A Review Study On Neutrosophic AH-Algebraic Structures

Mikail Bal, Gaziantep University, Turkey

Katy D. Ahmad, Islamic University Of Gaza, Palestine

Rozina Ali, Cairo university, Egypt

Corrsepondence:katyon765@gmail.com

Abstract: Neutrosophic Algebraic structures are rich fields for researchers to get many interesting generalizations of classical and fuzzy structures.  This Study is dedicated to give the interested reader some of special neutrosophic algebraic substructures of neutrosophic algebraic structures, especially AH-substructures in neutrosophic rings, spaces, modules, and their generalizations. 

Key words: Neutrosophic set, neutrosophic number,  neutrosophic integer, Neutrosophic Algebra.

1. Introduction

Neutrosophy is considered as a new generalization of fuzzy and intuitionistic fuzzy ideas proposed by Smarandache [1]. Neutrosophic sets were very applicable in many areas of pure and applied mathematic such as matrix theory, topology, number theory, and algebraic structures [2-12].

Neutrosophic algebra began with Smarandache et. al, where they have defined neutrosophic rings, groups, vector spaces, and modules [20-30, 60-75].

In the literature many researchers have studied and contributed to these structures and their generalizations, where we can find many generalizations such as refined neutrosophic structures [13-20, 36-40], and n-refined neutrosophic structures [46-54].

An interesting direction was opened by Abobala. Et. al, where they defined AH-substructures such as AH-subspaces, AH-homomorphisms, AH-ideals, and AH-submodules [6-10]. Recently, AH-substructures were used to build neutrosophic system of Euclidean geometry, especially AH-isometries[66], and Turiyam symbolic rings [78-90]. The Turiyam set given a way to represent the data set beyond the Non-Euclidean and NeutroGeoemtry [91-92].

Through this work, we review the basic concepts of AH-substructures in neutrosophic rings, refined neutrosophic rings, neutrosophic vector spaces, n-refined neutrosophic modules, and many other neutrosophic algebraic structures.

Main Discussion

Definition 1:

Let R(I) be a neutrosophic ring and P=P_0+P_1 I={a_0+a_1 I; a_0∈P_0,a_1∈P_1}.

(a) We say that P is an AH-ideal if P_0 andP_1 are ideals in the ring R.

(b) We say that P is an AHS-ideal if P_0=P_1.

Definition 2:

Let (R(I_1,I_2),+,×) be a refined neutrosophic ring, and P_0,P_1,P_2 be three ideals in the ring R then the set

P=(P_0,P_1 I_1,P_2 I_2 )={(a,bI_1,cI_2 );a∈P_0,b∈P_1,c∈P_2} is called a refined neutrosophic AH-ideal.

If P_0=P_1=P_2 then P is called a refined neutrosophic AHS-ideal.

Definition 3: 

Let V(I) = V+VI be a strong/weak neutrosophic vector space, the set

S=P+QI={x+yI;x∈P,y∈Q},where P and Q are subspaces of V is called an AH-subspace of V(I).

If P=Q then S is called an AHS-subspace of V(I).

Example 4:

We have V = R^2 is a vector space, P=<(0,1)>,Q=<(1,0)>, are two subspaces of V. The set

S=P+QI={(0,a)+(b,0)I;a,b∈R} is an AH-subspace of V(I).

The set L=P+PI={(0,a)+(0,b)I};a,b∈R is an AHS-subspace of V(I).

Theorem 5:

Let V(I) = V+VI be a neutrosophic weak vector space, and let S=P+QI be an AH-subspace of  V(I), i.e Q,P are supspaces of V, then S is a subspace by the classical meaning.

Proof:

Suppose that x=a+bI,y=c+dI∈S;a,c∈P,b,d∈Q, we have 

x+y=(a+c)+(b+d)I∈S. For each scalar m∈K we obtain m.x=m.a+(m.b)I∈S, since P and Q are subspaces; thus S=P+QI is a subspace of V(I) over the field K.

Theorem  6:

Let V(I) be a neutrosophic strong vector space over a neutrosophic field K(I), let S=P+PI be an AHS-subspace. S is a subspace of V(I).

Proof:

Suppose that x=a+bI,y=c+dI∈S;a,c,b,and c∈P, we have

x+y=(a+c)+(b+d)I∈S. Let  m=x+yI∈K(I) be a neutrosophic scalar, we find

m.x=(x.a)+(y.a+y.b+x.b)I∈S, since y.a+y.b+x.b∈P, thus we get the desired result.

Definition 7:

(a) Let V and W be two vector spaces, L_V:V→W be a linear transformation. The AHS-linear transformation can be defined as follows:

L:V(I)→W(I);L(a+bI)=L_V (a)+L_V (b)I. 

(b) If S = P+QI is an AH-subspace of V(I), L(S)=L_V (P)+L_V (Q)I.

(c) If S = P+QI is an AH-subspace of W(I), L^(-1) (S)=L_W^(-1) (P)+L_W^(-1) (Q)I.

(d) AH-Ker L =KerL_V+KerL_V  I={x+yI;x,y∈KerL_V}. 

Theorem 8:

Let W(I) and V(I) be two neutrosophic strong/weak vector spaces, and  L:V(I)→W(I) be an AHS-linear transformation, we have:

(a) AH-Ker L is an AHS-subspace of V(I).

(b) If  S = P+QI is an AH-subspace of V(I), L(S) is an AH-subspace of W(I).

(c) If S = P+QI is an AH-subspace of W(I), L^(-1) (S) is an AH-subspace of V(I).

Proof:

(a) Since KerL_V is a subspace of V, we find that AH-Ker L =KerL_V+KerL_V  I is an AHS-subspace of V(I).

