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On Refined Neutrosophic Vector Spaces I.

1 Introduction and Preliminaries

Neutrosophy is a new branch of philosophy that studies the origin, nature and scope of neutralities, as well as their interactions with different ideational spectra. The concept of neutrosophic logic/set was introduced by Smarandache in [20,22] as a generalization of fuzzy log/set [29] and respectively intuitionistic fuzzy logic/set [9]. In neutrosophic logic, each proposition has a degree of truth (T), a degree of indeterminancy (I) and a degree of falsity (F), where T, I, F are standard or non-standard subsets of [-0,1+]. In [21], Smarandache introduced the notion of refined neutrosophic components of the form < [T.sub.1], [T.sub.2,] ..., [T.sub.p]; [I.sub.1],[I.sub.2], *, [I.sub.r]; [F.sub.1], [F.sub.2], ..., [F.sub.s] > of the neutrosophic components < T, I, F >. The refinement has given rise to the introduction of refined neutrosophic set and the extension of neutrosophic numbers a + bI into refined neutrosophic numbers of the form (a + [b.sub.1][I.sub.1] + [b.sub.2][I.sub.2] + ... + [b.sub.n][I.sub.n]) where a, [b.sub.1], [b.sub.2], ..., [b.sub.n] are real or complex numbers. Agboola in [4] introduced the concept of refined neutrosophic algebraic structures and studied refined neutrosophic groups with their basic and fundamental properties. Since then, several neutrosophic researchers have studied this concept and a great deal of results have been published. Recently, Adeleke et al. studied refined neutrosophic rings and refined neutrosophic subrings in [1] and also in [2], they presented several results and examples on refined neutrosophic ideals and refined neutrosophic ring homomorphisms.

The concept of neutrosophic vector space was introduced by Vasantha Kandasamy and Florentin Smarandache in [23]. Further studies on neutrosophic vector spaces were carried out by Agboola and Akinleye in [8] where they generalized some properties of vector spaces and showed that every neutrosophic vector space over a neutrosophic field (resp. field) is a vector space. A comprehensive review of neutrosophic set, neutrosophic soft set, fuzzy set, neutrosophic topological spaces, neutrosophic vector spaces and new trends in neutrosophic theory can be found in [3,5-7,10-19,23-28].

In the present paper, we present the concept of a refined neutrosophic vector space. Weak(strong) refined neutrosophic vector spaces and subspaces, and, strong refined neutrosophic quotient vector spaces are studied. Several interesting results and examples are presented. It is shown that every weak (strong) refined neutrosophic vector space is a vector space and it is equally shown that every strong refined neutrosophic vector space is a weak refined neutrosophic vector space.

For the purposes of this paper, it will be assumed that I splits into two indeterminacies [I.sub.1] [contradiction (true (T) and false (F))] and [I.sub.2] [ignorance (true (T) or false (F))]. It then follows logically that:

[I.sub.1][I.sub.1] = [I.sup.2.sub.1] = [I.sub.1], [I.sub.2][I.sub.2] = [I.sup.2.sub.2] = [I.sub.2], and [I.sub.1][I.sub.1] = [I.sub.2][I.sub.1] = [I.sub.1].

Definition 1.1. [4] If * : X([I.sub.1], [I.sub.2]) * X([I.sub.1], [I.sub.2]) [??] X([I.sub.1], [I.sub.2]) is a binary operation defined on X([I.sub.1],[I.sub.2]), then the couple (X([I.sub.1],[I.sub.2]), *) is called a refined neutrosophic algebraic structure and it is named according to the laws (axioms) satisfied by *.

Definition 1.2. [4] Let (X([I.sub.1], [I.sub.2]), +, .) be any refined neutrosophic algebraic structure where + and . are ordinary addition and multiplication respectively.

For any two elements (a, b[I.sub.1], c[I.sub.2]), (d, e[I.sub.1], f[I.sub.2]) [member of] X ([I.sub.1], [I.sub.2]), we define

(a, b[I.sub.1], c[I.sub.2]) + (d e[I.sub.1], f[I.sub.2]) = (a + d, (b + e)[I.sub.1], (c + f)[I.sub.2]), (a, b[I.sub.1], c[I.sub.2]).(d, e[I.sub.1], f[I.sub.2]) = (ad, (ae + bd + be + bf + ce) [I.sub.1], (af + cd + cf) [I.sub.2]).

Definition 1.3. [4] If "+" and "." are ordinary addition and multiplication, [I.sub.[kappa]] with [kappa] = 1,2 have the following properties:

1. [I.sub.[kappa]] + [I.sub.[kappa]] + ... + [I.sub.[kappa]] = n[I.sub.[kappa]].

2. [I.sub.[kappa]] + (-[I.sub.[kappa]]) = 0.

3. [I.sub.[kappa]] * [I.sub.[kappa]] .... [I.sub.[kappa]] = [I.sup.n.sub.[kappa]] = [I.sub.[kappa]] for all positive integers n > 1.

4. 0 * [I.sub.[kappa]] = 0.

5. [I.sup.-1.sub.[kappa]] is undefined and therefore does not exist.

Definition 1.4. [4] Let (G, *) be any group. The couple (G([I.sub.1], [I.sub.2]), *) is called a refined neutrosophic group generated by G, [I.sub.1] and [I.sub.2]. (G([I.sub.1],[I.sub.2]), *) is said to be commutative if for all x, y [member of] G([I.sub.1],[I.sub.2]), we have x * y = y * x. Otherwise, we call (G([I.sub.1],[I.sub.2]), *) a non -commutative refined neutrosophic group.

Definition 1.5. [4] If (X([I.sub.1], [I.sub.2]), *) and (Y([I.sub.1],[I.sub.2]), *') are two refined neutrosophic algebraic structures, the mapping

[phi] : (X([I.sub.1], [I.sub.2]), *) (Y([I.sub.1], [I.sub.2]), *')

is called a neutrosophic homomorphism if the following conditions hold:

1. [phi]((a, b[I.sub.1], c[I.sub.2]) * (d, e[I.sub.1], f[I.sub.2])) = [phi]((a, b[I.sub.1], c[I.sub.2])) *' [phi]((d, e[I.sub.1], f[I.sub.2])).

2. [phi]([I.sub.[kappa]]) = [I.sub.[kappa]] for all (a, b[I.sub.1], c[I.sub.2]), (d, e[I.sub.1], f[I.sub.2]) [member of] X ([I.sub.1], [I.sub.2]) and [kappa] = 1,2.

Example 1.6. [4] Let

[Z.sub.2]([I.sub.1],[I.sub.2]) = {(0, 0, 0), (1, 0, 0), (0, [I.sub.1], 0) (0, 0, [I.sub.2]), (0, [I.sub.1], [I.sub.2]), (1, [I.sub.1], 0), (1, 0,[I.sub.2]), (1,[I.sub.1],[I.sub.2])}. Then ([Z.sub.2]([I.sub.1], [I.sub.2]), +) is a commutative refined neutrosophic group of integers modulo 2. Generally for a positive integer n [greater than or equal to] 2, ([Z.sub.n]([I.sub.1],[I.sub.2]), +) is a finite commutative refined neutrosophic group of integers modulo n.

