Printer Friendly

On Refined Neutrosophic Hypervector Spaces.

1 Introduction and Preliminaries

The concept of algebraic hyperstructure was first introduced by Marty [25]. He presented the definition of a hypergroup, studied its properties and applied them to study the groups of rational algebraic functions. Also, Marty used the new approach to solve several problems of the non-commutative algebra. Since then, several researchers have been working on this new field of modern algebra and developed it to a very large extent. M. Krasner [26], introduced the notions of hyperring and hyperfield and used them as technical tools in the study of the approximation of valued fields. There exist several types of hyperrings, some of which are: additive hyperring, multiplicative hyperring and general hyperrings. An important class of additive hyperrings is Krasner hyperrings [23,29,34].

A class of hyperrings (R, +, *) where "+" and "*" are hyperoperations was introduced by De Salvo [24]. This class of hyperrings has been further studied by Asokkumar [9], Asokkumar and Velrajan [10,11,28] and Davvaz and Leoreanu-Fotea [23]. Mittas in [27] introduced the theory of canonical hypergroups. J. Mittas was the first who studied them independently from their operations. Some connected hyperstructures with canonical hypergroups were introduced and analyzed by P. Corsini [21,22], P. Bonansinga [19,20], and K. Serafimidis in [32,33]. Further contributions to the theory of hyperstructures can be found in the books of P. Corsini [21], T. Vougiouklis, P. Corsini and V. Leoreanu [22], and Davvaz and V. Leoreanu [23]. The notion of hypervector spaces was introduced by M. Scafati Tallini. In the definition [31] of hypervector spaces, M. Scafati Tallini considered the field as the usual field. In [30], Sanjay Roy and T. K. Samanta generalized the notion of hypervector space by considering the hyperfield and considering the multiplication structure of a vector by a scalar as a hyperoperation like M. Scafati Tallini and they both called the hyperstructure a hypervector space. They established basic properties of hypervector space and thereafter the notions of linear combinations, linearly dependence, linearly independence, Hamel basis were introduced and several important properties like deletion theorem, extension theorem were developed.

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. Neutrosophic set and neutrosophic logic were introduced in 1995 by Smarandache as generalizations of fuzzy logic/set [43] and respectively intuitionistic fuzzy logic/set [13]. 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.sup.+][ as can be seen in [35,36]. A comprehensive review of neutrosophic set, neutrosophic soft set, neutrosophic topological spaces, neutrosophic algebraic structures and new trends in neutrosophic theory can be found in [3,14-18,37-42].

Agboola and Davvaz introduced and studied neutrosophic hypergroups and presented some of their elementary properties in [7] and in [8], they studied and presented basic properties of canonical hypergroups and hyperrings in a neutrosophic environment, Quotient neutrosophic canonical hypergroups and neutrosophic hyperrings were also presented. In [5], Agboola and Akinleye studied neutrosophic hypervector spaces and they presented their basic properties.

In [36], Smarandache introduced the concept of refined neutrosophic logic and neutrosophic set which allows for the splitting of the components < T, I, F > into 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] >. This refinement has given rise to 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]) are real or complex numbers which has led to the introduction of refined neutrosophic set. Refined neutrosophic set has been applied in the development of refined neutrosophic algebraic structures and refined neutrosophic hyperstructures. Agboola in [4] introduced the concept of refined neutrosophic algebraic structures and studied refined neutrosophic groups in particular. Since then, several researchers in this field have studied this concept and a great deal of results have been published. Recently for instance, Adeleke et al published results on refined neutrosophic rings, refined neutrosophic subring in [1] and in [2], they presented some results on refined neutrosophic ideals and refined neutrosophic homomorphism. The present paper is devoted to the study of refined neutrosophic hypervector space and presents some elementary properties of this structure.

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.2] = [I.sub.2][I.sub.1] = [I.sub.1].

