Soft neutrosophic group.

1 Introduction

The concept of neutrosophic set was first introduced by Smarandache [13, 16] which is a generalization of the classical sets, fuzzy set , intuitionistic fuzzy set  and interval valued fuzzy set . Soft Set theory was initiated by Molodstov as a new mathematical tool which is free from the problems of parameterization inadequacy. In his paper , he presented the fundamental results of new theory and successfully applied it into several directions such as smoothness of functions, game theory, operations research, Riemann-integration, Perron integration, theory of probability. Later on many researchers followed him and worked on soft set theory as well as applications of soft sets in decision making problems and artificial intelligence. Now, this idea has a wide range of research in many fields, such as databases [5, 6], medical diagnosis problem , decision making problem , topology , algebra and so on.Maji gave the concept of neutrosophic soft set in  and later on Broumi and Smarandache defined intuitionistic neutrosophic soft set. We have worked with neutrosophic soft set and its applications in group theory.

2 Preliminaries

2.1 Nuetrosophic Groups

Definition 1  Let (G, *) be any group and let <G [union] I> = {a + bl : a, b [member of] G}. Then neutrosophic group is generated by I and [member of] under * denoted by N (G) = {<G [union] I>, *}. I is called the neutrosophic element with the property [I.sup.2] = I. For an integer n, n + I and nI are neutrosophic elements and 0.I = 0.

[I.sup.-1], the inverse of I is not defined and hence does not exist.

Theorem 1  Let N (G) be a neutrosophic group. Then

1) N (G) in general is not a group;

2) N (G) always contains a group.

Definition 2 A pseudo neutrosophic group is defined as a neutrosophic group, which does not contain a proper subset which is a group.

Definition 3 Let N (G) be a neutrosophic group. Then,

1) A proper subset N (H) of N (G) is said to be a neutrosophic subgroup of N (G) if N (H) is a neutrosophic group, that is, N (H) contains a proper subset which is a group.

2) N (H) is said to be a pseudo neutrosophic subgroup if it does not contain a proper subset which is a group.

Example 1 (N(Z), +), (N(Q), +) (N(R), +) and (N (C), +) are neutrosophic groups of integer, rational, real and complex numbers, respectively.

Example 2 Let Z7 = {o, 1, 2, ..., 6} be a group under addition modulo 7.

N (G) = {([Z.sub.7] [union] I), '+ 'mod ulo7} is a neutrosophic group which is in fact a group. For N (G) = {a + bI : a, b [member of] [Z.sub.7]} is a group under + ' modulo 7.

Definition 4 Let N (G) be a finite neutrosophic group. Let P be a proper subset of N (G) which under the operations of N (G) is a neutrosophic group. If o (P) / o (N (G)) then we call P to be a Lagrange neutrosophic subgroup.

Definition 5 N (G) is called weakly Lagrange neutrosophic group if N(G) has at least one Lagrange neutrosophic subgroup.

Definition 6 N (G) is called Lagrange free neutrosophic group if N(G) has no Lagrange neutrosophic subgroup.

Definition 7 Let N (G) be a finite neutrosophic group. Suppose L is a pseudo neutrosophic subgroup of N (G) and if o (L) / o (N (G)) then we call L to be a pseudo Lagrange neutrosophic subgroup.

Definition 8 If N (G) has at least one pseudo Lagrange neutrosophic subgroup then we call N(G) to be a weakly pseudo Lagrange neutrosophic group.

Definition 9 If N (G) has no pseudo Lagrange neutrosophic subgroup then we call N (G) to be pseudo Lagrange free neutrosophic group.

Definition 10 Let N (G) be a neutrosophic group. We say a neutrosophic subgroup H of N (G) is normal if we can find x and y in N (G) such that H = xHy for all x, y [member of] N (G) (Note x = y or y = [x.sup.-1] can also occur).

Definition 11 A neutrosophic group N (G) which has no nontrivial neutrosophic normal subgroup is called a simple neutrosophic group.

Definition 12 Let N (G) be a neutrosophic group. A proper pseudo neutrosophic subgroup P of N(G) is said to be normal if we have P = xPy for all x, y [member of] N (G). A neutrosophic group is said to be pseudo simple neutrosophic group if N (G) has no nontrivial pseudo normal subgroups.

2.2 Soft Sets

Throughout this subsection U refers to an initial universe, E is a set of parameters, P (U) is the power set of U, and A [subset] E. Molodtsov  defined the soft set in the following manner:

Definition 13  A pair (F, A) is called a soft set over U where F is a mapping given by F : A [right arrow] P(U).

In other words, a soft set over U is a parameterized family of subsets of the universe U. For e [member of] A, F (e) may be considered as the set of e -elements of the soft set (F, A), or as the set of e-approximate elements of the soft set.