(b) We have L(S)=L_V (P)+L_V (Q)I; thus L(S) is an AH-subspace of W(I), since L_V (P),L_V (Q)are subspaces of W.

(c) By regarding L^(-1) (S)=L_W^(-1) (P)+L_W^(-1) (Q)I, L_W^(-1) (P)  and〖 L〗_W^(-1) (Q) are subspaces of V, we obtain that  L^(-1) (S) is an AH-subspace of V(I).

Theorem 9:

Let W(I) and V(I) be two neutrosophic strong vector spaces over a neutrosophic field K(I), and  L:V(I)→W(I) be an AHS-linear transformation, we have:

L(x+y)=L(x)+L(y),L(m.x)=m.L(x), for all x,y∈V(I),m∈K(I).

Proof:

Suppose x=a+bI,y=c+dI;a,b,c,d∈V,and m=s+tI∈K(I), we have

L(x+y)=L([a+c]+[b+d]I)=L_V (a+c)+L_V (b+d)I=

[L_V (a)+L_V (b)I]+[L_V (c)+L_V (d)I]=L(x)+L(y).

m.x=(s.a)+(s.b+t.a+t.b)I, L(m.x)=L_V (s.a)+L_V (s.b+t.a+t.b)I

= s.L_V (a)+[s.L_V (b)+t.L_V (a)+t.L_V (b)]I = (s+tI).(L_V (a)+L_V (b)I)=m.L(x).

Theorem 10:

Let S=P+QI be an AH-subspace of a neutrosophic weak vector space V(I) over a field K, suppose that 

X={x_i;1≤i≤n} is a bases of P and Y={y_j;1≤j≤m} is a bases of Q then X∪YI is a bases of S.

Proof:

 Let z=x+yI be an arbitrary element in S; x∈P,y∈Q. Since P and Q are subspaces of V we can write

x=a_1 x_1+a_2 x_2+⋯+a_n x_n; a_i∈K and x_i∈X, y=b_1 y_1+b_2 y_2+⋯+b_m y_m; b_i∈K,y_i∈Y.

Now we obtain z=(a_1 x_1+⋯+a_n x_n )+(b_1 y_1 I+⋯+b_m y_m I); thus X∪YI generates the subspace S.

X∪YI is linearly independent set. Assume that (a_1 x_1+⋯+a_n x_n )+(b_1 y_1 I+⋯+b_m y_m I)=0, this implies 

a_1 x_1+a_2 x_2+⋯+a_n x_n=0 and  〖(b〗_1 y_1+b_2 y_2+⋯+b_m y_m)I=0. Since X and Y are linearly independent sets over K, we get a_i=b_j=0 for all i,j and X∪YI is linearly independent then it is a bases of S.

Definition  11:

Let V(I) be a neutrosophic strong/weak vector space, S=P+QI be an AH-subspace of V(I), we define

 AH-Quotient as:

(V(I))⁄S=V⁄P+(V⁄(Q)) I= (x+P)+(y+Q)I;x,y∈V.

Theorem 12:

Let V(I) be a neutrosophic weak vector space over a field K, and S=P+QI be an AH-subspace of V(I). The AH-Quotient (V(I))⁄S is a vector space over the field K with respect to the following operations:

Addition: [(x+P)+(y+Q)I]+[(a+P)+(b+Q)]I=(x+a+P)+(y+b+Q)I;x,y,a,b∈V.

Multiplication by a scalar: (m).[(x+P)+(y+Q)I]=(m.x+P)+(m.y+Q)I;

x,y∈V and m∈K.

Definition 13:

Let (V,+,.) be a vector space over a field K, V_n (I) be the corresponding weak n-refined neutrosophic vector space over K. Consider the set {M_i;0≤i≤n}, where M_i is a subspace of V. We say: 

M_n (I)=M_0+M_1 I_1+⋯+M_n I_n={m_0+m_1 I_1+⋯+m_n I_n; m_i∈M_i} is a weak n-refined AH-subspace of the weak n-refined vector space V_n (I).

We say that M_n (I) is a weak n-refined AH-subspace if M_j=M_i  for all i,j.

Definition 14:

Let (V,+,.) be a vector space over a field K, V_n (I) be the corresponding strong n-refined neutrosophic vector space over the n-refined neutrosophic field K_n (I). Consider the set {M_i;0≤i≤n}, where M_i is a subspace of V. We say: 

M_n (I)=M_0+M_1 I_1+⋯+M_n I_n={m_0+m_1 I_1+⋯+m_n I_n; m_i∈M_i} is a strong n-refined AH-subspace of the strong n-refined vector space V_n (I).

We say that M_n (I) is a strong n-refined AH-subspace if M_j=M_i  for all i,j.

Theorem 15:

Let (V,+,.) be a vector space over a field K, V_n (I) be the corresponding weak n-refined neutrosophic vector space over K, M_n (I)=M_0+M_1 I_1+⋯+M_n I_n be a weak n-refined AH-subspace. Then

(a) M_n (I) is a vector subspace of  V_n (I).

(b) If X_i is a bases of M_i, X=⋃_(i=0)^n▒〖X_i I_i 〗 is a bases of M_n (I).

(c) dim⁡(M_n (I))=∑_(i=0)^n▒〖dim⁡(M_i)〗.

Proof:

(a) Let x=∑_(i=0)^n▒〖a_i I_i 〗,y=∑_(i=0)^n▒〖b_i I_i; b_i,a_i∈M_i 〗 be two arbitrary elements in M_n (I), r be an arbitrary element in K, we have:

x+y=∑_(i=0)^n▒〖(a_i+b_i)I_i 〗∈M_n (I), since a_i+b_i∈M_i because M_i is a subspace of V.

r.x= ∑_(i=0)^n▒〖〖ra〗_i I_i∈M_n (I)〗, since 〖ra〗_i∈M_i because M_i is a subspace of V. Thus M_n (I) is a vector subspace of V_n (I).

