut(p)∈H. Applying Proposition 6, for any anti(c*a)∈{anti(c*a)}, there exists p∈N such that anti(c*a)*(c*b)*neut(p)∈H. That is, (c*a) ≈ _H (c*b).

(3) Suppose a ≈ _H b. Then there exists p∈N such that anti(a)*b*neut(p)∈H, where anti(a)∈{anti(a)}. By Definition 3, Definition 6 (1), Definition 4 (1) and Definition 5,

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

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

= (anti(c)*(anti(a)*b*neut(p))*anti(anti(c)))*neut(c)∈H.  

By Proposition 4 (2), anti(c)*anti(a)∈{anti(a*c)}. Hence, there exists anti(a*c)∈{anti(a*c)} and p∈N such that anti(a*c)*(b*c)*neut(p)∈H. Applying Proposition 6, for any anti(a*c)∈{anti(a*c)}, there exists p∈N such that anti(a*c)*(b*c)*neut(p)∈H. That is, (a*c) ≈ _H(b*c).

(4) Combining (1), (2) and (3), one can get (4). □

5. Homomorphism Theorems of Perfect Neutrosophic Extended Tripet Groups (PNETGs)

Proposition 7.

Let (N, *) be a weak commutative NETG. Then the following conditions are equivalent:

(i)

for all a∈N, the opposite element of neut(a) is unique, that is, |{anti(neut(a))}| = 1;

(ii)

for all a∈N, and any anti(neut(a))∈{anti(neut(a))}, anti(neut(a)) = neut(a).

Proof. 

(i) ⇒ (ii): For all a∈N, by Proposition 1 (2) and Proposition 2 (2),

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

This means that neut(a)∈{anti(neut(a))}. Applying (i) we get anti(neut(a)) = neut(a).

(ii) ⇒ (i): Obviously. □

Definition 7.

Let (N, *) be a weak commutative NETG. Then N is called a perfect NETG if anti(neut(a)) = neut(a) for all a∈N.

Proposition 8.

Let (N, *) be a perfect NETG. Then neut(anti(a)) = neut(a) for all a∈N.

Proof. 

For all a∈N, and any anti(a)∈{anti(a)}, from anti(neut(a)) = anti(a*anti(a)), applying Proposition 4 (2) we have

anti(a)*anti(anti(a))∈{anti(neut(a))}.

Since anti(a)*anti(anti(a)) = neut(anti(a)), thus neut(anti(a))∈{anti(neut(a))}. By Definition 7, we get neut(anti(a)) = neut(a). □

The following examples show that there exists commutative NETG which is not perfect, and there exists non-commutative NETG which is perfect.

Example 9.

Let (N, *) be the commutative NETG in Example 1. Then N is not perfect, since 1 = neut(1), {anti(neut(1))} = {1, 2}, 2 ≠ neut(1).

Example 10.

Let N = {a, b, c, d, e, f, g}. The operation * on N is defined as

Table 7. Non-commutative perfect NETG.

*	a	b	c	d	e	f	g
a	a	b	c	d	e	f	a
b	b	a	f	e	d	c	b
c	c	e	a	f	b	d	c
d	d	f	e	a	c	b	d
e	e	c	d	b	f	a	e
f	f	d	b	c	a	e	f
g	a	b	c	d	e	f	g

. Then, (N, *) is a non- commutative perfect NETG.

Definition 8

([21,22]). Let (N₁, *) and (N₂, *) be two neutrosophic extended triplet groups (NETGs). A mapping f: N₁→N₂ is called a neutro-homomorphism if

∀x,y∈N₁, f(x*y) = f(x)* f(y).

The neutrosophic triplet kernel off is defined Ker(f) = {x∈N₁: there exists y∈N₂ such that f(x) = neut(y)}. A neutro-homomorphism f is called a neutro-monomorphism if it is only one to one (injective). A neutro-homomorphism f is called a neutro-epimorphism if it is only onto (surjective). If a neutro-homomorphism f: N₁→N₂ is one to one and onto, then f is called neutro-isomorphism, and N₁ and N₂ are called neutro- isomorphic and denoted N₁ ≅ N₂.

It is easy to verify that the following proposition is true (the proof is omitted).

Proposition 9.

Let (N₁, *) and (N₂, *) be two NETGs and f: N₁→N₂ be a neutro-homomorphism. Then

(1)

for any x∈N₁, f(neut(x)) = neut(f(x));

(2)

for any x∈N₁ and any anti(x)∈{anti(x)}, f(anti(x))∈{anti(f(x))}.

Theorem 2.

Let (N₁, *) and (N₂, *) be two perfect NETGs and f: N₁→N₂ be a neutro-homomorphism. Then

(1)

Ker(f) is a complete normal NT-subgroup of N₁;

(2)

g is neutro-epimorphism, where g: N₁→N₁/Ker(f); g(a) = [a]_Ker(f) for all a∈N₁.

Proof. 

(1) Assume a, b∈Ker(f), then there exists x, y∈N₂, f(a) = neut(x), f(b) = neut(y). Thus

f(a*b) = f(a)* f(b) = neut(x)* neut(y) = neut(y*x). (By Proposition 4 (1))

This means that a*b∈Ker(f).

For any anti(a)∈{anti(a)}, by Proposition 9 (2), f(anti(a))∈{anti(f(a))} = {anti(neut(x))}. Applying Definition 7, {anti(neut(x))} = {neut(x)}. Thus, f(anti(a)) = neut(x). This means that anti(a)∈Ker(f).