Example 3 Suppose that U is the set of shops. E is the set of parameters and each parameter is a word or sentence. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let us consider a soft set (F, A) which describes the attractiveness of shops that Mr. Z is taking on rent. Suppose that there are five houses in the universe

U = {[h.sub.1], [h.sub.2], [h.sub.4], [h.sub.5]} under consideration, and that

A = {[e.sub.1], [e.sub.2], [e.sub.3]} be the set of parameters where

[e.sub.1] stands for the parameter 'high rent,

[e.sub.2] stands for the parameter 'normal rent,

[e.sub.3] stands for the parameter 'in good condition.

Suppose that

F ([e.sub.1]) = {[h.sub.1], [h.sub.4]},

F ([e.sub.2]) = {[h.sub.2], [h.sub.5]},

F ([e.sub.3]) = {[h.sub.3], [h.sub.4], [h.sub.5]}.

The soft set (F, A) is an approximated family

{F([e.sub.i]), i = 1, 2, 3} of subsets of the set U which gives us a collection of approximate description of an object. Thus, we have the soft set (F, A) as a collection of approximations as below:

(F, A) = {high rent = {[h.sub.1], [h.sub.4]}, normal rent = {[h.sub.2], [h.sub.5]}, in good condition = {[h.sub.3], [h.sub.4], [h.sub.5]}}.

Definition 14 . For two soft sets (F, A) and (H, B) over U, (F, A) is called a. soft subset of (H, B) if

1) A [subset or equal to] B and

2) F(e) [subset or equal to] H (e), for all e [member of] A.

This relationship is denoted by (F, A) [subset] (H, B). Similarly (F, A) is called a soft superset of (H, B) if (H, B) is a soft subset of (F, A) which is denoted by (F, A) [subset] (H, B).

Definition 15 . Two soft sets (F, A) and (H, B) over U are called soft equal if (F, A) is a soft subset of (H, B) and (H, B) is a soft subset of (F, A).

Definition 16 Let  (F, A) and (G, B) be two soft sets over a common universe U such that A [intersection] B [not equal to] [phi]. Then their restricted intersection is denoted by (F, A) PR (G, B) = (H, C) where (H, C) is defined as H(c) = F(c) [intersection] G(c) for all c [member of] C = A [intersection] B.

Definition 17  The extended intersection of two soft sets (F, A) and (G, B) over a common universe U is the soft set (H, C), where C = A U B, and for all e [member of] C, H (e) is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We write (F, A) [[intersection].sub.3] (G, B) = (H, C).

Definition 18  The restricted union of two soft sets (F, A) and (G, B) over a common universe U is the soft set (H, C), where C = A [union] B, and for all e [member of] C, H (e) is defined as the soft set (H, C) = (F, A) [[union].sub.R] (G, B) where C = A [intersection] B and H(c) = F(c) [union] G(c) for all c [member of] C.

Definition 19  The extended union of two soft sets (F, A) and (G, B) over a common universe U is the soft set (H, C), where C= A [union] B, and for all e [member of] C, H (e) is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We write (F, A) [[union].sub.[epsilon]] (G, B) = (H, C).

2.3 Soft Groups

Definition 20  Let (F, A) be a soft set over G. Then (F, A) is said to be a soft group over [member of] if and only if F (x) < [member of] forall X [member of] A. Example 4 Suppose that

G = A = [S.sub.3] = {e, (12), (13), (23), (123), (132)} . Then (F, A) is a soft group over [S.sub.3] where

F (e) = {e},

F (12) = {e (12)}'

F (13) = {e (13)}'

F (23) = {e (23)}'

F (123) = F (132) = {e, (123), (132)}.

Definition 21  Let (F, A) be a soft group over G. Then

1) (F, A) is said to be an identity soft group over G if F (x) = {e} for all x [member of] A, where e is the identity element of [member of] and

2) (F, A) is said to be an absolute soft group if F (x) = [member of] for all X [member of] A.

Definition 22 The restricted product (H, C) of two soft groups (F, A) and (K, B) over [member of] is denoted by the soft set (H, C) = [(F, A).sup.^.sub.o](K, B) where C = A [intersection] B and H is a set valued function from C to P (G) and is defined as H (c) = F (c)K (c) for all c [member of] C. The soft set (H, C) is called the restricted soft product of (F, A) and (K, B) over G.

3 Soft Neutrosophic Group

Definition 23 Let N (G) be a neutrosophic group and (F, A) be soft set over N (G).Then (F, A) is called soft neutrosophic group over N(G) if and only if F (x) < N (G), for all x [member of] A.

Example 5 Let

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be a neutrosophic group under addition modulo 4. Let A = {[e.sub.1], [e.sub.2], [e.sub.3], [e.sub.4]} be the set of parameters, then (F, A) is soft neutrosophic group over N ([Z.sub.4]) where

F ([e.sub.1]) = {0, 1, 2, 3}, F ([e.sub.2]) = {0, 1, 21, 3I},

F ([e.sub.3]) = {0, 2, 2I, 2 + 2I},

F ([e.sub.4]) = {0, 1, 2I, 3I, 2, 2 + 2I, 2 + I, 2 + 3I}.