(b) Suppose that X_0={x_1^((0) ),…,x_(s_0)^((0) ) },X_1={x_1^((1) ),…,x_(s_1)^((1) ) },〖……,X〗_n={x_1^((n)),…,x_(s_n)^((n))}, let x=∑_(i=0)^n▒〖a_i I_i 〗 be an arbitrary element of M_n (I), since X_i is a basis of M_i for all i. We can write:

a_i=∑_(j=0)^(s_i)▒t_j^((i))  x_j^((i)); t_j∈K, so x=∑_(j=0)^(s_0)▒t_j^((0))  x_j^((0))+∑_(j=0)^(s_1)▒t_j^((1))  x_j^((1)) I_1+⋯+∑_(j=0)^(s_n)▒t_j^((n))  x_j^((n)) I_n . This implies that X is a generating set of M_n (I).

Now we prove that X is linearly independent. For our purpose we assume

∑_(j=0)^(s_0)▒t_j^((0))  x_j^((0))+∑_(j=0)^(s_1)▒t_j^((1))  x_j^((1)) I_1+⋯+∑_(j=0)^(s_n)▒t_j^((n))  x_j^((n)) I_n=0, by definition of n-refined vector space we find

∑_(j=0)^(s_i)▒t_j^((i))  x_j^((i)) for all i, hence t_j^((i))=0 for all i,j, since each X_i is linearly independent itself. Thus our proof is complete.

(c) It holds directly from (b).

Theorem 16:

Let (V,+,.) be a vector space over a field K, V_n (I) be the corresponding strong n-refined neutrosophic vector space over the n-refined neutrosophic field K_n (I), M_n (I)=M+MI_1+⋯+MI_n be a strong n-refined AHS-subspace. Then:

(a) M_n (I) is a submodule of  V_n (I).

(b) If Y is a bases of M, X=⋃_(i=0)^n▒〖YI_i 〗 is a bases of M_n (I).

(c) dim⁡(M_n (I))=∑_(i=0)^n▒〖dim⁡(M)〗=n.dim⁡(M).

Remark 17:

If V_n (I) is a strong n-refined neutrosophic vector space over the n-refined neutrosophic field K_n (I), and 

M_n (I)=M_0+M_1 I_1+⋯+M_n I_n is a strong n-refined AH-subspace, then it is not supposed to be a submodule.

Example18 :

Let V=R^2 be a vector space over R, V_2 (I)=R_2^2 (I)={(a,b)+(c,d)I_1+(e,f)I_2;a,b,c,d,e,f∈R} be the corresponding strong 2-refined neutrosophic vector space over the neutrosophic field R_2 (I).

M=<0,1>,N=<(1,0)> are two subspaces of V, T=M+NI_1+NI_2 is a strong AH-subspace of V_2 (I).

x=(0,1)+(2,0) I_1+(1,0) I_2∈T,r=1+1.I_1+1.I_2∈R_2 (I),

r.x=1.(0,1)+1.(0,1) I_1+1.(0,1) I_2+1.(2,0) I_1 I_1+1.(2,0) I_1+1.(1,0)I_1 I_2+1.(0,1) I_2+1.(2,0) I_1 I_2+1.(2,0) I_2 I_2=(0,1)+[(0,1)+(2,0)+(1,0)+(2,0)] I_1+[(0,1)+(0,1)+(2,0)]I_2=

(0,1)+(5,1) I_1+(2,2)I_2 , r.x does not belong to T, thus T is not a submodule.

Definition 19:

Let V_n (I) be a weak/strong n-refined neutrosophic vector space, M_n (I)=M_0+M_1 I_1+⋯+M_n I_n,

W_n (I)=W_0+W_1 I_1+⋯+W_n I_n be two weak/strong AH-subspaces of V_n (I), we define:

(a) M_n (I)∩W_n (I)=(M_0∩W_0 )+(M_1∩W_1 ) I_1+⋯+(M_n∩W_n)I_n.

(b) M_n (I)+W_n (I)=(M_0+W_0 )+(M_1+W_1 ) I_1+⋯+(M_n+W_n)I_n.

Theorem 20:

Let V_n (I) be a weak n-refined neutrosophic vector space, M_n (I)=M_0+M_1 I_1+⋯+M_n I_n,

W_n (I)=W_0+W_1 I_1+⋯+W_n I_n be two weak AH-subspaces of V_n (I). Then:

M_n (I)∩W_n (I),M_n (I)+W_n (I) are two weak AH-subspaces of V_n (I).

Proof:

Since M_i∩W_i,M_i+W_i are subspaces of V for all i, we obtain the proof.

Theorem 21:

Let V_n (I) be a strong n-refined neutrosophic vector space, M_n (I)=M_0+M_1 I_1+⋯+M_n I_n,

W_n (I)=W_0+W_1 I_1+⋯+W_n I_n be two strong AH-subspaces of V_n (I). Then:

(a) M_n (I)∩W_n (I) is a strong AH-subspaces of V_n (I).

(b) M_n (I)+W_n (I) is not supposed to be a strong AH-subspace of V_n (I).

Definition 22:

Let V,W be two vector spaces over the field K, f_i:V→W;0≤i≤n+1 be n+1 linear transformations,  V_n (I),W_n (I) be the corresponding weak n-refined neutrosophic vector spaces over the field K respectively. We say:

(a) f:V_n (I)→W_n (I);f(∑_(i=0)^n▒〖a_i I_i)=f_0 (a_0 )+f_1 (a_1 ) I_1+⋯+f_n (a_n ) I_n=∑_(i=0)^n▒〖f_i (a_i)I_i 〗〗 is a weak AH-linear transformation.