Example 1.7. [4] Let (G([I.sub.1],[I.sub.2]), *) and and (H([I.sub.1],[I.sub.2]), *') be two refined neutrosophic groups. Let 4 : G([I.sub.1],[I.sub.2]) * H([I.sub.1],[I.sub.2]) [right arrow] G([I.sub.1], [I.sub.2]) be a mapping defined by [phi](x, y) = x and let [psi] : G([I.sub.1], [I.sub.2]) * H([I.sub.1], [I.sub.2]) [right arrow] H([I.sub.1],[I.sub.2]) be a mapping defined by [psi](x, y) = y. Then [phi] and [psi] are refined neutrosophic group homomorphisms.

Definition 1.8. [1] Let (R, +, .) be any ring. The abstract system (R([I.sub.1],[I.sub.2]), +, .) is called a refined neutrosophic ring generated by R, [I.sub.1],[I.sub.2]. (R([I.sub.1], [I.sub.2]), +,.) is called a commutative refined neutrosophic ring if for all x, y [member of] R([I.sub.1],[I.sub.2]), we have xy = yx. If there exists an element e = (1,0,0) [member of] R([I.sub.1],[I.sub.2]) such that ex = xe = x for all x [member of] R([I.sub.1], [I.sub.2]), then we say that (R([I.sub.1], [I.sub.2]), +,.) is a refined neutrosophic ring with unity.

Definition 1.9. 1 Let (R([I.sub.1],[I.sub.2]), +,.) be a refined neutrosophic ring and let n [member of] [Z.sup.+].

(i) If nx = 0 for all x [member of] R([I.sub.1],[I.sub.2]), we call (R([I.sub.1],[I.sub.2]), +,.) a refined neutrosophic ring of characteristic n and n is called the characteristic of (R([I.sub.1],[I.sub.2]), +,.).

(ii) (R([I.sub.1],[I.sub.2]), +,.) is call a refined neutrosophic ring of characteristic zero if for all x [member of] R([I.sub.1],[I.sub.2]), nx = 0 is possible only if n = 0.

Example 1.10. [1]

(i) Z([I.sub.1], [I.sub.2]), Q([I.sub.1], [I.sub.2]), K([I.sub.1], [I.sub.2]), C([I.sub.1], [I.sub.2]) are commutative refined neutrosophic rings with unity of characteristics zero.

(ii) Let [Z.sub.2]([I.sub.1], [I.sub.2]) = {(0,0,0), (1,0,0), (0, [I.sub.1], 0), (0,0,[I.sub.2]), (0,[I.sub.1],[I.sub.2]), (1,[I.sub.1],0), (1,0, [I.sub.2]), (1,[I.sub.1],[I.sub.2])}. Then ([Z.sub.2]([I.sub.1] , [I.sub.2]), +,.) is a commutative refined neutrosophic ring of integers modulo 2 of characteristic 2. Generally for a positive integer n [greater than or equal to] 2, ([Z.sub.n] ([I.sub.1], [I.sub.2]), +, .) is a finite commutative refined neutrosophic ring of integers modulo n of characteristic n.

Example 1.11. [1] Let [mathematical expression not reproducible] be a refined neutrosophic set of all n * n matrix.

Then ([M.sup.R.sub.n*n] ([I.sub.1], [I.sub.2]), +,.) is a non-commutative refined neutrosophic ring under matrix multiplication.

Theorem 1.12. 1 Let (R([I.sub.1], [I.sub.2]), +,.) be any refined neutrosophic ring. Then (R([I.sub.1], [I.sub.2]), +,.) is a ring.

2 Formulation of a Refined Neutrosophic Vector Space

In this section, we develop the concept of refined neutrosophic vector space and its subspaces and also present some of their basic properties.

Definition 2.1. Let (V, +,.) be any vector space over a field K. Let V ([I.sub.1], [I.sub.2]) =< V [union] ([I.sub.1], [I.sub.2]) > be a refined neutrosophic set generated by V, [I.sub.1] and [I.sub.2]. We call the triple (V([I.sub.1], [I.sub.2]), +,.) a weak refined neutrosophic vector space over a field K, if V ([I.sub.1], [I.sub.2]) is a refined neutrosophic vector space over a refined neutrosophic field K([I.sub.1], [I.sub.2]), then V([I.sub.1], [I.sub.2]) is called a strong refined neutrosophic vector space. The elements of V([I.sub.1][I.sub.2]) are called refined neutrosophic vectors and the elements of K([I.sub.1], [I.sub.2]) are called refined neutrosophic scalars.

If u = a + b[I.sub.1] + c[I.sub.2], v = d + e[I.sub.1] + f[I.sub.2] [member of] V([I.sub.1], [I.sub.2]) where a, b, c, d, e and f are vectors in V and [alpha] = k + m[I.sub.1] + n[I.sub.2] [member of] K([I.sub.1], [I.sub.2]) where k, m and n are scalars in K, we define:

u + v = (a + b[I.sub.1] + c[I.sub.2]) + (d + e[I.sub.1] + f[I.sub.2]) = (a + d) + (b + e)[I.sub.1] + (c + f)[I.sub.2],

and

[alpha]u = (k + m[I.sub.1] + n[I.sub.2]).(a + b[I.sub.1] + c[I.sub.2]) = k.a + (k.b + m.a + m.b + m.c + n.b)[I.sub.1] + (k.c + n.a + n.c)[I.sub.2].

Example 2.2. Let [R.sup.2]([I.sub.1], [I.sub.2]) denote the refined set of all ordered pairs (x, y) where x and y are refined neutrosophic real numbers given as x = a + b[I.sub.1] + c[I.sub.2] and y = d + e[I.sub.1] + f[I.sub.2].

Define addition and scalar multiplication on [R.sup.2] ([I.sub.1], [I.sub.2]) by

[mathematical expression not reproducible]

For a = (k + m[I.sub.1] + n[I.sub.2]) e R([I.sub.1], [I.sub.2]) [mathematical expression not reproducible]

Then [R.sup.2] ([I.sub.1], [I.sub.2]) is strong refined neutrosophic vector space over R([I.sub.1], [I.sub.2]).

And if [alpha] [member of] R with scalar multiplication defined as

[alpha](x, y) = [alpha](a + b[I.sub.1] + c[I.sub.2], d + e[I.sub.1] + f[I.sub.2]) = ([alpha].a + [alpha].b[I.sub.1] + [alpha].c[I.sub.2], [alpha].d + [alpha].e[I.sub.1] + [alpha].f[I.sub.2]) = ([alpha].x, [alpha].y), then [R.sup.2]([I.sub.1], [I.sub.2]) is weak refined neutrosophic vector space over R.

Example 2.3. [M.sub.m*n]([I.sub.1], [I.sub.2]) = {[aj] : aij [member of] Q([I.sub.1], [I.sub.2])} is a weak refined neutrosophic vector space over a field Q and it is a strong refined neutrosophic vector space over a refined neutrosophic field Q([I.sub.1], [I.sub.2]).