Theorem 2 Let (F, A) and (H, A) be two soft neutrosophic groups over N(G). Then their intersection (F, A) [intersection] (H, A) is again a soft neutrosophic group over N(G).

Proof The proof is straightforward.

Theorem 3 Let (F, A) and (H, B) be two soft neutrosophic groups over N(G). If A [intersection] B = [phi], then (F, A) [union] (H, B) is a soft neutrosophic group over N (G).

Theorem 4 Let (F, A) and (H, A) be two soft neutrosophic groups over N (G). If F (e) [subset or equal to] H (e) for all e [member of] A, then (F, A) is a soft neutrosophic sub-group of (H, A).

Theorem 5 The extended union of two soft neutrosophic groups (F, A) and (K, B) over N (G) is not a soft neutrosophic group over N(G).

Proof Let (F, A) and (K, B) be two soft neutrosophic groups over N (G). Let C = A [union] B, then for all e [member of] C, (F, A) [[union].sub.[epsilon]] (K, B) = (H, C) where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As union of two subgroups may not be again a subgroup. Clearly if e [member of] C = A [intersection] B, then H (e) may not be a subgroup of N(G). Hence the extended union (H, C) is not a soft neutrosophic group over N(G).

Example 6 Let (F, A) and (K, B) be two soft neutrosophic groups over N([Z.sub.2]) under addition modulo 2, where

F ([e.sub.1]) = {0, 1}, F ([e.sub.2]) = {0, I}

And

K ([e.sub.2]) = {0, 1}, K ([e.sub.3]) = {0, 1 + I}.

Then clearly their extended union is not a soft neutrosophic group as

H ([e.sub.2]) = F ([e.sub.2]) [union] K ([e.sub.2]) = {0, 1, I} is not a subgroup of N([Z.sub.2]).

Theorem 6 The extended intersection of two soft neutrosophic groups over N(G) is soft neutrosophic group over N(G).

Theorem 7 The restricted union of two soft neutrosophic groups (F, A) and (K, B) over N(G) is not a soft neutrosophic group over N(G).

Theorem 8 The restricted intersection of two soft neutro sophic groups over N(G) is soft neutrosophic group over N(G).

Theorem 9 The restricted product of two soft neutrosophic groups (F, A) and (K, B) over N(G) is a soft neutrosophic group over N(G).

Theorem 10 The and operation of two soft neutrosophic groups over N(G) is soft neutrosophic group over N(G).

Theorem 11 The OR operation of two soft neutrosophic groups over N(G) may not be a soft neutrosophic group.

Definition 24 A soft neutrosophic group which does not contain a proper soft group is called soft pseudo neutrosophic group.

Example 7 Let

N([Z.sub.2]) = <[Z.sub.2] [union] I> = {0, 1, 1, 1 + I} be a neutrosophic group under addition modulo 2. Let A = {[e.sub.1], [e.sub.2], [e.sub.3]} be the set of parameters, then (F, A) is a soft pseudo neutrosophic group over N (G) where

F ([e.sub.1]) = {0, 1},

F ([e.sub.2]) = {0, I},

F ([e.sub.3]) = {0, 1 + I}.

Theorem 12 The extended union of two soft pseudo neutrosophic groups (F, A) and (K, B) over N(G) is not a soft pseudo neutrosophic group over N(G).

Example 8 Let

N([Z.sub.2]) = <[Z.sub.2] U I> = {0, 1, 1, 1 + I} be a neu trosophic group under addition modulo 2. Let (F, A) and (K, B) be two soft pseudo neutrosophic groups over N (G), where

F ([e.sub.1]) = {0, 1}, F ([e.sub.2]) = {0, I},

F ([e.sub.3]) = {0, 1 + I}.

And

K([e.sub.1]) = {0, 1 + I}, K ([e.sub.2]) = {0, 1}.

Clearly their restricted union is not a soft pseudo neutrosophic group as union of two subgroups is not a subgroup.

Theorem 13 The extended intersection of two soft pseudo neutrosophic groups (F, A) and (K, B) over

N(G) is again a soft pseudo neutrosophic group over N(G).

Theorem 14 The restricted union of two soft pseudo neutrosophic groups (F, A) and (K, B) over N(G) is not a soft pseudo neutrosophic group over N(G).

Theorem 15 The restricted intersection of two soft pseudo neutrosophic groups (F, A) and (K, B) over N(G) is again a soft pseudo neutrosophic group over N (G).

Theorem 16 The restricted product of two soft pseudo neutrosophic groups (F, A) and (K, B) over N(G) is a soft pseudo neutrosophic group over N (G).

Theorem 17 The AND operation of two soft pseudo neutrosophic groups over N(G) soft pseudo neutrosophic soft group over N(G).

Theorem 18 The OR operation of two soft pseudo neutrosophic groups over N(G) may not be a soft pseudo neutrosophic group.