(b) If f_i=f_j  for all i,j, we call f a weak AHS-linear transformation.

Example 23:

(a) Let V=R^3,W=R^2 be two vector spaces over the field R, V_2 (I)=R_2^3 (I)={ (x_0,y_0 z_0 )+(x_1,y_1,z_1 ) I_1+(x_2,y_2,z_2 ) I_2; x_i,y_i,z_i∈R},

W_2 (I)={(x_0,y_0 )+(x_1,y_1 ) I_1+(x_2,y_2 ) I_2; x_i,y_i∈R} be the corresponding weak 2-refined neutrosophic vector spaces. We have g:V→W;g(a,b,c)=(b,c),h:V→W;h(a,b,c)=(2a,0)

s:V→W;s(a,b,c)=(2b,3c) are three linear transformations. 

(b) f:V_2 (I)→W_2 (I);f(m+nI_1+qI_2 )=g(m)+h(n) I_1+s(q) I_2;m,n,q∈V is a weak AH-linear transformation.

(c) We clarify f as follows:

x=(1,2,2)+(1,0,1) I_1+(3,-1,0)I_2∈V_2 (I),

f(x)=g(1,2,2)+[h(1,0,1) 〖]I〗_1+[s(3,-1,0)] I_2=(2,2)+(2,0) I_1+(-2,0)I_2.

(d) k:V_2 (I)→W_2 (I);k(m+nI_1+qI_2 )=g(m)+g(n) I_1+g(q) I_2;m,n,q∈V is a weak AHS-linear transformation.

Definition 24:

Let V,W be two vector spaces over the field K, f_i:V→W;0≤i≤n+1 be n+1 linear transformations,  V_n (I),W_n (I) be the corresponding strong n-refined neutrosophic vector spaces over the n-refined neutrosophic field K_n (I) respectively. We say:

(a) f:V_n (I)→W_n (I);f(∑_(i=0)^n▒〖a_i I_i)=f_0 (a_0 )+f_1 (a_1 ) I_1+⋯+f_n (a_n ) I_n=∑_(i=0)^n▒〖f_i (a_i)I_i 〗〗 is a strong AH-linear transformation.

(b) If f_i=f_j  for all i,j, we call f a strong AHS-linear transformation.

Example 25:

(a) Let V=R^3,W=R^2 be two vector spaces over the field R, V_2 (I)=R_2^3 (I)={(x_0,y_0 z_0 )+(x_1,y_1,z_1 ) I_1+(x_2,y_2,z_2 ) I_2; x_i,y_i,z_i∈R},

W_2 (I)={(x_0,y_0 )+(x_1,y_1 ) I_1+(x_2,y_2 ) I_2; x_i,y_i∈R} be the corresponding strong 2-refined neutrosophic vector spaces over the 2-refined neutrosophic field R_2 (I). We have g:V→W;g(a,b,c)=(b,c),h:V→W;h(a,b,c)=(2a,0),

s:V→W;s(a,b,c)=(2b,3c) are three linear transformations. 

(b) f:V_2 (I)→W_2 (I);f(m+nI_1+qI_2 )=g(m)+h(n) I_1+s(q) I_2;m,n,q∈V is a strong AH-linear transformation.

(c) We clarify f as follows:

x=(1,2,2)+(1,0,1) I_1+(3,-1,0)I_2∈V_2 (I),

f(x)=g(1,2,2)+[h(1,0,1) 〖]I〗_1+[s(3,-1,0)] I_2=(2,2)+(2,0) I_1+(-2,0)I_2.

(d) k:V_2 (I)→W_2 (I);k(m+nI_1+qI_2 )=g(m)+g(n) I_1+g(q) I_2;m,n,q∈V is a strong AHS-linear transformation.

Definition 26:

Let V_n (I),W_n (I) be two weak/strong n-refined neutrosophic vector spaces,

f:V_n (I)→W_n (I);f(∑_(i=0)^n▒〖a_i I_i)=f_0 (a_0 )+f_1 (a_1 ) I_1+⋯+f_n (a_n ) I_n=∑_(i=0)^n▒〖f_i (a_i)I_i 〗〗 be a weak/strong AH-linear transformation. We define:

(a) AH-Ker(f)=Ker(f_0 )+Ker(f_1 ) I_1+⋯+Ker(f_n)I_n.

(b) AH-Im(f)=Im(f_0 )+Im(f_1 ) I_1+⋯+Im(f_n ) I_n.

Theorem 27:

Let V_n (I),W_n (I) be two weak n-refined neutrosophic vector spaces,

f:V_n (I)→W_n (I);f(∑_(i=0)^n▒〖a_i I_i)=f_0 (a_0 )+f_1 (a_1 ) I_1+⋯+f_n (a_n ) I_n=∑_(i=0)^n▒〖f_i (a_i)I_i 〗〗 be a weak AH-linear transformation. Then:

(a) AH-Ker(f) is a weak AH-subspace of V_n (I).

(b) AH-Im(f) is a weak AH-subspace of W_n (I).

(c) If M_n (I)=M_0+M_1 I_1+⋯+M_n I_n is a weak AH-subspace of V_n (I), f(M_n (I)) is a weak AH-subspace of W_n (I).

Proof:

(a) Since Ker(f_i) is a subspace of V, we find that

 AH-Ker(f)=Ker(f_0 )+Ker(f_1 ) I_1+⋯+Ker(f_n)I_n is a weak AH-subspace of V_n (I).

(b) Since Im(f_i) is a subspace of W, we find that AH-Im(f)=Im(f_0 )+Im(f_1 ) I_1+⋯+Im(f_n ) I_n is a weak AH-subspace of W_n (I).