Example 2.4. Let V = Q([I.sub.1], [I.sub.2])([square root of 2]) = {a + (b[I.sub.1] + c[I.sub.2])[square root of 2] : a, b, c [member of] Q}. Then V is a weak refined neutrosophic vector space over Q. If u = a + (b[I.sub.1] + c[I.sub.2])[square root of 2] and v = d + (e[I.sub.1] + f[I.sub.2])[square root of 2] then u + v = (a + d) + (b + e)[I.sub.1] [square root of 2] + (c + f)[I.sub.2][square root of 2] is again in V. Also, for [alpha] [member of] Q, [alpha]u = [alpha](a + (b[I.sub.1] + c[I.sub.2])[square root of 2]) = [alpha].a + ([alpha].b[I.sub.1] + [alpha].c[I.sub.2])[square root of 2] is in V.

Proposition 2.5. Every strong refined neutrosophic vector space is a weak refined neutrosophic vector space.

Proof. Suppose that V([I.sub.1],[I.sub.2]) is a strong refined neutrosophic vector space over a refined neutrosophic field K([I.sub.1], [I.sub.2]) say. Since K [subset or equal to] K([I.sub.1],[I.sub.2]) for every field K, we have that V([I.sub.1], [I.sub.2]) is also a weak refined neutrosophic vector space.

Proposition 2.6. Every weak (strong) refined neutrosophic vector space is a vector space.

Proof. Suppose that V([I.sub.1],[I.sub.2]) is a strong refined neutrosophic vector space over a refined neutrosophic field K([I.sub.1], [I.sub.2]). That (V([I.sub.1],[I.sub.2]), +) is an abelian group can be established easily. Let u = a + b[I.sub.1] + c[I.sub.2], v = d + e[I.sub.1] + f[I.sub.2] [member of] V([I.sub.1],[I.sub.2]), [alpha] = k + m[I.sub.1] + n[I.sub.2], [beta] = p + q[I.sub.1] + r[I.sub.2] [member of] K([I.sub.1],[I.sub.2]) where a, b, c, d, e, f [member of] V and k, m, n, p, q, r [member of] K. Then:

1. [mathematical expression not reproducible]

2. [mathematical expression not reproducible]

3. [mathematical expression not reproducible]

4. For 1 = 1 + 0[I.sub.1] + 0[I.sub.2] [member of] K([I.sub.1][I.sub.2]), we have 1u = (1 + 0[I.sub.1] + 0[I.sub.2])(a + b[I.sub.1] + c[I.sub.2]) = a + (b + 0 + 0 + 0 + 0)[I.sub.1] + (c + 0 + 0)[I.sub.2] = a + b[I.sub.1] + c[I.sub.2]. Accordingly, V([I.sub.1], [I.sub.2]) is a vector space.

Example 2.7. Let [P.sub.[infinity]]([I.sub.1], [I.sub.2]) be the set of refined neutrosophic formal power series in variable x of the form [[summation].sup.[infinity].sub.n=0] [a.sub.n]x.sup.n = [a.sub.0] + [a.sub.1]x + [a.sub.2][x.sup.2] + ... + [a.sub.n][x.sup.n] + ..., with [a.sub.n] [member of] R([I.sub.1], [I.sub.2]) and [a.sub.n] = [p.sub.n] + [q.sub.n][I.sub.1] + [r.sub.n][I.sub.2] for n = 1, 2, 3 ...

If addition and scalar multiplication (for [alpha] [member of] K([I.sub.1], [I.sub.2])) are defined as:

[mathematical expression not reproducible]

Then [P.sub.[infinity]]([I.sub.1],[I.sub.2]) is a strong refined neutrosophic vector space over K([I.sub.1],[I.sub.2]).

Note 1. A refined neutrosophic formal power series can be loosely thought of as an object that is like a refined neutrosophic polynomial. Alternatively, one may think of a refined neutrosophic formal power series as a refined neutrosophic power series in which we ignore the questions of convergence by not assuming that the variable x denotes any numerical value (not even an unknown value). Thus, we do not regard these refined neutrososphic formal power series as infinite sum in [P.sub.x] of refined neutrosophic monomials.

Example 2.8. Let P([I.sub.1], [I.sub.2]) be the set of all refined neutrosophic polynomial in variable x with coefficients in R[[I.sub.1], [I.sub.2]]. Let p, q [member of] P([I.sub.1], [I.sub.2]) and [alpha] = (k + [I.sub.1] + v[I.sub.2]) [member of] K([I.sub.1], [I.sub.2]).

p = ([a.sub.0] + [b.sub.0][I.sub.1] + [c.sub.0][I.sub.2]) + ([a.sub.1] + [b.sub.1][I.sub.1] + [c.sub.1][I.sub.2])x + ... + [(.sub.an] + [b.sub.n][I.sub.1] + [c.sub.n][I.sub.2])[x.sup.n],

and

q = ([a'.sub.0] + [b'.sub.0][I.sub.1] + [c'.sub.0][I.sub.2]) + ([a'.sub.1] + [b'.sub.1][I.sub.1] + [c'.sub.1][I.sub.2])x + ... + ([a'.sub.m] + [b'.sub.m][I.sub.1] + [c'.sub.m][I.sub.2])[x.sup.m].

If m [greater than or equal to] n, the sum of p and q is given by

[mathematical expression not reproducible]

A similar definition is given if m < n.

The product of p and a scalar a is given by

[alpha]*p = (k+u[I.sub.1]+v[I.sub.2])([a.sub.0]+[b.sub.0][I.sub.1]+[c.sub.0][I.sub.2])+(k+u[I.sub.1]+v[I.sub.2])([a.sub.1]+[b.sub.1 ][I.sub.1]+[c.sub.1][I.sub.2])x+...+(k+u[I.sub.1]+v[I.sub.2])([a.sub.n]+[b.sub.n][I.sub.1][+.sub.cn][I.sub.2])[x.sup.n] . Then (P([I.sub.1],[I.sub.2]), +, *) is a strong refined vector space over a refined neutrosophic field K([I.sub.1],[I.sub.2]).

Proposition 2.9. Let F ([I.sub.1],[I.sub.2]) be a refined neutrosophic field and (R([I.sub.1], [I.sub.2]),[phi]) a refined neutrosophic F-algebra, where R([I.sub.1],[I.sub.2]) is a refined neutrosophic ring with identity. Then R([I.sub.1],[I.sub.2]) is a strong refined neutrosophic vector space over F([I.sub.1],[I.sub.2]) with addition being that in R([I.sub.1],[I.sub.2]) and scalar multiplication defined by ar = [phi](a)r. Here [phi] : F([I.sub.1],[I.sub.2]) [right arrow] R([I.sub.1],[I.sub.2]) is a neutrosophic homomorphism such that [phi]((1 + 0[I.sub.1] + 0[I.sub.2])) = (1 + 0[I.sub.1] + 0[I.sub.2]) and [phi]([I.sub.[kappa]]) = [I.sub.[kappa]].