Theorem 19 Every soft pseudo neutrosophic group is a soft neutrosophic group.

Proof The proof is straight forward.

Remark 1 The converse of above theorem does not hold.

Example 9 Let N([Z.sub.4]) be a neutrosophic group and (F, A) be a soft neutrosophic group over N ([Z.sub.4]).

Then

F ([e.sub.1]) = {0, 1, 2, 3}, F ([e.sub.2]) = {0, I, 2I, 3I},

F ([e.sub.3]) = {0, 2, 21, 2 + 2I}.

But (F, A) is not a soft pseudo neutrosophic group as (H, B) is clearly a proper soft subgroup of (F, A).

where

H K) = {0, 2}, H([e.sub.2]) = {0, 2}.

Theorem 20 (F, A) over N(G) is a soft pseudo neutrosophic group if N (G) is a pseudo neutrosophic group.

Proof Suppose that N(G) be a pseudo neutrosophic group, then it does not contain a proper group and for all e [member of] A, the soft neutrosophic group (F, A) over N (G) is such that F (e) < N (G). Since each F (e) is a pseudo neutrosophic subgroup which does not contain a proper group which make (F, A) is soft pseudo neutrosophic group.

Example 10 Let

N([Z.sub.2]) = <[Z.sub.2] [union] I> = {0, 1, 1, 1 + I} be a pseudo neutrosophic group under addition modulo 2. Then clearly (F, A) a soft pseudo neutrosophic soft group over N([Z.sub.2]), where

F ([e.sub.1]) = {0, 1}, F ([e.sub.2]) = {0 I},

F ([e.sub.3]) = {0, 1 + I}.

Definition 25 Let (F, A) and (H, B) be two soft neutrosophic groups over N (G). Then (H, B) is a soft neutrosophic subgroup of (F, A), denoted as (H, B) < (F, A), if

1) B [subset] A and

2) H (e) < F (e), for all e [member of] A.

Example 11 Let N([Z.sub.4]) = [[Z.sub.4] [union] I) be a soft neutrosophic group under addition modulo 4, that is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let (F, A) be a soft neutrosophic group over N([Z.sub.4]), then

F ([e.sub.1]) = {0, 1, 2, 3}, F ([e.sub.2]) = {0, I, 2I, 3I},

F ([e.sub.3]) = {0, 2, 2I, 2 + 2I},

F ([e.sub.4]) = {0, I, 2I, 3I, 2, 2 + 2I, 2 + I, 2 + 3I}.

(H, B) is a soft neutrosophic subgroup of (F, A), where

H ([e.sub.1]) = {0, 2}, H ([e.sub.2]) = {0, 2I},

H ([e.sub.4]) = {0, 1, 21, 3I}.

Theorem 21 A soft group over [member of] is always a soft neutrosophic subgroup of a soft neutrosophic group over

N (G) if A [subset] B.

Proof Let (F, A) be a soft neutrosophic group over N(G) and (H, B) be a soft group over G. As G [subset] N (G) and for all b [member of] B, H (b) < [member of] [subset] N (G). This implies H (e) < F (e), for all e [member of] A as B [subset] A. Hence (H, B(F, A).

Example 12 Let (F, A) be a soft neutrosophic group over N([Z.sub.4]), then

F ([e.sub.1]) = {0, 1, 2, 3}, F ([e.sub.2]) = {0, I, 2I, 3I},

F ([e.sub.3]) = {0, 2, 21, 2 + 2I}.

Let B = {[e.sub.1], [e.sub.3]} such that (H, B) < (F, A), where

H ([e.sub.1]) = {0, 2}, H ([sub.3]) = {0, 2}.

Clearly B [subset] A and H (e) < F (e) for all e [member of] B.

Theorem 22 A soft neutrosophic group over always contains a soft group over G.

Proof The proof is followed from above Theorem.

Definition 26 Let (F, A) and (H, B) be two soft pseudo neutrosophic groups over N(G). Then (H, B) is called soft pseudo neutrosophic subgroup of (F, A), denoted as (H, B)- (F, A), if

1) B [subset] A

2) H (e) < F (e), for all e [member of] A.

Example 13 Let (F, A) be a soft pseudo neutrosophic group over N([Z.sub.4]), where

F ([e.sub.1]) = {0, I, 2I, 3I}, F ([e.sub.2]) = {0, 2I}.

Hence (H, B) < (F, A) where

H ([e.sub.1]) = {n, 2I}.

Theorem 23 Every soft neutrosophic group (F, A) over N(G) has soft neutrosophic subgroup as well as soft pseudo neutrosophic subgroup.

Proof Straightforward.

Definition 27 Let (F, A) be a soft neutrosophic group over N(G), then (F, A) is called the identity soft neutrosophic group over N(G) if F (x) = {e}, for all x [member of] A, where e is the identity element of G.

Definition 28 Let (H, B) be a soft neutrosophic group over N (G), then (H, B) is called Full-soft neutrosophic group over N(G) if F (x) = N (G), for all x [member of] A.