(c) It is known that f_i (M_i) is a subspace of W, hence

 f(M_n (I))=f_0 (M_0 )+f_1 (M_1 ) I_1+⋯+f_n (M_n)I_n is a weak AH-subspace of W_n (I).

Theorem 28:

Let V_n (I),W_n (I) be two strong n-refined neutrosophic vector spaces over the n-refined neutrosophic field K_n (I),

f:V_n (I)→W_n (I);f(∑_(i=0)^n▒〖a_i I_i)=f_0 (a_0 )+f_1 (a_1 ) I_1+⋯+f_n (a_n ) I_n=∑_(i=0)^n▒〖f_i (a_i)I_i 〗〗 be a strong AH-linear transformation. Then:

(a) AH-Ker(f) is a strong AH-subspace of V_n (I).

(b) AH-Im(f) is a strong AH-subspace of W_n (I).

(c) If M_n (I)=M_0+M_1 I_1+⋯+M_n I_n is a strong AH-subspace of V_n (I), f(M_n (I)) is a strong AH-subspace of W_n (I).

Theorem 29 :

Let V_n (I),W_n (I) be two weak n-refined neutrosophic vector spaces over the field K,

f:V_n (I)→W_n (I);f(∑_(i=0)^n▒〖a_i I_i)=f_0 (a_0 )+f_1 (a_1 ) I_1+⋯+f_n (a_n ) I_n=∑_(i=0)^n▒〖f_i (a_i)I_i 〗〗 be a weak AH-linear transformation. Then:

f(x+y)=f(x)+f(y),f(r.x)=r.f(x) for all x,y∈V_n (I),r∈K.

Proof:

Let x=∑_(i=0)^n▒〖a_i I_i 〗,y=∑_(i=0)^n▒〖b_i I_i 〗 be two arbitrary elements in V_n (I), r∈K be any element in the field K, we have:

f(x+y)=f(∑_(i=0)^n▒〖(a_i+b_i ) I_i 〗)=∑_(i=0)^n▒〖f_i (a_i+b_i)I_i 〗=∑_(i=0)^n▒〖f_i (a_i)I_i 〗+∑_(i=0)^n▒〖f_i (b_i)I_i 〗=f(x)+f(y).

f(r.x)=f(∑_(i=0)^n▒〖ra_i I_i 〗)=∑_(i=0)^n▒〖f_i (〖ra〗_i)I_i 〗=r.∑_(i=0)^n▒〖f_i (a_i ) I_i=r.f(x)〗. 

Theorem 30:

Let V_n (I),W_n (I) be two strong n-refined neutrosophic vector spaces over the n-refined neutrosophic field  K_n (I),

f:V_n (I)→W_n (I);f(∑_(i=0)^n▒〖a_i I_i)=f_0 (a_0 )+f_1 (a_1 ) I_1+⋯+f_n (a_n ) I_n=∑_(i=0)^n▒〖f_i (a_i)I_i 〗〗 be a strong AH-linear transformation. Then:

f(x+y)=f(x)+f(y),f(r.x)=r.f(x) for all x,y∈V_n (I),r∈K_n (I).

Proof:

Let x=∑_(i=0)^n▒〖a_i I_i 〗,y=∑_(i=0)^n▒〖b_i I_i 〗 be two arbitrary elements in V_n (I), r=∑_(i=0)^n▒〖r_i I_i 〗∈K_n (I) be any element in the n-refined neutrosophic field K_n (I), we have:

f(x+y)=f(∑_(i=0)^n▒〖(a_i+b_i ) I_i 〗)=∑_(i=0)^n▒〖f_i (a_i+b_i)I_i 〗=∑_(i=0)^n▒〖f_i (a_i)I_i 〗+∑_(i=0)^n▒〖f_i (b_i)I_i 〗=f(x)+f(y).

For the proof of the second proposition we use induction on n. If n=0, the theorem is true clearly.

Suppose that it is true for n-1, we must prove it for n.

f(r.x)=f(∑_(i,j=0)^n▒〖r_i a_j I_i 〗 I_j )=f(∑_(i,j=0)^(n-1)▒〖r_i a_j I_i 〗 I_j+(∑_(i=0)^n▒〖r_i I_i 〗) a_n I_n), we can write

 ∑_(i,j=0)^(n-1)▒〖r_i a_j I_i 〗 I_j=m_0+m_1 I_1+⋯+m_(n-1) I_(n-1),

(∑_(i=0)^n▒〖r_i I_i 〗) a_n I_n=r_1 a_n I_1+r_2 a_n I_2+⋯+(r_0 a_n+r_n a_n)I_n,

r.x=∑_(i,j=0)^(n-1)▒〖r_i a_j I_i 〗 I_j+(∑_(i=0)^n▒〖r_i I_i 〗) a_n I_n=m_0+(m_1+r_1 a_n ) I_1+(m_2+r_2 a_n ) I_2+⋯+(r_0 a_n+r_n a_n)I_n,

f(r.x)=f_0 (m_0 )+f_1 (m_1+r_1 a_n ) I_1+f_2 (m_2+r_2 a_n ) I_2+⋯f_n (r_0 a_n+r_n a_n)I_n=

f_0 (m_0 )+[f_1 (m_1 )+r_1 f_1 (a_n )] I_1+⋯+[r_0 f_n (a_n )+r_n f_n (a_n )] I_n=r.f(x).

Theorem 31:

Let V_n (I),W_n (I),U_n (I) be three weak n-refined neutrosophic vector spaces over the field K,

f:W_n (I)→U_n (I);f(∑_(i=0)^n▒〖a_i I_i)=f_0 (a_0 )+f_1 (a_1 ) I_1+⋯+f_n (a_n ) I_n=∑_(i=0)^n▒〖f_i (a_i)I_i 〗〗,

g:V_n (I)→W_n (I);g(∑_(i=0)^n▒〖a_i I_i)=g_0 (a_0 )+g_1 (a_1 ) I_1+⋯+g_n (a_n ) I_n=∑_(i=0)^n▒〖g_i (a_i)I_i 〗〗,

 be two weak AH-linear transformations. Then:

(a) fog=∑_(i=0)^n▒〖(f_i og_i)〗.