Definition 1.1. Let (F, +, *) be any field. The triple (F(I), +, *) is called a neutrosophic field generated by F and I. (Q(I), +, *) and (R(I), +, *) are examples of neutrosophic fields.

Definition 1.2. [6] Let (V, +, *) be any vector space over a field K and let V (I) = < V [union] I > be a neutrosophic set generated by V and I. The triple (V (I), +, *) is called a weak neutrosophic vector space over a field K. If V (I) is a neutrosophic vector space over a neutrosophic field K(I), then V (I) is called a strong neutrosophic vector space. The elements of V (I) are called neutrosophic vectors and the elements of K(I) are called neutrosophic scalars.

If u = a + bI, v = c + dI [member of] V (I) where a, b, c and d are vectors in V and [alpha] = k + mI [member of] K(I) where k and m are scalars in K, then :

u + v = (a + bI) + (c + dI) = (a + c) + (b + d)I,

and

[alpha]u = (k + mI) * (a + bI) = k * a + (k * b + m * a + m * b)I.

Definition 1.3. [23] Let H be a non-empty set and o : H x H [right arrow] [P.sup.*](H) be a hyperoperation. The couple (H, o) is called a hypergroupoid. For any two non-empty subsets A and B of H and x [member of] H, we define

A o B = [union over (a[member of]A,b[member of]B)] a o b, A o x = A o {x} and x o B = {x} o B.

Definition 1.4. [23] A hypergroupoid (H, o) is called a semihypergroup if for all a, b, c of H we have (a o b) o c = a o (b o c), which means that

[union over (u[member of]aob)] u o c = [union over (v[member of]boc)] a o v.

A hypergroupoid (H, o) is called a quasihypergroup if for all a [member of] H we have a o H = H o a = H. This condition is also called the reproduction axiom.

Definition 1.5. [23] A hypergroupoid (H, o) which is both a semihypergroup and a quasi- hypergroup is called a hypergroup.

Definition 1.6. [23] Let (H, o) and (H', o') be two hypergroupoids. A map [phi] : H [right arrow] H', is called

1. an inclusion homomorphism if for all x, y of H, we have [phi](x o y) [subset or equal to] [phi](x) o' [phi](y);

2. a good homomorphism if for all x, y of H, we have [phi](x o y) = [phi](x) o' [phi](y).

Definition 1.7. [23] Let ([H.sub.1], *1) and ([H.sub.2], *2) be any two refined hypergroupoids and let f : [H.sub.1] [right arrow] [H.sub.2] be a map. We say that :

1. f is a homomorphism if for all x, y of [H.sub.1],

f(x *1 y) [subset] f(x) *2 f(y);

2. f is a good homomorphism if for all x, y of [H.sub.1],

f(x *1 y) = f(x) *2 f(y);

3. f is a strong homomorphism on the left if

f(x) [member of] f(y) *2 f(z) [??] [there exists] y' [member of] [H.sub.1] [??] f(y) = f(y') and x [member of] y' *1 z.

Similarly, we can define a homomorphism, which is strong on the right. If f is strong on the right and on the left we say that f is a strong homomorphism.

Definition 1.8. [23] Let H be a non-empty set and let + be a hyperoperation on H. The couple (H, +) is called a canonical hypergroup if the following conditions hold:

1. x + y = y + x, for all x, y [member of] H,

2. x + (y + z) = (x + y) + z, for all x, y, z [member of] H,

3. there exist a neutral element 0 [member of] H such that x + 0 = {x} = 0 + x, for all x [member of] H,

4. for every x [member of] H, there exist a unique element -x [member of] H such that 0 [member of] x + (-x) [intersection] (-x) + x,

5. z [member of] x + y implies y [member of] -x + z and x [member of] z - y, for all x, y, z [member of] H.

Definition 1.9. [23] A hyperring is a triple (R, +, *) satisfying the following axioms:

1. (R, +) is a canonical hypergroup.

2. (R, *) is a semihypergroup such that x * 0 = 0 * x = 0 for all x [member of] R, that is, 0 is a bilaterally absorbing element.