Example 14 Let

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is a neutrosophic real group where R is set of real numbers and [I.sup.2] = I, therefore [I.sup.n] = I, for n a positive integer. Then (F, A) is a Full-soft neutrosophic real group where

F(e) = N(R), for all e [member of] A

Theorem 24 Every Full-soft neutrosophic group contain absolute soft group.

Theorem 25 Every absolute soft group over [member of] is a soft neutrosophic subgroup of Full-soft neutrosophic group over N(G).

Theorem 26 Let N(G) be a neutrosophic group. If order of N(G) is prime number, then the soft neutrosophic group (F, A) over N (G) is either identity soft neutrosophic group or Full-soft neutrosophic group.

Proof Straightforward.

Definition 29 Let (F, A) be a soft neutrosophic group over N (G). If for all e [member of] A, each F (e) is Lagrange neutrosophic subgroup of N(G), then (F, A) is called soft Lagrange neutrosophic group over N(G).

Example 15 Let N ([Z.sub.3] / {0})= {1, 2, 1, 21} is a neutrosophic group under multiplication modulo 3. Now {l, 2}, {1, I} are subgroups of N ([Z.sub.3] / {0}) which divides order of N ([Z.sub.3] / {0}). Then the soft neutrosophic group

(F, A) = {F ([e.sub.1]) = {1, 2}, F ([e.sub.2]) = {1, I}} is an example of soft Lagrange neutrosophic group.

Theorem 27 If N(G) is Lagrange neutrosophic group, then (F, A) over N(G) is soft Lagrange neutrosophic group but the converse is not true in general.

Theorem 28 Every soft Lagrange neutrosophic group is a soft neutrosophic group.

Proof Straightforward.

Remark 2 The converse of the above theorem does not hold.

Example 16 Let N(G) = {1, 2, 3, 4, 1, 21, 31, 41} be a neutrosophic group under multiplication modulo 5 and (F, A) be a soft neutrosophic group over N (G), where

F([e.sub.1]) = {1, 4, I, 2I, 3I, 4I}, F ([e.sub.2]) = {1, 2, 3, 4},

F([e.sub.3]) = {1, 1, 21, 31, 41}.

But clearly it is not soft Lagrange neutrosophic group as F ([e.sub.1]) which is a subgroup of N (G) does not divide order of N(G).

Theorem 29 If N (G) is a neutrosophic group, then the soft Lagrange neutrosophic group is a soft neutrosophic group.

Proof Suppose that N (G) be a neutrosophic group and (F, A) be a soft Lagrange neutrosophic group over N(G). Then by above theorem (F, A) is also soft neutrosophic group.

Example 17 Let N([Z.sub.4]) be a neutrosophic group and F, A) is a soft Lagrange neutrosophic group over N([Z.sub.4]) under addition modulo 4, where

F ([e.sub.1]) = {0, 1, 2, 3}, F ([e.sub.2]) = {0, I, 2I, 3I},

F ([e.sub.3]) = {0, 2, 21, 2 + 2I}.

But (F, A) has a proper soft group (H, B), where

H ([e.sub.1]) = {0, 2}, H ([e.sub.3]) = {0, 2}.

Hence (F, A) is soft neutrosophic group.

Theorem 30 Let (F, A) and (K, B) be two soft Lagrange neutrosophic groups over N(G). Then

1) Their extended union (F, A)[[intersection].sub.[epsilon]] (K, B) over N (G) is not soft Lagrange neutrosophic group over N(G).

2) Their extended intersection (F, A) [[union].sub.R] (K, B) over N(G) is not soft Lagrange neutrosophic group over N(G).

3) Their restricted union (F, A) [[union].sub.R] (K, B) over N(G) is not soft Lagrange neutrosophic group over N(G).

4) Their restricted intersection (F, A) [[intersection].sub.R] (K, B) over N(G) is not soft Lagrange neutrosophic group over N(G).

5) Their restricted product (F, A) (K, B) over N(G) is not soft Lagrange neutrosophic group over N(G).

Theorem 31 Let (F, A) and (H, B) be two soft Lagrange neutrosophic groups over N (G).Then

1) Their AND operation (F, A)a(K, B) is not soft Lagrange neutrosophic group over N(G).

2) Their OR operation (F, A) [disjunction] (K, B) is not a soft Lagrange neutrosophic group over N(G).

Definition 30 Let (F, A) be a soft neutrosophic group over N(G). Then (F, A) is called soft weakly Lagrange neutrosophic group if atleast one F(e) is a Lagrange neutrosophic subgroup of N (G), for some e [member of] A.

Example 18 Let N(G) = {1, 2, 3, 4, 1, 21, 31, 41} be a neutrosophic group under multiplication modulo 5, then (F, A) is a soft weakly Lagrange neutrosophic group over N(G), where

F ([e.sub.1]) = {1, 4, I, 2I, 3I, 4I}, F ([e.sub.2]) = {1, 2, 3, 4},

F ([e.sub.3]) = {1, 1, 21, 31, 41}.