Proof. 1. That (R([I.sub.1], [I.sub.2]), +) is a neutrosophic abelian group can be easily established.

Let x = a + b[I.sub.1] + c[I.sub.2], y = d + e[I.sub.1] + f[I.sub.2] [member of] R([I.sub.1], [I.sub.2]), [alpha] = [kappa] + m[I.sub.1] + n[I.sub.2], [beta] = u + v[I.sub.1] + W[I.sub.2] [member of] F([I.sub.1], [I.sub.2]). Where a, b, c, d, e, f [member of] R and k, m, n, u, v, w [member of] F.

2. [mathematical expression not reproducible]

3. [mathematical expression not reproducible], since [phi] is a refined neutrosophic homomorphism.

4. [mathematical expression not reproducible]

5. For 1 = 1 + 0[I.sub.1] + 0[I.sub.2] [member of] F([I.sub.1][I.sub.2]), we have 1x = (1 + 0[I.sub.1] + 0[I.sub.2])(a + b[I.sub.1] + c[I.sub.2]) = [phi](1 + 0[I.sub.1] + 0[I.sub.2])(a + b[I.sub.1] + c[I.sub.2]). Accordingly, R([I.sub.1], [I.sub.2]) is a strong refined neutrosophic vector space over F([I.sub.1],[I.sub.2]).

Lemma 2.10. Let V([I.sub.1], [I.sub.2]) be a strong refined neutrosophic vector space over a refined neutrosophic field K ([I.sub.1], [I.sub.2]) and let x = a + b[I.sub.1] + c[I.sub.2], y = d + e[I.sub.1] + f[I.sub.2], z = k + m[I.sub.1] + n[I.sub.2], [member of] V ([I.sub.1], [I.sub.2]), [beta] = u + v[I.sub.1] + w[I.sub.2] [member of] K([I.sub.1], [I.sub.2]). Then

1. x + z = y + z implies x = y.

2. 0x = 0.

3. [beta]0 = 0.

4. (-[beta])x = [beta](-x) = -([beta]x).

Proof. 1. x + z = y + z

[mathematical expression not reproducible]

2. Consider [mathematical expression not reproducible]

3. Since [beta] [member of] K([I.sub.1], [I.sub.2]), [mathematical expression not reproducible]

4. Let [beta] [member of] K([I.sub.1],[I.sub.2]) and x [member of] V([I.sub.1],[I.sub.2]).

So [mathematical expression not reproducible]

Then [beta]x + (-[beta])x = 0 [??] (-[beta])x = -[beta]x.

So [mathematical expression not reproducible]

Then [beta]x + [beta](-x) = 0 [??] [beta](-x) = -[beta]x.

Definition 2.11. Let V([I.sub.1], [I.sub.2]) be a strong refined neutrosophic vector space over a refined neutrosophic field K ([I.sub.1], [I.sub.2]) and let W ([I.sub.1], [I.sub.2]) be a nonempty subset of V ([I.sub.1]; [I.sub.2]). W ([I.sub.1], [I.sub.2]) is called a strong refined neutrosophic subspace of V([I.sub.1], [I.sub.2]) if W([I.sub.1], [I.sub.2]) is itself a strong refined neutrosophic vector space over K([I.sub.1], [I.sub.2]). It is essential that W([I.sub.1],[I.sub.2]) contains a proper subset which is a vector space.

Definition 2.12. Let V([I.sub.1], [I.sub.2]) be a weak refined neutrosophic vector space over a field K and let W([I.sub.1],[I.sub.2]) be a nonempty subset of V([I.sub.1],[I.sub.2]). W([I.sub.1],[I.sub.2]) is called a weak refined neutrosophic subspace of V([I.sub.1],[I.sub.2]) if W([I.sub.1],[I.sub.2]) is itself a weak refined neutrosophic vector space over K. It is essential that W([I.sub.1],[I.sub.2]) contains a proper subset which is a vector space.

Proposition 2.13. Let V ([I.sub.1],[I.sub.2]) be a strong refined neutrosophic vector space over a refined neutrosophic field K([I.sub.1], [I.sub.2]) and let W([I.sub.1], [I.sub.2]) be a nonempty subset of V([I.sub.1] ,[I.sub.2]). W([I.sub.1],[I.sub.2]) is a strong refined neutrosophic subspace of V ([I.sub.1],[I.sub.2]) if and only if the following conditions hold:

1. u, v [member of] W([I.sub.1], [I.sub.2]) implies u + v [member of] W([I.sub.1], [I.sub.2]).

2. u [member of] W([I.sub.1],[I.sub.2]) implies [alpha]u [member of] W([I.sub.1],[I.sub.2]) for all [alpha] [member of] K([I.sub.1],[I.sub.2]).

3. W([I.sub.1],[I.sub.2]) contains a proper subset which is a vector space.

Example 2.14. Let V([I.sub.1],[I.sub.2]) be a weak (strong) refined neutrosophic vector space. V([I.sub.1],[I.sub.2]) is a weak (strong) refined neutrosophic subspace called a trivial weak (strong) refined neutrosophic subspace.

Example 2.15. Let K([I.sub.1],[I.sub.2]) = R([I.sub.1],[I.sub.2]) be a refined neutrosophic field and V([I.sub.1],[I.sub.2]) = [R.sup.3]([I.sub.1],[I.sub.2]) be a strong refined neutrosophic vector space. Take W([I.sub.1], [I.sub.2]) to be the set of all vectors in V([I.sub.1],[I.sub.2]) whose last component is 0 = 0 + 0[I.sub.1] + 0[I.sub.2]. Then W([I.sub.1], [I.sub.2]) is strong refined neutrosophic subspace of V([I.sub.1], [I.sub.2]).

Proof. To see this, let

W([I.sub.1], [I.sub.2]) = {(x = a + b[I.sub.1] + c[I.sub.2], y = d + e[I.sub.1] + f[I.sub.2], 0 = 0 + 0[I.sub.1] + 0[I.sub.2]) [member of] V([I.sub.1], [I.sub.2]) : a, b, c, d, e, f [member of] V}.

1. Given that u, v [member of] W([I.sub.1], [I.sub.2]), where u = (x, y, 0) and v = (x', y', 0). Then [mathematical expression not reproducible]. Hence we have that u + v [member of] W ([I.sub.1], [I.sub.2]).

2. Given u [member of] W([I.sub.1], [I.sub.2]) and scalar [alpha] [member of] K([I.sub.1], [I.sub.2]) with [alpha] = r + s[I.sub.1] + t[I.sub.2]. Then [mathematical expression not reproducible].

3. Since W [subset] W([I.sub.1], [I.sub.2]) is a proper subset which is a vector space, W([I.sub.1], [I.sub.2]) is strong refined neutrosophic subspace.

Example 2.16. Let V ([I.sub.1],[I.sub.2]) = [R.sup.2] ([I.sub.1] ,[I.sub.2]) be a strong refined neutrosophic vectors space over a refined neutrosophic field R([I.sub.1], [I.sub.2]) let

W([I.sub.1], [I.sub.2]) = {(x = a + b[I.sub.1] + c[I.sub.2], y = d + e[I.sub.1] + f[I.sub.2]) [member of] V([I.sub.1], [I.sub.2]) : x = y with a, b, c, d, e, f [member of] V}.