3. For all x, y, z [member of] R

(a) x * (y + z) = x * y + x * z and

(b) (x + y) * z = x * z + y * z.

That is, the hyperoperation * is distributive over the hyperoperation +.

Definition 1.10. [5] Let P(V) be the power set of a set V, [P.sup.*](V) = P(V) - {0} and let K be a field. The quadruple (V, +, *, K) is called a hypervector space over a field K if:

1. (V, +) is an abelian group.

2. * : K * V [right arrow] [P.sup.*](V) is a hyperoperation such that for all k, m [member of] K and u, v [member of] V, the following conditions hold:

(a) (k + m) * u [subset or equal to] (k * u) + (m * u),

(b) k * (u + v) [subset or equal to] (k * u) + (k * v),

(c) k * (m * u) = (km) * u, where k * (m * u) = {k * v : v [member of] m * u},

(d) (-k) * u = k * (-u),

(e) u [member of] 1 * u.

A hypervector space is said to be strongly left distributive (resp. strongly right distributive) if equality holds in (a) (resp. in (b)). (V, +, *, K) is called a strongly distributive hypervector space if it is both strongly left and strongly right distributive.

Definition 1.11. [12] Let V and W be hypervector spaces over K. A mapping T : V [right arrow] W is called 1. weak linear transformation iff

T(x + y) = T(x) + T(y) and T(a o x) [intersection] a o T(x) [not equal to] 0, [for all] x, y [member of] V, a [member of] K,

2. linear transformation iff

T(x + y) = T(x) + T(y) and T(a o x) [subset or equal to] a o T(x), [for all] x, y [member of] V, a [member of] K,

3. good linear transformation iff

T(x + y) = T(x) + T(y) and T(a o x) = a o T(x), [for all] y [member of] V, a [member of] K.

Definition 1.12. [7] Let (H, *) be any hypergroup and let < H [union] I > = {x = (a, bI) : a, b [member of] H}. The couple N(H) = (< H [union] I >, *) is called a neutrosophic hypergroup generated by H and I under the hyperoperation *. The part a is called the non-neutrosophic part of x and the part b is called the neutrosophic part of x.

If x = (a, bI) and y = (c, dI) are any two elements of N(H), where a, b, c, d [member of] H, then x * y = (a, bI) * (c, dI) = {(u, vI)|u [member of] a * c, v [member of] a * d [union] b * c [union] b * d} = (a * c, (a * d [union] b * c [union] b * d)I). Note that a * c [subset or equal to] H and (a * d [union] b * c [union] b * d) [subset or equal to] H.

Definition 1.13. [8] A neutrosophic hyperring is a triple (N(R), +, *) satisfying the following axioms :

1. (N(R), +) is a neutrosophic canonical hypergroup.

2. (N(R), *) is a neutrosophic semihypergroup.

For all (a, bI),(c, dI),(e, fI) [member of] N(R),

(a) (a, bI) * ((c, dI) + (e, fI)) = (a, bI) * (c, dI) + (a, bI) * (e, fI) and

(b) ((c, dI) + (e, fI)) * (a, bI) = (c, dI) * (a, bI) + (e, fI) * (a, bI).

Definition 1.14. [6] Let (V, +, *, K) be any strongly distributive hypervector space over a field K and let V(I) = < V [union] I > = {u = (a, bI) : a, b [member of] V} be a set generated by V and I. The quadruple (V(I), +, *, K) is called a weak neutrosophic strongly distributive hypervector space over a field K.

For every u = (a, bI), v = (c, dI) [member of] V (I) and k [member of] K, then

u + v = (a + c,(b + d)I) [member of] V (I),

k * u = {(x, yI) : x [member of] k * a, y [member of] k * b}.