As F ([e.sub.1]) and F ([e.sub.3]) which are subgroups of N(G) do not divide order of N(G).

Theorem 32 Every soft weakly Lagrange neutrosophic group (F, A) is soft neutrosophic group.

Remark 3 The converse of the above theorem does not hold in general.

Example 19 Let N ([Z.sub.4]) be a neutrosophic group under addition modulo 4 and A = {[e.sub.1], [e.sub.2]} be the set ofparameters, then (F, A) is a soft neutrosophic group over N ([Z.sub.4]), where

F ([e.sub.1]) = {0, I, 2I, 3I}, F ([e.sub.2]) = {0, 2I}. But not soft weakly Lagrange neutrosophic group over N ([Z.sub.4]).

Definition 31 Let (F, A) be a soft neutrosophic group over N(G). Then (F, A) is called soft Lagrange free neutrosophic group if F(e) is not Lagrangeneu-trosophic subgroup of N(G), for all e [member of] A.

Example 20 Let N(G) = {1, 2, 3, 4, 1, 21, 31, 4I} be a neutrosophic group under multiplication modulo 5 and then (F, A) be a soft Lagrange free neutrosophic group over N(G), where

F ([e.sub.1]) ={1, 4, I, 2I, 3I, 4I}, F ([e.sub.2]) = {1, I, 2I, 3I, 4I}.

As F ([e.sub.1]) and F ([e.sub.2]) which are subgroups of N(G) do not divide order of N(G).

Theorem 33 Every soft Lagrange free neutrosophic group (F, A) over N(G) is a soft neutrosophic group but the converse is not true.

Definition 32 Let (F, A) be a soft neutrosophic group over N (G). If for all e [member of] A, each F (e) is a pseudo Lagrange neutrosophic subgroup of N (G), then (F, A) is called soft pseudo Lagrange neutrosophic group over N(G).

Example 21 Let N([Z.sub.4]) be a neutrosophic group under addition modulo 4 and A = {[e.sub.1], [e.sub.2]} be the set of parameters, then (F, A) is a soft pseudo Lagrange neutrosophic group over N([Z.sub.4]) where

F([e.sub.1]) = {0, I, 2I, 3I}, F ([e.sub.2]) = {0, 2I}.

Theorem 34 Every soft pseudo Lagrange neutrosophic group is a soft neutrosophic group but the converse may not be true.

Proof Straightforward.

Theorem 35 Let (F, A) and (K, B) be two soft pseudo Lagrange neutrosophic groups over N(G). Then

1) Their extended union (F, A)[[union].sub.[epsilon]] (K, B) over N(G) is not a soft pseudo Lagrange neutrosophic group over N(G).

2) Their extended intersection (F, A) [[intersection].sub.[epsilon]] (K, B) over N(G) is not pseudo Lagrange neutrosophic soft group over N(G).

3) Their restricted union (F, A) [[union].sub.R] (K, B) over N(G) is not pseudo Lagrange neutrosophic soft group over N(G).

4) Their restricted intersection (F, A) [[intersection].sub.R] (K, B) over N(G) is also not soft pseudo Lagrange neutrosophic group over N(G).

5) Their restricted product [(F, A).sup.^.sub.o] (K, B) over N(G) is not soft pseudo Lagrange neutrosophic group over N(G).

Theorem 36 Let (F, A) and (H, B) be two soft pseudo Lagrange neutrosophic groups over N(G). Then

1) Their and operation (F, A) a(K, B) is not soft pseudo Lagrange neutrosophic group over N(G).

2) Their OR operation (F, A)V (K, B) is not a soft pseudo Lagrange neutrosophic soft group over N(G).

Definition 33 Let (F, A) be a soft neutrosophic group over N (G). Then (F, A) is called soft weakly pseudo Lagrange neutrosophic group if atleast one F (e) is a pseudo Lagrange neutrosophic subgroup of N (G), for some e [member of] A.

Example 22 Let N (G) = {1, 2, 3, 4, 1, 21, 31, 41} be a neutrosophic group under multiplication modulo 5 Then (F, A) is a soft weakly pseudo Lagrange neutrosophic group over N(G), where

F ([e.sub.1]) = {1, I, 2I, 3I, 41}, F ([e.sub.2]) = {1, I}.

As F([e.sub.1]) which is a subgroup of N(G) does not divide order of N(G).

Theorem 37 Every soft weakly pseudo Lagrange neutrosophic group (F, A) is soft neutrosophic group.

Remark 4 The converse of the above theorem is not true in general.

Example 23 Let N([Z.sub.4]) be a neutrosophic group under addition modulo 4 and A = {[e.sub.1], [e.sub.2]} be the set of parameters, then (F, A) is a soft neutrosophic group over N ([Z.sub.4]), where

F([e.sub.1]) = {), I, 2I, 3I}, F ([e.sub.2]) = {0, 2I}.