(b) fog is a weak AH-linear transformation between V_n (I),U_n (I).

Proof:

(a) Let x=∑_(i=0)^n▒〖a_i I_i 〗∈V_n (I), fog(x)=f(∑_(i=0)^n▒〖g_i (a_i ) I_i 〗)=f(g_0 (a_0 )+g_1 (a_1 ) I_1+⋯+g_n (a_n ) I_n)=

f_0 (g_0 (a_0 ))+f_1 (g_1 (a_1 )) I_1+⋯+f_n (g_n (a_n )) I_n=∑_(i=0)^n▒(f_i og_i ) (a_i)I_i.

(b) Since f_i og_i is a linear transformation for all i, then we get the proof.

Theorem 32:

Let V_n (I),W_n (I),U_n (I) be three strong n-refined neutrosophic vector spaces over the n-refined neutrosophic field K,

f:W_n (I)→U_n (I);f(∑_(i=0)^n▒〖a_i I_i)=f_0 (a_0 )+f_1 (a_1 ) I_1+⋯+f_n (a_n ) I_n=∑_(i=0)^n▒〖f_i (a_i)I_i 〗〗,

g:V_n (I)→W_n (I);g(∑_(i=0)^n▒〖a_i I_i)=g_0 (a_0 )+g_1 (a_1 ) I_1+⋯+g_n (a_n ) I_n=∑_(i=0)^n▒〖g_i (a_i)I_i 〗〗,

 be two strong AH-linear transformations. Then:

(a) fog=∑_(i=0)^n▒〖(f_i og_i)〗.

(b) fog is a strong AH-linear transformation between V_n (I),U_n (I).

Definition 33:

Let M(I_1,I_2) be a strong refined neutrosophic module over the refined neutrosophic ring R(I_1,I_2), P,Q,S be three submodules of M. The set N = (P,QI_1,SI_2 )={(a,bI_1,cI_2 );a∈P,b∈Q,c∈S} is called a strong AH-submodule of the strong refined neutrosophic module M(I_1,I_2).

If P = Q = S, we call N a strong AHS-submodule.

Theorem 34:

Let M(I_1,I_2) be a strong refined neutrosophic module over the refined neutrosophic ring R(I_1,I_2),

N = (P,PI_1,PI_2 ) be a strong AHS-submodule. Then N is a submodule by classical meaning.

Definition 35:

Let M,W be two modules over the ring R, M(I_1,I_2 )  and W(I_1,I_2) be the corresponding strong refined neutrosophic modules over the refined neutrosophic ring R(I_1,I_2). Let f,g,h:M→W be three homomorphisms, then [f,g,h]:M(I_1,I_2 )→W(I_1,I_2 );[f,g,h](a,bI_1,cI_2 )=(f(a),g(b) I_1,h(c) I_2) is called a strong AH-homomorphism. If f=g=h, we get the strong AHS-homomorphism.

Definition 36:

Let M(I_1,I_2 ),W(I_1,I_2) be two strong refined neutrosophic modules over the refined neutrosophic ring R(I_1,I_2), [f,g,h]:M(I_1,I_2 )→W(I_1,I_2 ) be a strong AH-homomorphism, we define

(a) AH-Ker[f,g,h]=(Ker(f),Ker(g) I_1,Ker(h) I_2 )={(a,bI_1,cI_2 );a∈Ker(f),b∈Ker(g),c∈Ker(h)}.

(b) AH-Im[f,g,h]=(Im(f),Im(g) I_1,Im(h) I_2).

Theorem 37:

Let M(I_1,I_2 ),W(I_1,I_2) be two strong refined neutrosophic modules over the refined neutrosophic ring R(I_1,I_2), [f,g,h]:M(I_1,I_2 )→W(I_1,I_2 ) be a strong AH-homomorphism.

(a) If N = (P,QI_1,SI_2 ) is a strong AH-submodule of M(I_1,I_2 ), then [f,g,h](N) is a strong AH-submodule of W(I_1,I_2 ).

(b) [f,g,h] is a classical module homomorphism.

(c) AH-Ker[f,g,h] is a strong AH-submodule of M(I_1,I_2 ).

(d) AH-Im[f,g,h] is a strong AH-submodule of W(I_1,I_2).

Proof:

(a) Since f(P),g(Q),h(S) are submodules of N, we find that [f,g,h](N)=(f(P),g(Q) I_1,h(S) I_2) is a strong AH-submodule of W(I_1,I_2 ).

(b) Let m=(x,yI_1,zI_2 ),n=(a,bI_1,cI_2) be two arbitrary elements in M(I_1,I_2 ), r=(t,uI_1,vI_2) be any element in R(I_1,I_2),

m+n=(x+a,[y+b] I_1,[z+c]I_2), r.m=(tx,[xu+yt+yu+yv+zu] I_1,[xv+zt+zv] I_2),

[f,g,h](m+n)=(f(x+a),g([y+b])I_1,h([z+c])I_2)=(f(x),g(y)I_1,h(z)I_2 )+(f(a),g(b) I_1,h(c) I_2 )=[f,g,h](m)+[f,g,h](n).

[f,g,h](r.m)=(f(tx),g([xu+yt+yu+yv+zu] 〖)I〗_1,h([xv+zt+zv]) I_2 )=

(t,uI_1,vI_2 ).(f(x),g(y)I_1,h(z)I_2 )=r.[f,g,h](m). Thus [f,g,h] is a classical homomorphism.