If K is a neutrosophic field, that is, K = K(I), then the quadruple (V (I), +, *, K(I)) is called a strong neutrosophic strongly distributive hypervector space over a neutrosophic field K(I). For every u = (a, bI), v = (c, dI) [member of] V (I) and [alpha] = (k, mI) [member of] K(I), we define

u + v = (a + c,(b + d)I) [member of] V (I),

[alpha] * u = {(x, yI) : x [member of] k * a, y [member of] k * b [union] m * a [union] m * b}.

The zero neutrosophic vector of V(I), (0, 0I), is denoted by [theta], the zero element 0 [member of] K is represented by (0, 0I) in K(I) and 1 [member of] K is represented by (1, 0I) in K(I).

Definition 1.15. [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 1.16. [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 1.17. [4] If "+" and "*" are ordinary addition and multiplication, [I.sub.k] with k = 1, 2 have the following properties:

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

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

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

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

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

2 Formulation of Refined Neutrosophic Hypervector Space

This section shows the formulation of refined neutrosophic hypervector space and present some of its properties.

Definition 2.1. Let (V, +, *, K) be any strongly distributive hypervector space over a field K and let

V ([I.sub.1], [I.sub.2]) = < V [union] ([I.sub.1], [I.sub.2]) > = {u = (a, b[I.sub.1], c[I.sub.2]) : a, b, c [member of] V}

be a set generated by V, [I.sub.1] and [I.sub.2]. The quadruple (V ([I.sub.1], [I.sub.2]), +, *, K) is called a weak refined neutrosophic strongly distributive hypervector space over a field K. For every element 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]), and k [member of] K we define

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

k * u = {(x, y[I.sub.1], z[I.sub.2]) : x [member of] k * a, y [member of] k * b, z [member of] k * c}.

If K is a refined neutrosophic field, that is, K = K([I.sub.1], [I.sub.2]), then the quadruple (V ([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) is called a strong refined neutrosophic strongly distributive hypervector space over a refined neutrosophic field K([I.sub.1], [I.sub.2]).

For every element 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]), and [alpha] = (k, m[I.sub.1], n[I.sub.2]) [member of] K([I.sub.1], [I.sub.2]), 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]),

[alpha] * u = {(x, y[I.sub.1], z[I.sub.2]) : (x [member of] k * a, y [member of] k * b [union] m * a [union] m * b [union] m * c [union] n * b, z [member of] k * c [union] n * a [union] n * c)}.

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. The zero refined neutrosophic vector of V ([I.sub.1], [I.sub.2]), (0, 0[I.sub.1], 0[I.sub.2]), is denoted by [theta], the zero element 0 [member of] K is represented by (0, 0[I.sub.1], 0[I.sub.2]) in K([I.sub.1], [I.sub.2]) and 1 [member of] K is represented by (1, 0[I.sub.1], 0[I.sub.2]) [member of] K([I.sub.1], [I.sub.2]).

Example 2.2. 1. Let r be a fixed positive integer and let

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

[alpha] * u = {(x, y[square root of (r)][I.sub.1], z[square root of (r)][I.sub.2]) : x [member of] [alpha] * a, y [member of] [alpha] * b, z [member of] [alpha] * c} [member of] V.

2. Let V ([I.sub.1], [I.sub.2]) = R([I.sub.1], [I.sub.2]) and let K = R. For all 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]) and k [member of] K, define:

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

k * u = {(x, y[I.sub.1], z[I.sub.2]) : x [member of] k * a, y [member of] k * b, z [member of] k * c}.

Then (V ([I.sub.1], [I.sub.2]), +, *, K) is a weak neutrosophic strongly distributive hypervector space over the field K.