For any p∈N₁, by Proposition 9 (1), f(neut(p)) = neut(f(p)), then neut(p)∈Ker(f). This means that {neut(p): p∈N₁}⊆Ker(f).

Moreover, for any p∈N₁, by Definition 8, Proposition 9, Definitions 3 and 7, Proposition 4, we have

f(anti(p)*a*p)    

= f(anti(p))*f(a)*f(p)  

= f(anti(p))*neut(x)*f(p)

= f(anti(p))*f(p)*neut(x)

= f(anti(p)*p)*neut(x) 

= f(neut(p))*neut(x)  

= neut(f(p))*neut(x)  

= neut(x*f(p)).    

It follows that anti(p)*a*p∈Ker(f).

Combining above results, we get that Ker(f) is a complete normal NT-subgroup of N₁.

(2) By (1) and Theorem 1 (4) we get (2). □

Remark 3.

Under the conditions of the above theorem, even if f is bijective, the related isomorphism theorem cannot be obtained. An example is given below.

Example 11.

Let N = {1, 2, 3, 4, 5}. Define operation * on N as following

Table 8. Commutative perfect NETG.

*	1	2	3	4	5
1	2	1	4	3	1
2	1	2	3	4	2
3	4	3	1	2	3
4	3	4	2	1	4
5	1	2	3	4	5

. Then, (N, *) is a perfect NETG.

Define f: N→N; for any a∈N, f(a) = a. Obviously, f is a neutro-isomorphism, Ker(f) = {2, 5}, N/Ker(f) = {{1}, {2, 5}, {3}, {4}}. It is easy to verify that g is neutro-epimorphism, where g: N→N/Ker(f); g(a) = [a]_Ker(f) for all a∈N. But, N/Ker(f) is not isomorphic to N.

Example 12.

Let N₁ = {1, 2, 3, 4, 5, 6, 7} and N₂ = {a, b, c, d}. The operations *₁, *₂ on N₁, N₂ are defined as

Table 9. The operation *₁ on commutative perfect NETG N₁.

*1	1	2	3	4	5	6	7
1	2	1	4	3	1	1	1
2	1	2	3	4	2	2	2
3	4	3	1	2	3	3	3
4	3	4	2	1	4	4	4
5	1	2	3	4	5	6	2
6	1	2	3	4	6	5	2
7	1	2	3	4	2	2	7

and

Table 10. The operation *₂ on commutative perfect NETG N₂.

*2	a	b	c	d
a	b	a	d	c
b	a	b	c	d
c	d	c	a	b
d	c	d	b	a

. Then, (N₁, *₁) and (N₂, *₂) are perfect NETGs.

Define mapping ϕ: N₁ → N₂; 1 $\mapsto$ a, 2 $\mapsto$ b, 3 $\mapsto$ c, 4 $\mapsto$ d, 5 $\mapsto$ b, 6 $\mapsto$ b, 7 $\mapsto$ b. Then ϕ is a neutro-homomorphism, ker(ϕ) = {2, 5, 6, 7}, and N₁/ker(ϕ) = {{1}, {2, 5, 6, 7}, {3}, {4}}. Moreover, we can verify that N₁/Ker(ϕ) is isomorphic to N₂.

Example 13.

Let N₁ = {1, 2, 3, 4, 5, 6, 7, 8} and N₂ = {a, b, c, d, e, f}. The operations *₁, *₂ on N₁, N₂ are defined as

Table 11. The operation *₁ on non-commutative perfect NETG N₁.

*1	1	2	3	4	5	6	7	8
1	1	2	3	4	5	6	1	1
2	2	1	6	5	4	3	2	2
3	3	5	1	6	2	4	3	3
4	4	6	5	1	3	2	4	4
5	5	3	4	2	6	1	5	5
6	6	4	2	3	1	5	6	6
7	1	2	3	4	5	6	8	7
8	1	2	3	4	5	6	7	8

and

Table 12. The operation *₂ on non-commutative perfect NETG N₂.

*2	a	b	c	d	e	f
a	a	b	c	d	e	f
b	b	a	f	e	d	c
c	c	e	a	f	b	d
d	d	f	e	a	c	b
e	e	c	d	b	f	a
f	f	d	b	c	a	e

. Then, (N₁, *₁) and (N₂, *₂) are non-commutative perfect NETGs.

Define mapping ϕ: N₁ → N₂; 1 $\mapsto$ a, 2 $\mapsto$ b, 3 $\mapsto$ c, 4 $\mapsto$ d, 5 $\mapsto$ e, 6 $\mapsto$ f, 7 $\mapsto$ a, 8 $\mapsto$ a. Then ϕ is a neutro-homomorphism, ker(ϕ) = {1, 7, 8}, and N₁/ker(ϕ) = {{1, 7, 8}, {2}, {3}, {4}, {5}, {6}}. Moreover, we can verify that N₁/Ker(ϕ) is isomorphic to N₂.

6. Conclusions

This paper has further studied the neutrosophic extended triplet group (NETG) and obtained some important results. First, examples are given to show that some results in [21] are not true. Second, some new notions of normal NT-group and complete normal NT-group are introduced, and a quotient structure theorem of weak commutative NETG is proved. Third, the concept of perfect neutrosophic extended triplet group (PNETG)is proposed, a homomorphism theorem of PNETG is established. All these results are interesting for exploring the structure characterizations of NETG. As future research topics, we will discuss the integration of the related uncertainty theory, such as the combination of neutrosophic set, fuzzy set, rough set and logic algebras (see [23,24,25,26]).