Then W ([I.sub.1], [I.sub.2]) is a strong refined neutrosophic subspace of V ([I.sub.1], [I.sub.2]).

Example 2.17. Let V([I.sub.1],[I.sub.2]) = [M.sub.n*n]([I.sub.1],[I.sub.2]) = {[[a.sub.ij]] : [a.sub.ij] [member of] R([I.sub.1],[I.sub.2])} be a strong refined neutrosophic vector space over n([I.sub.1], [I.sub.2]) and let W([I.sub.1], [I.sub.2]) = [A.sub.n*n]([I.sub.1], [I.sub.2]) = {[[b.sub.ij]] : [b.sub.ij] [member of] R([I.sub.1], [I.sub.2]) and trace(A)= 0}. Then W ([I.sub.1], [I.sub.2]) is a strong refined neutrosophic subspace of V ([I.sub.1], [I.sub.2]).

Example 2.18. Let V([I.sub.1], [I.sub.2]) = [R.sup.3]([I.sub.1], [I.sub.2]) be a strong refined neutrosophic vectors space of column refined neutrosophic vectors of length 3 over a refined neutrosophic field R([I.sub.1], [I.sub.2]). Consider

[mathematical expression not reproducible]

W ([I.sub.1], [I.sub.2]) consisting of all refined neutrosophic vectors with 0 = 0 + 0[I.sub.1] + 0[I.sub.2] in the last entry. Then W ([I.sub.1], [I.sub.2]) is a strong refined neutrosophic subspace of V ([I.sub.1], [I.sub.2]).

Proposition 2.19. Lei V([I.sub.1],[I.sub.2]) be a strong refined neutrosophic vector space over a refined neutrosophic field K([I.sub.1], [I.sub.1]) and Let [{[U.sub.n]([I.sub.1],[I.sub.2])}.sub.n[member of][lambda]] be a family of strong refined neutrosophic subspaces of V ([I.sub.1]; [I.sub.2]). Then [[intersection].sub.n[member of][lambda]] [U.sub.n]([I.sub.1], [I.sub.2]) is a strong refined neutrosophic subspace of V([I.sub.1]; [I.sub.2]).

Proof. Consider the collection of strong refined neutrosophic subspaces {[U.sub.n]([I.sub.1], [I.sub.2]) : n [member of] [lambda]} of V([I.sub.1], [I.sub.2]). Take u = a + b[I.sub.1] + c[I.sub.2], v = d + e[I.sub.1] + f[I.sub.2], [alpha] = [kappa] + p[I.sub.1] + q[I.sub.2] and [beta] = r + s[I.sub.1] + t[I.sub.2].

Let u, v [member of] [[intersection].sub.n[member of][lambda]] [U.sub.n]([I.sub.1]; [I.sub.2]) then u, v [member of] [U.sub.n]([I.sub.1]; [I.sub.2]) [for all] n [member of] [lambda]. Now for all scalars [alpha], [beta] [member of] K([I.sub.1], [I.sub.2]) we have that [alpha]u + [beta]v = (k + p[I.sub.1] + q[I.sub.2])(a + b[I.sub.1] + c[I.sub.2]) + (r + s[I.sub.1] + t[I.sub.2])(d + e[I.sub.1] + f[I.sub.2]) = (ka + (kb + pa + pb + pc + qb)[I.sub.1] + (kc + qa + qc)[I.sub.2]) + (rd + (re + sd + se + sf + te)[I.sub.1] + (rf + td + tf)[I.sub.2]) = (ka + rd) + (kb + pa + pb + pc + qb + re + sd + se + sf + te)[I.sub.1] + (kc + qa + qc + rf + td + tf)[I.sub.2]. Therefore [alpha]u + [beta]v [member of] [U.sub.n]([I.sub.1], [I.sub.2]) [for all] n [member of] [lambda] [??] [alpha]u + [beta]v [member of] [[intersection].sub.n[member of][lambda]] [U.sub.n]([I.sub.1], [I.sub.2]). Lastly, since [U.sub.n]([I.sub.1], [I.sub.2]) for all n [member of] [lambda] contains a proper subset [U.sub.n] which is vector space, we have that [[intersection].sub.n[member of][lambda]] [U.sub.n]([I.sub.1], [I.sub.2]) is a strong refined neutrosophic subspace.

Proposition 2.20. Let V([I.sub.1],[I.sub.2]) be a strong refined neutrosophic vector space over the neutrosophic field K([I.sub.1],[I.sub.2]) and let [U.sub.1]([I.sub.1], [I.sub.2]), [U.sub.2]([I.sub.1], [I.sub.2]) be any strong refined neutrosophic subspaces of V([I.sub.1],[I.sub.2]). Then [U.sub.1]([I.sub.1],[I.sub.2]) [union] [U.sub.2]([I.sub.1],[I.sub.2]) is a strong refined neutrosophic subspaces if and only if [U.sub.1]([I.sub.1],[I.sub.2]) [subset or equal to] U2([I.sub.1],[I.sub.2]) or [U.sub.1]([I.sub.1],[I.sub.2]) [??] [U.sub.2]([I.sub.1],[I.sub.2]).

Proof. Let [U.sub.1]([I.sub.1], [I.sub.2]) and [U.sub.2]([I.sub.1], [I.sub.2]) be any strong refined neutrosophic subspaces of V([I.sub.1], [I.sub.2]). [??] Now, suppose [U.sub.1]([I.sub.1], [I.sub.2]) [subset or equal to] [U.sub.2]([I.sub.1], [I.sub.2]) or [U.sub.1]([I.sub.1], [I.sub.2]) [??] [U.sub.2]([I.sub.1], [I.sub.2]) then we shall show the [U.sub.1]([I.sub.1],[I.sub.2]) [union] [U.sub.2] ([I.sub.1], [I.sub.2]) is a strong refined neutrosophic subspaces of V ([I.sub.1], [I.sub.2]). Without loss of generality, suppose that [U.sub.1]([I.sub.1], [I.sub.2]) [subset or equal to] [U.sub.2]([I.sub.1], [I.sub.2]). Then we have that [U.sub.1]([I.sub.1],[I.sub.2]) [union] [U.sub.2]([I.sub.1], [I.sub.2]) = [U.sub.2]([I.sub.1],[I.sub.2]). But [U.sub.2]([I.sub.1],[I.sub.2]) is defined to be a strong refined neutrosophic subspace of V([I.sub.1], [I.sub.2]), so we can say that [U.sub.1]([I.sub.1], [I.sub.2]) [union] [U.sub.2]([I.sub.1], [I.sub.2]) is a strong refined neutrosophic subspace of V ([I.sub.1], [I.sub.2]).