Example 2.3. 1. Let V ([I.sub.1], [I.sub.2]) = [R.sup.3]([I.sub.1], [I.sub.2]) and let K = R([I.sub.1], [I.sub.2]). For all u = ((a, b[I.sub.1], c[I.sub.2]),(d, e[I.sub.1], f[I.sub.2]),(g, h[I.sub.1], j[I.sub.2])), v = ((a', b' [I.sub.1], c'[I.sub.2]), (d', e'[I.sub.1], f'[I.sub.2]),(g', h'[I.sub.1], j'[I.sub.2])) [member of] V ([I.sub.1], [I.sub.2]) and [alpha] = (k, m[I.sub.1], n[I.sub.2]) [member of] K([I.sub.1], [I.sub.2]), define :

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

[alpha] * u = {(([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])) : [x.sub.1] [member of] k * a, [y.sub.1] [member of] k * b [union] m * a [union] m * b [union] m * c [union] n * b, [z.sub.1] [member of] k * c [union] n * a [union] n * c, [x.sub.2] [member of] k * d, [y.sub.2] [member of] k * e [union] m * d [union] m * e [union] m * f [union] n * e, [z.sub.2] [member of] k * f [union] n * d [union] n * f [x.sub.3] [member of] k * g, [y.sub.3] [member of] k * h [union] m * g [union] m * h [union] m * j [union] n * h, [z.sub.3] [member of] k * j [union] n * g [union] n * j}.

Then (V ([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) is a strong refined neutrosophic hypervector space over the refined neutrosophic field K([I.sub.1], [I.sub.2]).

2. Let V ([I.sub.1], [I.sub.2]) = [R.sup.2]([I.sub.1], [I.sub.2]) and K = R define for all x = (u, v) [member of] V ([I.sub.1], [I.sub.2]) with u = (a, b[I.sub.1], c[I.sub.2]), v = (d, e[I.sub.1], f[I.sub.2]) and [alpha] [member of] K

[mathematical expression not reproducible].

Then (V ([I.sub.1], [I.sub.2]), +, *, K) is a weak refined neutrosophic strongly distributive hypervector space. From now on, every weak(strong) refined neutrosophic strongly distributive hypervector space will simply be called a weak(strong) refined neutrosophic hypervector space.

Lemma 2.4. Let V ([I.sub.1], [I.sub.2]) be a weak refined neutrosophic hypervector space over a field K. Then for all k [member of] K and u = (a, b[I.sub.1], c[I.sub.2]) [member of] V ([I.sub.1], [I.sub.2]), we have

1. k * [theta] = {[theta]}.

2. k * u = {[theta]} implies that k = [theta] or u = [theta].

3. -u [member of] (-1) * u

Proof. 1. k * [theta] = k * (0 * [theta]) = (k.0) * [theta] = 0 * [theta] = [theta]

2. Let k [member of] K and u [member of] V be such that k * u = {[theta]}.

If k = 0, then 0 * u = [theta].

If k [not equal to] 0, then [k.sup.-1] [member of] K. Therefore k * u = [theta] [??] [k.sup.-1] * (k * u) = [k.sup.-1] * [theta] [??] ([k.sup.-1].k) * u = [theta] [??] [1.sub.K] * u = [theta] [??] u = [theta].

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

Proof. Suppose that V ([I.sub.1], [I.sub.2]) is a strong refined neutrosophic hypervector 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, then we have that V ([I.sub.1], [I.sub.2]) is also a weak refined neutrosophic hypervector space.

Proposition 2.6. Every weak refined neutrosophic hypervector space is a strongly distributive hypervector space.

Proof. Suppose that V ([I.sub.1], [I.sub.2]) is a weak refined neutrosophic hypervector space over a field K. Obviously, (V ([I.sub.1], [I.sub.2]), +) is an abelian group. 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]) and k, m [member of] K be arbitrary. Then

(1) k * u + m * u = {(p, q[I.sub.1], r[I.sub.2]) : p [member of] k * a, q [member of] k * b, r [member of] k * c}+ {(s, t[I.sub.1], w[I.sub.2]) : s [member of] m * a, t [member of] m * b, w [member of] m * c} = {(p + s,(q + t)[I.sub.1],(r + w)[I.sub.2]) : p + s [member of] k * a + m * a, q + t [member of] k * b + m * b, r + w [member of] k * c + m * c}.