(c) Since Ker(f),Ker(g),Ker(h) are submodules of M, we get AH-Ker[f,g,h]=(Ker(f),Ker(g) I_1,Ker(h) I_2 ) as a strong AH-submodule of  M(I_1,I_2 ).

(d) Since Im(f),Im(g),Im(h) are submodules of W, we get AH-Im[f,g,h]=(Im(f),Im(g) I_1,Im(h) I_2) as a strong AH-submodule of  W(I_1,I_2 ).

Example 38:

(a) Let M=R^2,W=R be two modules over the ring R,

f:M→W;f(x,y)=2x,g:M→W;g(x,y)=3y,h:M→W;h(x,y)=x+y are three homomorphisms.

(b) [f,g,h]:M(I_1,I_2 )→W(I_1,I_2 );[f,g,h]((x,y),(z,t) I_1,(s,m) I_2 )=(f(x,y),g(z,t) I_1,h(s,m) I_2)=

(2x,3tI_1,[s+m] I_2) is a strong AH-homomorphism, where x,y,z,t,s,m∈R.

(c) P={(0,x);x∈R},Q={(x,0);x∈R} are two submodules of M,

N=(P,PI_1,QI_2 )={((0,x),(0,y) I_1,(z,0) I_2;x,y,z∈R} is a strong AH-submodule of M(I_1,I_2 ).

(d) f(P)={0},g(P)={3y;y∈R}=R,h(Q)={z;z∈R}=R,

[f,g,h](N)=(f(P),g(P) I_1,h(Q) I_2 )=(0,RI_1,RI_2 )={(0,xI_1,yI_2 );x,y∈R} is a strong AH-submodule of W(I_1,I_2).

(e) Ker(f)={(0,x);x∈R},Ker(g)={(x,0);x∈R},Ker(h)={(y,-y);y∈R},

AH-Ker[f,g,h]=(ker⁡(f),Ker(g) I_1,Ker(h) I_2 )={(0,x),(y,0) I_1,(z,-z) I_2 );x,y,z∈R}.

Definition 39: 

Let M(I) = M+MI be a strong/weak neutrosophic module, the set

S=P+QI={x+yI;x∈P,y∈Q},where P and Q are submodules of V is called an AH-submodule of M(I).

If P=Q, S is called an AHS-submodule of M(I).

Example 40:

We have M = Z^2=Z×Z is a module over R, P=<(0,1)>,Q=<(1,0)>, are two submodules of M. The set

S=P+QI={(0,a)+(b,0)I;a,b∈Z} is an AH-submodule of M(I).

The set L=P+PI={(0,a)+(0,b)I};a,b∈Z is an AHS-submodule of M(I).

Theorem 41:

Let M(I) = M+MI be a neutrosophic weak module over the ring R, and let S=P+QI be an AH-submodule of M(I), then S is a submodule.

Proof:

Suppose that x=a+bI,y=c+dI∈S;a,c∈P,b,d∈Q, 

x+y=(a+c)+(b+d)I∈S. For each scalar m∈R we obtain m.x=m.a+(m.b)I∈S, since P and Q are submodules; thus S=P+QI is a submodule of M(I) over the ring R.

Theorem 42:

Let M(I) be a neutrosophic strong module over a neutrosophic ring R(I), let S=P+PI be an AHS-submodule. Then  S is a submodule of M(I).

Definition 43:

(a) Let M and W be two modules, L_M:M→W be a homomorphism. The AHS-homomorphism can be defined as follows:

L:M(I)→W(I);L(a+bI)=L_M (a)+L_M (b)I. 

(b) If S = P+QI is an AH-submodule of M(I), L(S)=L_M (P)+L_M (Q)I.

(c) If S = P+QI is an AH-submodule of W(I), L^(-1) (S)=L_W^(-1) (P)+L_W^(-1) (Q)I.

(d) AH-Ker(L) =Ker(L_M)+Ker(L_M) I={x+yI;x,y∈Ker(L_M)}. 

Theorem 44:

Let W(I) and M(I) be two neutrosophic strong/weak modules, and  L:M(I)→W(I) be an AHS-homomorphism:

(a) AH-Ker(L) is an AHS-submodule of M(I).

(b) If  S = P+QI is an AH-submodule of M(I), L(S) is an AH-submodule of W(I).

(c) If S = P+QI is an AH-submodule of W(I), L^(-1) (S) is an AH-submodule of M(I).

Proof:

(a) Since Ker〖(L〗_M) is a submodule of M, we find that AH-Ker(L) =Ker(L_M)+Ker(L_M)I is an AHS-submodule of M(I).

(b) We have L(S)=L_M (P)+L_M (Q)I; thus L(S) is an AH-submodule of W(I), since L_M (P),L_M (Q)are submodules of W.

(c) By regarding that L^(-1) (S)=L_W^(-1) (P)+L_W^(-1) (Q)I, 〖 L〗_W^(-1) (P)  and〖 L〗_W^(-1) (Q) are submodules of M, we obtain that  L^(-1) (S) is an AH-subModule of M(I).

Theorem 45:

Let W(I) and M(I) be two neutrosophic strong modules over a neutrosophic ring R(I), and  L:M(I)→W(I) be an AHS-homomorphism. Then

L(x+y)=L(x)+L(y),L(m.x)=m.L(x), for all x,y∈M(I),m∈R(I).