But it is not soft weakly pseudo Lagrange neutrosophic group.

Theorem 38 Let (F, A) and (K, B) be two soft weakly pseudo Lagrange neutrosophic groups over N (G). Then

1) Their extended union (F, A)[[union].sub.[epsilon]] (K, B) over N (G) is not soft weakly pseudo Lagrange neutrosophic group over N(G).

2) Their extended intersection (F, A) [[intersection].sub.[epsilon]] (K, B) over N(G) is not soft weakly pseudo Lagrange neutrosophic group over N(G).

3) Their restricted union (F, A) [[union].sub.R] (K, B) over N(G) is not soft weakly pseudo Lagrange neutrosophic group over N(G).

4) Their restricted intersection (F, A) [[intersection].sub.R] (K, B) over N(G) is not soft weakly pseudo Lagrange neutrosophic group over N(G).

5) Their restricted product [(F, A).sup.^.sub.o] (K, B) over N(G) is not soft weakly pseudo Lagrange neutrosophic group over N(G).

Definition 34 Let (F, A) be a soft neutrosophic group over N(G). Then (F, A) is called soft pseudo Lagrange free neutrosophic group if F(e) is not pseudo Lagrange neutrosophic subgroup of N (G), for all e [member of] A.

Example 24 Let N(G) = {1, 2, 3, 4, 1, 21, 31, 41} be a neutrosophic group under multiplication modulo 5 Then (F, A) is a soft pseudo Lagrange free neutrosophic group over N(G), where

F ([e.sub.1]) = {1, I, 2I, 3I, 4I}, F ([e.sub.2]) = {1, I, 2I, 3I, 4I}.

As F ([e.sub.1]) and F ([e.sub.2]) which are subgroups of N(G) do not divide order of N(G).

Theorem 39 Every soft pseudo Lagrange free neutrosophic group (F, A) over N(G) is a soft neutrosophic group but the converse is not true.

Theorem 40 Let (F, A) and (K, B) be two soft pseudo Lagrange free neutrosophic groups over N(G). Then

1) Their extended union (F, A)[[union].sub.[epsilon]] (K, B) over N(G) is not soft pseudo Lagrange free neutrosophic group over N(G).

2) Their extended intersection (F, A) [[intersection].sub.[epsilon]] (K, B) over N(G) is not soft pseudo Lagrange free neutrosophic group over N(G).

3) Their restricted union (F, A)[[union].sub.R] (K, B) over N(G) is not pseudo Lagrange free neutrosophic soft group over N(G).

4) Their restricted intersection (F, A) [[intersection].sub.R] (K, B) over N(G) is not soft pseudo Lagrange free neutrosophic group over N(G).

5) Their restricted product [(F, A).sup.^.sub.o] (K, B) over N(G) is not soft pseudo Lagrange free neutrosophic group over N(G).

Definition 35 A soft neutrosophic group (F, A) over N (G) is called soft normal neutrosophic group over N(G) if F(e) is a normal neutrosophic subgroup of N (G), for all e [member of] A.

Example 25 Let N (G) = {e, a, b, c, I, aI, bI, cI} be a neutrosophic group under multiplication where [a.sup.2] = [b.sup.2] = [c.sup.2] = e, bc = cb = a, ac = ca = b, ab = ba = c. Then (F, A) is a soft normal neutrosophic group over N (G) where

F ([e.sub.1]) = {e, a, I, aI},

F ([e.sub.2]) = {e, b, I, bI},

F ([e.sub.3]) = {e, c, I, cI}.

Theorem 42 Every soft normal neutrosophic group (F, A) over N(G) is a soft neutrosophic group but the converse is not true.

Theorem 42 Let (F, A) and (H, B) be two soft normal neutrosophic groups over N(G). Then

1) Their extended union (F, A)[[union].sub.[epsilon]] (K, B) over N(G) is not soft normal neutrosophic group over N(G).

2) Their extended intersection (F, A) [[intersection].sub.[epsilon]] (K, B) over N(G) is soft normal neutrosophic group over N(G).

3) Their restricted union (F, A)[[union].sub.R] (K, B) over N(G) is not soft normal neutrosophic group over N(G).

4) Their restricted intersection (F, A) [[intersection].sub.R] (K, B) over N(G) is soft normal neutrosophic group over N(G).

5) Their restricted product (F, A) (K, B) over N(G) is not soft normal neutrosophic soft group over N(G).

Theorem 43 Let (F, A) and (H, B) be two soft normal neutrosophic groups over N (G). Then

1) Their AND operation (F, A) [conjunction] (K, B) is soft normal neutrosophic group over N(G).

2) Their OR operation (F, A) [disjunction] (K, B) is not soft normal neutrosophic group over N(G).