And, (k + m) * u = {(x, y[I.sub.1], z[I.sub.2]) : x [member of] (k + m) * a, y [member of] (k + m) * b, z [member of] (k + m) * c} = {(x, y[I.sub.1], z[I.sub.2]) : x [member of] k * a + m * a, y [member of] k * b + m * b, z [member of] k * c + m * c} = k * u + m * u.

(2) k * u + k * v = {(p, q[I.sub.1], r[I.sub.2]) : p [member of] k * a, q [member of] k * b, r [member of] k * c}+ {(s, t[I.sub.1], w[I.sub.2]) : s [member of] k * d, t [member of] k * e, w [member of] k * f} = {(p + s,(q + t)[I.sub.1],(r + w)[I.sub.2]) : p + s [member of] k * a + k * d, q + t [member of] k * b + k * e, r + w [member of] k * c + k * f}.

And,

k * (u + v) = k * (a + d,(b + e)[I.sub.1],(c + f)[I.sub.2]) = {(x, y[I.sub.1], z[I.sub.2]) : x [member of] k * (a + d), y [member of] k * (b + e), z [member of] k * (c + f)} = {(x, y[I.sub.1], z[I.sub.2]) : x [member of] k * a + k * d, y [member of] k * b + k * e, z [member of] k * c + k * f} = k * u + k * v.

(3) k * (m * u) = k * {(x, y[I.sub.1], z[I.sub.2]) : x [member of] m * a, y [member of] m * b, z [member of] m * c} = {(p, q[I.sub.1], r[I.sub.2]) : p [member of] k * x, q [member of] k * y, r [member of] k * z} = {(p, q[I.sub.1], r[I.sub.2]) : p [member of] k * (m * a), q [member of] k * (m * b), r [member of] k * (m * c)} = {(p, q[I.sub.1], r[I.sub.2]) : p [member of] (km) * a, q [member of] (km) * b, r [member of] (km) * c} = (km) * (a, b[I.sub.1], c[I.sub.2]) = (km) * u.

(4) (-k) * u = {(x, y[I.sub.1], z[I.sub.2]) : x [member of] (-k) * a, y [member of] (-k) * b, z [member of] (-k) * c} = {(x, y[I.sub.1], c[I.sub.2]) : x [member of] k * (-a), y [member of] k * (-b), z [member of] k * (-c)} = k * (-a, -b[I.sub.1], -c[I.sub.2]) = k * (-u).

(5) 1 * u = {(x, y[I.sub.1], z[I.sub.2]) : x [member of] 1 * a, y [member of] 1 * b, z [member of] 1 * c} = {(x, y[I.sub.1], z[I.sub.2]) : x [member of] {a}, y [member of] {b}, z [member of] {c}} = {(a, b[I.sub.1], c[I.sub.2])}. [??] u [member of] 1 * u.

Accordingly, V ([I.sub.1], [I.sub.2]) is a strongly distributive hypervector space.

Corollary 2.7. Every weak refined neutrosophic hypervector space which is strongly right distributive is strongly left distributive.

Proof. The proof follows from the proof of Proposition 2.6.

Proposition 2.8. Let ([V.sub.1]([I.sub.1], [I.sub.2]), +1, *1, K([I.sub.1], [I.sub.2])) and ([V.sub.2]([I.sub.1], [I.sub.2]), +2, *2, K([I.sub.1], [I.sub.2])) be two strong refined neutrosophic hypervector spaces over a refined neutrosophic field K([I.sub.1], [I.sub.2]). Let [mathematical expression not reproducible].

Then ([V.sub.1]([I.sub.1], [I.sub.2]) * [V.sub.2]([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) is a strong neutrosophic hypervector space.

Proof. Suppose that V1([I.sub.1], [I.sub.2]) and V2([I.sub.1], [I.sub.2]) are strong refined neutrosophic hypervector spaces over a refined neutrosophic field K([I.sub.1], [I.sub.2]).

Let [mathematical expression not reproducible] be arbitrary.