Proof:

Suppose x=a+bI,y=c+dI;a,b,c,d∈M,and m=s+tI∈K(I), we have

L(x+y)=L([a+c]+[b+d]I)=L_M (a+c)+L_M (b+d)I=[L_M (a)+L_M (b)I]+[L_M (c)+L_M (d)I]=L(x)+L(y). 

m.x=(s.a)+(s.b+t.a+t.b)I, L(m.x)=L_M (s.a)+L_M (s.b+t.a+t.b)I

= s.L_M (a)+[s.L_M (b)+t.L_M (a)+t.L_M (b)]I = (s+tI).(L_M (a)+L_M (b)I)=m.L(x).

Theorem 46:

Let S=P+QI be an AH-submodule of a neutrosophic weak module M(I) over a ring R, suppose that 

X={x_i;1≤i≤n} is a bases of P and Y={y_j;1≤j≤m} is a bases of Q then X∪YI is a bases of S.

Proof:

 Let z=x+yI be an arbitrary element in S; x∈P,y∈Q. Since P and Q are submodules of M we can write

x=a_1 x_1+a_2 x_2+⋯+a_n x_n; a_i∈R and x_i∈X, y=b_1 y_1+b_2 y_2+⋯+b_m y_m; b_i∈K,y_i∈Y.

Now we obtain z=(a_1 x_1+⋯+a_n x_n )+(b_1 y_1 I+⋯+b_m y_m I); thus X∪YI generates the subspace S.

X∪YI is linearly independent set. Assume that (a_1 x_1+⋯+a_n x_n )+(b_1 y_1 I+⋯+b_m y_m I)=0, this implies 

a_1 x_1+a_2 x_2+⋯+a_n x_n=0 and  〖(b〗_1 y_1+b_2 y_2+⋯+b_m y_m)I=0. Since X and Y are linearly independent sets over R, we get a_i=b_j=0 for all i,j and X∪YI is linearly independent then it is a basis of S.

Result 47:

Let S=P+QI be an AH-submodule of a neutrosophic weak module M(I) with finite dimension over a ring R, from Theorem 3.8 and the fact that X∩YI=∅, we find dim⁡(S)=dim⁡(P)+dim⁡(Q).

Example 48:

Let M =Z^3=Z×Z×Z is a module over the ring Z, P=<(0,0,1)>,Q=<(0,1,0)> be two submodules of M,

(a) S=P+QI={(0,0,m)+(0,n,0)I; m,n∈Z }is an AH-submodule of M(I).

(b) The set {(0,0,1),(0,1,0)I} is a bases of S, dim⁡(S)=dim⁡(P)+dim⁡(Q)=1+1=2.

(c) L_M:M→M; L_M (x,y,z)=(x+y,y,z)for all x,y,z∈Z is a homomorphism, the corresponding AHS-homomorphism is 

L:M(I)→M(I);L[(x,y,z)+(a,b,c)I]=L_M (x,y,z)+L_M (a,b,c)I =(x+y,y,z)+(a+b,b,c)I.

(d) L(S)=L_M (P)+L_M (Q)=L_M {(0,0,m)}+L_M {(0,n,0)}I={(0,0,m)+(n,n,0)I}; m,n∈Z, which is an AH-submodule of M(I).

Example 49:

Let M=Z^2=Z×Z, W=Z^3=Z×Z×Z be two modules over the ring Z, L_M:M→W; L_M (x,y)=(x+y,x+y,x+y) is a homomorphism. The corresponding AHS-homomorphism  is

L:M(I)→W(I);L[(x,y)+(a,b)I]=(x+y,x+y,x+y)+(a+b,a+b,a+b)I. 

KerL_M=<(1,-1)>, AH-Ker(L)=Ker(L_M)+Ker(L_M) I=<(1,-1)>+<(1,-1)>I =

{(a,-a)+(b,-b)I;a,b∈Z} which is an AHS-submodule of M(I).

We find dim⁡(Ker(L))=1+1=2.

Definition 50:

Let M(I) be a neutrosophic strong/weak module, S=P+QI be an AH-submodule of M(I), we define

 AH-Quotient module as:

(M(I))⁄S=M⁄P+(M⁄(Q)) I= (x+P)+(y+Q)I;x,y∈M.

Theorem 51:

Let M(I) be a neutrosophic weak module over a ring R, and S=P+QI be an AH-submodule of M(I). The AH-Quotient (M(I))⁄S is a module with respect to the following operations:

Addition: [(x+P)+(y+Q)I]+[(a+P)+(b+Q)]I=(x+a+P)+(y+b+Q)I;x,y,a,b∈M.

Multiplication by a scalar: (m).[(x+P)+(y+Q)I]=(m.x+P)+(m.y+Q)I;

x,y∈M and m∈R.

Proof:

It is easy to check that operations are well defined, and ((M(I))⁄S,+) is abelian group.

Let z=[(x+P)+(y+Q)I]∈(M(I))⁄S, we have 1.z=z.

Assume that m,n∈R, we have m.(n.z)=m.[(n.x+P)+(n.y+Q)I]=(m.n.x+P)+(m.n.y+Q)I=(m.n).z.

(m+n).z=[(m+n).x+P]+[(m+n).y+Q]I=m.z+n.z.

Let h=[(a+P)+(b+Q)I]∈(M(I))⁄S, z+h=(x+a+P)+(y+b+Q)I,

m.(z+h)=(m.x+m.a+P)+(m.y+m.b+Q)m.z+m.h.

Conclusion

In this review, we have discussed many neutrosophic algebraic AH-structures, such as AH-subspaces, AH-submodules, AH-ideals and n-refined AH-ideals and spaces.

This work maybe very useful in the future studies in Turiyam algebraic AH-structures, especially spaces and modules, where they can be built as new generalizations of corresponding neutrosophic ones.

Acknowledgements: Author thanks the editorial team for the valuable time. 

Funding :Author declares that, there is no funding for this paper. 

Conflicts of Interest: Author declares that, there is no conflict of interest.

Ethics approval: This article does not contain any studies with human or animals participants.

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