Definition 36 Let (F, A) be a soft neutrosophic group over N(G). Then (F, A) is called soft pseudo normal neutrosophic group if F(e) is a pseudo normal neutrosophic subgroup of N(G), for all e [member of] A

Example 26 Let

N ([Z.sub.2]) = <[Z.sub.2] [union] I> = {0, 1, I, 1 + I} be a neutrosophic group under addition modulo 2 and let A = {[e.sub.1], [e.sub.2]} be the set of parameters, then (F, A) is soft pseudo normal neutrosophic group over N (G), where

F ([e.sub.1]) = {0, I}, F ([e.sub.2]) = {0, 1 + I}.

As F([e.sub.1]) and F([e.sub.2]) are pseudo normal subgroup of N(G).

Theorem 44 Every soft pseudo normal neutrosophic group (F, A) over N(G) is a soft neutrosophic group but the converse is not true.

Theorem 45 Let (F, A) and (K, B) be two soft pseudo normal neutrosophic groups over N(G). Then

1) Their extended union (F, A)[[union].sub.[epsilon]] (K, B) over N(G) is not soft pseudo normal neutrosophic group over N(G).

2) Their extended intersection (F, A) [[intersection].sub.[epsilon]] (K, B over N(G) is soft pseudo normal neutrosophic group over N(G).

3) Their restricted union (F, A) [[union].sub.R] (K, B) over N(G) is not soft pseudo normal neutrosophic group over N(G).

4) Their restricted intersection (F, A) [[union].sub.R] (K, B) over N(G) is soft pseudo normal neutrosophic group over N(G).

5) Their restricted product [(F, A).sup.^.sub.o] (K, B) over N(G) is not soft pseudo normal neutrosophic group over

Theorem 46 Let (F, A) and (K, B) be two soft pseudo normal neutrosophic groups over N (G). Then

1) Their AND operation (F, A) [conjunction] (K, B) is soft pseudo normal neutrosophic group over N(G).

2) Their OR operation (F, A) [disjunction] (K, B) is not soft pseudo normal neutrosophic group over N(G).

Definition 37 Let N(G) be a neutrosophic group. Then (F, A) is called soft conjugate neutrosophic group over N (G) if and only if F (e) is conjugate neutrosophic subgroup of N(G), for all e [member of] A.

Example 27 Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

be a neutrosophic group under addition modulo 6 and let P = {0, 3, 3I, 3 + 3I} and K = {0, 2, 4, 2 + 2I, 4 + 4I, 2I, 4I} are conjugate neutrosophic subgroups of N(G). Then (F, A) is soft conjugate neutrosophic group over N (G), where

F ([e.sub.1]) = {0, 3, 3I, 3 + 3I},

F ([e.sub.2]) = {0:2:4:2 + 2I, 4 + 4I, 2I, 4I}.

Theorem 47 Let (F, A) and (K, B) be two soft conjugate neutrosophic groups over N(G). Then

1) Their extended union (F, A)[[union].sub.[epsilon]] (K, B) over N (G) is not soft conjugate neutrosophic group over N(G).

2) Their extended intersection (F, A) [[intersection].sub.[epsilon]] (K, B) over N(G) is again soft conjugate neutrosophic group over N(G).

3) Their restricted union (F, A) [[union].sub.R] (K, B) over N(G) is not soft conjugate neutrosophic group over N(G).

4) Their restricted intersection (F, A) Pr (K, B) over N(G) is soft conjugate neutrosophic group over N(G).

5) Their restricted product (F, A) (K, B) over N(G) is not soft conjugate neutrosophic group over N(G).

Theorem 48 Let (F, A) and (K, B) be two soft conjugate neutrosophic groups over N (G). Then

1) Their AND operation (F, A) [conjunction] (K, B) is again soft conjugate neutrosophic group over N(G).

2) Their OR operation (F, A) [disjunction] (K, B) is not soft conjugate neutrosophic group over N(G).

Conclusion

In this paper we extend the neutrosophic group and subgroup, pseudo neutrosophic group and subgroup to soft neutrosophic group and soft neutrosophic subgroup and respectively soft pseudo neutrosophic group and soft pseudo neutrosophic subgroup. The normal neutrosophic subgroup is extended to soft normal neutrosophic subgroup. We showed all these by giving various examples in order to illustrate the soft part of the neutrosophic notions used.

Accepted: November 28, 2013.

Acknowledgements

We would like to thank Editors and referees for their kind suggestions that helped to improve this paper.

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Muhammad Shabir (1), Mumtaz Ali (2), Munazza Naz (3), and Florentin Smarandache (4)

(1, 2) Department of Mathematics, Quaid-i-Azam University, Islamabad, 44000, Pakistan. E-mail: mshabirbhatti@yahoo.co.uk, mumtazali770@yahoo.com (3) Department of Mathematical Sciences, Fatima Jinnah Women University, The Mall, Rawalpindi, 46000, Pakistan. E-mail: munazzanaz@yahoo.com (4) University of New Mexico, 705 Gurley Ave., Gallup, New Mexico 87301, USA E-mail: fsmarandache@gmail.com
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