[??] We want to show that if [U.sub.1]([I.sub.1], [I.sub.2]) [union] [U.sub.2]([I.sub.1], [I.sub.2]) is a strong refined neutrosophic subspace of V([I.sub.1], [I.sub.2]) then either [U.sub.1]([I.sub.1], [I.sub.2]) [subset or equal to] [U.sub.2]([I.sub.1], [I.sub.2]) or [U.sub.1]([I.sub.1], [I.sub.2]) [??] [U.sub.2]([I.sub.1], [I.sub.2]).

Now suppose that [U.sub.1] ([I.sub.1], [I.sub.2]) [union] [U.sub.2]([I.sub.1], [I.sub.2]) is a strong refined neutrosophic subspace of V ([I.sub.1], [I.sub.2]) and suppose by contradiction that [U.sub.1]([I.sub.1], [I.sub.2]) [??] [U.sub.2]([I.sub.1],[I.sub.2]) or [U.sub.1]([I.sub.1], [I.sub.2]) [??] [U.sub.2]([I.sub.1],[I.sub.2]). Thus there exist elements [x.sub.1] = [a.sub.1] + [b.sub.1][I.sub.1] + [c.sub.1][I.sub.2] [member of] [U.sub.1]([I.sub.1], [I.sub.2]) [??] [U.sub.2]([I.sub.1], [I.sub.2]) and [x.sub.2] = [a.sub.2] + [b.sub.2][I.sub.1] + [c.sub.2][I.sub.2] [member of] [U.sub.2]([I.sub.1], [I.sub.2])\[U.sub.1]([I.sub.1], [I.sub.2]). So we have that [x.sub.1], [x.sub.2] [member of] [U.sub.1]([I.sub.1], [I.sub.2]) [union] [U.sub.2]([I.sub.1],[I.sub.2]), since [U.sub.1]([I.sub.1], [I.sub.2]) [union] [U.sub.2]([I.sub.1], [I.sub.2]) is a strong refined neutrosophic subspace, we must have that [x.sub.1] + [x.sub.2] = [x.sub.3] [member of] [U.sub.1]([I.sub.1], [I.sub.2]) [union] [U.sub.2]([I.sub.1], [I.sub.2]). Therefore [x.sub.1] + [x.sub.2] = [x.sub.3] [member of] [U.sub.1]([I.sub.1], [I.sub.2]) or [x.sub.1] + [x.sub.2] = [x.sub.3] [member of] [U.sub.2]([I.sub.1], [I.sub.2]) [??] [x.sub.2] = [x.sub.3] - [x.sub.1] [member of] [U.sub.1]([I.sub.1], [I.sub.2]) or [x.sub.1] = [x.sub.3] - [x.sub.2] [member of] [U.sub.2]([I.sub.1], [I.sub.2]) which is a contradiction. Hence [U.sub.1]([I.sub.1], [I.sub.2]) [subset or equal to] [U.sub.2]([I.sub.1],[I.sub.2]) or [U.sub.1]([I.sub.1], [I.sub.2]) [??] [U.sub.2]([I.sub.1],[I.sub.2]) as required.

Remark 2.21. Let V([I.sub.1], [I.sub.2]) be a strong refined neutrosophic vector space over a refined neutrosophic field K ([I.sub.1], [I.sub.2]) and let [W.sub.1]([I.sub.1], [I.sub.2]) and [W.sub.2]([I.sub.1], [I.sub.2]) be two distinct strong refined neutrosophic subspaces of V ([I.sub.1], [I.sub.2]). [W.sub.1]([I.sub.1], [I.sub.2]) [union] [W.sub.2](I) is a strong refined neutrosophic subspace of V([I.sub.1], [I.sub.2]) iff [W.sub.1]([I.sub.1],[I.sub.2]) C[subset or equal to] [W.sub.2]([I.sub.1],[I.sub.2]) or [W.sub.2]([I.sub.1], [I.sub.2]) [subset or equal to] [W.sub.1]([I.sub.1], [I.sub.2]).

Definition 2.22. Let U([I.sub.1], [I.sub.2]) and W([I.sub.1], [I.sub.2]) be any two strong refined neutrosophic subspaces of a strong refined neutrosophic vector space V([I.sub.1], [I.sub.2]) over a neutrosophic field K([I.sub.1], [I.sub.2]).

1. The sum of U([I.sub.1], [I.sub.2]) and W([I.sub.1], [I.sub.2]) denoted by U([I.sub.1], [I.sub.2]) + W([I.sub.1], [I.sub.2]) is defined by the set {u + w : u [member of] U([I.sub.1], [I.sub.2]), w [member of] W([I.sub.1],[I.sub.2])}.

2. V([I.sub.1], [I.sub.2]) is said to be the direct sum of U([I.sub.1], [I.sub.2]) and W([I.sub.1], [I.sub.2]) written V([I.sub.1], [I.sub.2]) = U([I.sub.1], [I.sub.2]) [direct sum] W([I.sub.1], [I.sub.2]) if every element v [member of] V([I.sub.1], [I.sub.2]) can be written uniquely as v = u + w where u [member of] U([I.sub.1], [I.sub.2]) and w [member of] W([I.sub.1], [I.sub.2]).

Proposition 2.23. Let U ([I.sub.1], [I.sub.2]) and W ([I.sub.1], [I.sub.2]) be any two strong refined neutrosophic subspaces of a strong refined neutrosophic vector space V ([I.sub.1],[I.sub.2]), over a refined neutrosophic field K ([I.sub.1], [I.sub.2]). Then U([I.sub.1], [I.sub.2]) + W([I.sub.1], [I.sub.2]) is a strong refined neutrosophic subspace of V([I.sub.1], [I.sub.2]).

Proof. Since U([I.sub.1], [I.sub.2]) and W([I.sub.1], [I.sub.2]) are nonempty strong refined neutrosophic subspaces, U ([I.sub.1], [I.sub.2])+ W ([I.sub.1],[I.sub.2]) [not equal to] {}. Obviously U([I.sub.1], [I.sub.2]) + W([I.sub.1], [I.sub.2]) contains a proper subset U + W which is a vector space. Now let x, y [member of] U ([I.sub.1], [I.sub.2]) + W ([I.sub.1], [I.sub.2]) and [alpha], [beta] [member of] K ([I.sub.1], [I.sub.2]). Then x = ([u.sub.1] + [u.sub.2][I.sub.1] + [u.sub.3][I.sub.2]) + ([w.sub.1] + [w.sub.2][I.sub.1] + [w.sub.3][I.sub.2]), y = ([u.sub.4] + [u.sub.5][I.sub.1] + [u.sub.6][I.sub.2]) + ([w.sub.4] + [w.sub.5][I.sub.1] + [w.sub.6][I.sub.2]) where [u.sub.i] [member of] U, [w.sub.i] [member of] W, with i = 1, 2, 3, 4, 5, 6. [alpha] = k + m[I.sub.1] + n[I.sub.2], [beta] = p + q[I.sub.1] + r[I.sub.2] where k, m, n, p, q, r [member of] K ([I.sub.1],[I.sub.2]).

Then, [mathematical expression not reproducible]. Accordingly U ([I.sub.1], [I.sub.2]) + W ([I.sub.1], [I.sub.2]) is a strong refined neutrosophic subspace of V ([I.sub.1], [I.sub.2]).

Proposition 2.24. Let U ([I.sub.1], [I.sub.2]) and W ([I.sub.1], [I.sub.2]) be strong refined neutrosophic subspaces of a strong refined neutrosophic vector space V ([I.sub.1], [I.sub.2]) over a refined neutrosophic field K ([I.sub.1], [I.sub.2]). V ([I.sub.1],[I.sub.2]) = U ([I.sub.1], [I.sub.2]) [direct sum] W ([I.sub.1], [I.sub.2]) if and only if the following conditions hold:

1. V ([I.sub.1],[I.sub.2]) = U ([I.sub.1],[I.sub.2]) + W ([I.sub.1], [I.sub.2]) and

2. U ([I.sub.1], [I.sub.2]) [intersection] W ([I.sub.1], [I.sub.2]) = {0}.

Proof. The proof is similar to the proof in classical case.

Example 2.25. Let V([I.sub.1], [I.sub.2]) = [R.sup.3]([I.sub.1], [I.sub.2]) be a strong refined neutrosophic vector space over a refined neutrosophic field R([I.sub.1], [I.sub.2]) and let U ([I.sub.1], [I.sub.2]) = {(u, 0, w) : u = a + b[I.sub.1] + c[I.sub.2], w = g + h[I.sub.1] + k[I.sub.2] [member of] R([I.sub.1], [I.sub.2])} and W([I.sub.1], [I.sub.2]) = {(0, v, 0) : v = d + e[I.sub.1] + f[I.sub.2] [member of] R([I.sub.1], [I.sub.2])}, be strong refined neutrosophic subspaces of V([I.sub.1], [I.sub.2]). Then V([I.sub.1], [I.sub.2]) = U([I.sub.1], [I.sub.2]) [direct sum] W([I.sub.1], [I.sub.2]).

To see this, let x = (u, v, w) [member of] V([I.sub.1], [I.sub.2]), then x = (u, 0, w) + (0, v, 0), so x [member of] U([I.sub.1], [I.sub.2]) + W([I.sub.1], [I.sub.2]). Hence V ([I.sub.1] J2) = U ([I.sub.1], [I.sub.2]) + W ([I.sub.1], [I.sub.2]). To show that U([I.sub.1], [I.sub.2]) [intersection] W([I.sub.1], [I.sub.2]) = {0}, let x = (u, v, w) [member of] U([I.sub.1], [I.sub.2]) [intersection] W([I.sub.1], [I.sub.2]). Then v = 0, i.e d + e[I.sub.1] + f[I.sub.2] = 0 + 0[I.sub.1] + 0[I.sub.2] because x lies in U([I.sub.1], [I.sub.2]), and u = w = 0 i.e a + b[I.sub.1] + c[I.sub.2] = g + h[I.sub.1] + k[I.sub.2] = 0 + 0[I.sub.1] + 0[I.sub.2] because x lies in W([I.sub.1], [I.sub.2]). Thus x = (0,0,0) = 0, so 0 = 0 + 0[I.sub.1] + 0[I.sub.2] is the only refined neutrosophic vector in U([I.sub.1], [I.sub.2]) [intersection] W([I.sub.1], [I.sub.2]). So U([I.sub.1], [I.sub.2]) [intersection] W([I.sub.1], [I.sub.2]) = {0 + 0[I.sub.1] + 0[I.sub.2]} = {0}. Hence, V([I.sub.1], [I.sub.2]) = U([I.sub.1], [I.sub.2]) [direct sum] W([I.sub.1], [I.sub.2]).

Example 2.26. In the strong refined neutrosophic vector space V([I.sub.1], [I.sub.2]) = [R.sup.5]([I.sub.1],[I.sub.2]), consider the strong refined neutrosophic subspaces

U([I.sub.1], [I.sub.2]) = {(a, b, c, 0, 0)|a = [x.sub.1] + [y.sub.1][I.sub.1] + [z.sub.1][I.sub.2], b = [x.sub.2] + [y.sub.2][I.sub.1] + [z.sub.2][I.sub.2], and c = [x.sub.3] + [y.sub.3][I.sub.1]+ [z.sub.3][I.sub.2] [member of] V([I.sub.1], [I.sub.2])}

and

W = {0,0,0, d, e)|d = [x.sub.4] + [y.sub.4][I.sub.1] + [z.sub.4][I.sub.2], e = [x.sub.5] + [y.sub.5][I.sub.1] + [z.sub.5][I.sub.2] [member of] V([I.sub.1],[I.sub.2])}.

Then V([I.sub.1], [I.sub.2]) = U([I.sub.1], [I.sub.2]) [direct sum] W([I.sub.1],[I.sub.2]).

To see this, let x = (a, b, c, d, e) be any refined neutrosophic vector in V([I.sub.1], [I.sub.2]), then x = (a, b, c, 0, 0) + (0, 0, 0, d, e), so x lies in U([I.sub.1], [I.sub.2]) + W([I.sub.1], [I.sub.2]). Hence V([I.sub.1], [I.sub.2]) = U([I.sub.1], [I.sub.2]) + W([I.sub.1], [I.sub.2]). To show that U([I.sub.1], [I.sub.2]) [intersection] W([I.sub.1], [I.sub.2]) = {0}, let x = (a, b, c, d, e) lie in U([I.sub.1],[I.sub.2]) [intersection] W([I.sub.1],[I.sub.2]). Then d = e = 0 i.e [x.sub.4] + [y.sub.4][I.sub.1] + [z.sub.4][I.sub.2] = [x.sub.5] + [y.sub.5][I.sub.1] + [z.sub.5][I.sub.2] = 0 + 0[I.sub.1] + 0[I.sub.2] because x lies in U([I.sub.1], [I.sub.2]), and

a = b = c = 0 i.e, [x.sub.1] + [y.sub.1][I.sub.1] + [z.sub.1][I.sub.2] = [x.sub.2] + [y.sub.2][I.sub.1] + [z.sub.2][I.sub.2] = [x.sub.3] + [y.sub.3][I.sub.1] + [z.sub.3][I.sub.2] = 0 + 0[I.sub.1] + 0[I.sub.2] because x lies in W ([I.sub.1],[I.sub.2]).

Thus x = (0, 0, 0, 0, 0) = 0, so 0 = 0 + 0[I.sub.1] + 0[I.sub.2] is the only refined neutrosophic vector in u([I.sub.1], [I.sub.2]) [intersection] W([I.sub.1], [I.sub.2]). So U([I.sub.1], [I.sub.2]) [intersection] W([I.sub.1], [I.sub.2]) = {0 + 0[I.sub.1] + 0[I.sub.2]}. Hence, V([I.sub.1], [I.sub.2]) = U([I.sub.1], [I.sub.2]) [direct sum] W([I.sub.1],[I.sub.2]).

Example 2.27. Let V([I.sub.1], [I.sub.2]) = [P.sub.2n]([I.sub.1], [I.sub.2]) be the strong refined neutrosophic vector space over a neutrosophic field K ([I.sub.1], [I.sub.2]). Then let

[mathematical expression not reproducible]

be strong refined neutrosophic subspaces of [P.sub.2n]([I.sub.1], [I.sub.2]).

Then [P.sub.2n]([I.sub.1], [I.sub.2]) = [U.sub.1]([I.sub.1], [I.sub.2]) [direct sum] [U.sub.2]([I.sub.1], [I.sub.2]).

Proposition 2.28. Let U([I.sub.1],[I.sub.2]) and V([I.sub.1], [I.sub.2]) be strong refined neutrosophic vector spaces over a refined neutrosophic field K ([I.sub.1], [I.sub.2]). Then

U ([I.sub.1], [I.sub.2]) * V ([I.sub.1],[I.sub.2]) = {(u,v) : u [member of] U ([I.sub.1],[I.sub.2]), v [member of] V ([I.sub.1], [I.sub.2])}

is a strong refined neutrosophic vector space over K([I.sub.1], [I.sub.2]) where addition and multiplication are defined by:

(u, v) + (u', v') = (u + u', v + v'), [alpha](u, v) = ([alpha]u, [alpha]v).

Proof. 1. We want to show that (U([I.sub.1], [I.sub.2]) * V([I.sub.1], [I.sub.2]), +) is refined neutrosophic abelian group.

(a) Clearly (U([I.sub.1], [I.sub.2]) * V([I.sub.1], [I.sub.2], +) is closed, since for (u, v), (u, v') [member of] (U([I.sub.1], [I.sub.2]) * V([I.sub.1], [I.sub.2])) where (u, v) = ((a + b[I.sub.1] + c[I.sub.2]), (d + e[I.sub.1] + f[I.sub.2])) we have that [mathematical expression not reproducible].

(b) Let (u, v), (u', v') and (u", v") [member of] U([I.sub.1], [I.sub.2]) * V([I.sub.1], [I.sub.2]). Then [mathematical expression not reproducible].

Then we say that "+" is associative.

(c) The identity in U ([I.sub.1], [I.sub.2]) * V ([I.sub.1], [I.sub.2])) is ([mathematical expression not reproducible]) where [mathematical expression not reproducible] is the identity in U([I.sub.1], [I.sub.2]) and [mathematical expression not reproducible] is the identity in V([I.sub.1], [I.sub.2]) then [mathematical expression not reproducible].

(d) For each (u,v) [member of] U([I.sub.1],[I.sub.2]) * V([I.sub.1],[I.sub.2])) the inverse is (-u, -v) where -u [member of] U([I.sub.1], [I.sub.2]) and -v [member of] V([I.sub.1], [I.sub.2]) is the inverse of u and v respectively. Then [mathematical expression not reproducible].

(e) For (u, v), (u', v') [member of] U([I.sub.1], [I.sub.2]) * V([I.sub.1], [I.sub.2]) we have that [mathematical expression not reproducible].

Then we say that "+" is commutative.

Let [alpha] = k + m[I.sub.1] + n[I.sub.2], [beta] = r + s[I.sub.1] + t[I.sub.2] [member of] K([I.sub.1], [I.sub.2]), then

2. [mathematical expression not reproducible]

3. [mathematical expression not reproducible]

4. [mathematical expression not reproducible]

5. For (1 + 1[I.sub.1] + I[I.sub.2]) [member of] K([I.sub.1],[I.sub.2]), we have [mathematical expression not reproducible].

Proposition 2.29. Lei U ([I.sub.1], [I.sub.2]) be weak refined neutrosophic vector spaces over a field and V be a vector space over afield K. Then

U ([I.sub.1], [I.sub.2]) * V = {(u, v) : u = (a + b[I.sub.1] + c[I.sub.2]) [member of] U ([I.sub.1], [I.sub.2]), v [member of] V}

is a weak refined neutrosophic vector space over K where addition and multiplication are defined by:

(u, v) + (u', v') = (u + u', v + v'), [alpha](u, v) = ([alpha]u, [alpha]v).

Proof. The proof follows the same approach as in the proof of Proposition 2.28

Definition 2.30. Let W ([I.sub.1], [I.sub.2]) be a strong refined neutrosophic subspace of a strong refined neutrosophic vector space V([I.sub.1], [I.sub.2]) over a refined neutrosophic field K([I.sub.1], [I.sub.2]). The quotient V([I.sub.1], [I.sub.2])/W([I.sub.1], [I.sub.2]) is defined by the set

{v + W([I.sub.1], [I.sub.2]) : v [member of] V([I.sub.1], [I.sub.2])}.

Proposition 2.31. The quotient V([I.sub.1], [I.sub.2])/W([I.sub.1], [I.sub.2]) is a strong refined neutrosophic vector space over a refined neutrosophic field K([I.sub.1], [I.sub.2]) if addition and multiplication are defined for all u + W([I.sub.1], [I.sub.2]), v + W ([I.sub.1], [I.sub.2]) [member of] V ([I.sub.1], [I.sub.2])/W ([I.sub.1], [I.sub.2]) and [alpha] [member of] K ([I.sub.1], [I.sub.2]) as follows:

(u + W ([I.sub.1], [I.sub.2])) + (v + W ([I.sub.1], [I.sub.2])) = (u + v) + W ([I.sub.1], [I.sub.2]), [alpha](u + W ([I.sub.1], [I.sub.2])) = [alpha]u + W ([I.sub.1], [I.sub.2]).

This strong refined neutrosophic vector space (V ([I.sub.1], [I.sub.2])/W ([I.sub.1], [I.sub.2]), +,.) over a neutrosophic field K ([I.sub.1], [I.sub.2]) is called a strong refined neutrosophic quotient space.

3 Conclusion

In this paper, we have presented the concept of refined neutrosophic vector spaces. Weak(strong) refined neutrosophic vector spaces and subspaces, and, strong refined neutrosophic quotient vector spaces were studied. Several interesting results and examples were presented. It was shown that every weak (strong) refined neutrosophic vector space is a vector space and it was equally shown that every strong refined neutrosophic vector space is a weak refined neutrosophic vector space. This work will be continued in our next paper titled "On Refined Neutrosophic Vector Spaces II".

Doi: 10.5281/zenodo.3884059

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(1) M.A. Ibrahim, (2) A.A.A. Agboola, (3) B.S. Badmus, (4) S.A. Akinleye

(1,2,4) Department of Mathematics, Federal University of Agriculture, Abeokuta, Nigeria.

(3) Department of Physics, Federal University of Agriculture, Abeokuta, Nigeria.

muritalaibrahim40@gmail.com (1), agboolaaaa@funaab.edu.ng (2), badmusbs@yahoo.com (3), smakinleye@yahoo.com4

Received: March 27, 2020 Revised: May 03, 2020 Accepted: May 28, 2020
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Author:Ibrahim, M.A.; Agboola, A.A.A.; Badmus, B.S.; Akinleye, S.A.
Publication:International Journal of Neutrosophic Science
Date:Jul 1, 2020
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