1. We can easily show that (V1([I.sub.1], [I.sub.2]) * V2([I.sub.1], [I.sub.2]), +) is an abelian group.

2. Now we want to show that ([alpha] + [beta]) * u [subset or equal to] [alpha] * u + [beta] * u. Consider

[mathematical expression not reproducible]

Now if we take x = [s.sub.1] + [s'.sub.1], y = [t.sub.1] + [t'.sub.1], z = [w.sub.1] + [w'.sub.1], p = [s.sub.2] + [s'.sub.2], q = [t.sub.2] + [t'.sub.2] and r = [w.sub.1] + [w'.sub.2] then we have

[mathematical expression not reproducible].

Then ([alpha] + [beta]) * u [subset or equal to] [alpha] * u + [beta] * u.

3. Now we want to show that [alpha] * (u + v) [subset or equal to] [alpha] * u + [alpha] * v

[mathematical expression not reproducible].

If we take x = [s.sub.1] + [s'.sub.1], y = [t.sub.1] + [t'.sub.1], z = [w.sub.1] + [w'.sub.1], p = [s.sub.2] + [s'.sub.2], q = [t.sub.2] + [t'.sub.2] and r = [w.sub.1] + [w'.sub.2] then we have

[mathematical expression not reproducible].

Then we have that [alpha] * (u + v) [subset or equal to] [alpha] * u + [alpha] * v.

4. [mathematical expression not reproducible].

5. [mathematical expression not reproducible].

6. 1 * u = {(x, y[I.sub.1], z[I.sub.2])(p, q[I.sub.1], r[I.sub.2])) : x [member of] 1 *[a.sub.1], y [member of] 1*[b.sub.1], z [member of] 1 * [c.sub.1], p [member of] 1 * [a.sub.2], q [member of] 1 * [b.sub.2], r [member of] 1 * [c.sub.2]} = {([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.2])([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2])) : [a.sub.1] [member of] 1 * [a.sub.1], [b.sub.1] [member of] 1 * [b.sub.1], [c.sub.1] [member of] 1 * [c.sub.1], [a.sub.2] [member of] 1*[a.sub.2], [b.sub.2] [member of] 1*[b.sub.2], [c.sub.2] [member of] 1 * [c.sub.2]}, which shows that u [member of] 1 * u.

Accordingly, V1([I.sub.1], [I.sub.2]) * V2([I.sub.1], [I.sub.2]) is a strong refined neutrosophic hypervector space.

Proposition 2.9. Let (V ([I.sub.1], [I.sub.2]), [direct sum], *1, K) and (H, +H, *H, K) be a weak refined neutrosophic hypervector spaces and a hypervector space, respectively. Let

V ([I.sub.1], [I.sub.2]) * H = {((a, b[I.sub.1], c[I.sub.2]), h) : (a, b[I.sub.1], c[I.sub.2]) [member of] [V.sub.1]([I.sub.1], [I.sub.2]), h [member of] H}. For all [mathematical expression not reproducible]. Then ([V.sub.1]([I.sub.1], [I.sub.2]) * H, +, *, K) is a weak neutrosophic hypervector space.

Proof. The proof follows from the same pattern as the proof of Proposition 2.8.

Definition 2.10. Let (V ([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) be a strong refined neutrosophic hypervector 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 said to be a subhypervector space of V ([I.sub.1], [I.sub.2]) if (W[[I.sub.1], [I.sub.2]], +, *, K([I.sub.1], [I.sub.2])) is also a refined neutrosophic hypervector space over the refined neutrosophic field 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 hypervector space over a field K.

Example 2.11. Let V ([I.sub.1], [I.sub.2]) = [R.sup.2]([I.sub.1], [I.sub.2]) and K = R([I.sub.1], [I.sub.2]) then ([R.sup.2]([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) is a strong refined neutrosophic hypervector space over refined neutrosophic field K = R([I.sub.1], 12), where the hyperoperations + and * are defined [for all] u = (([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.1]),([a.sub.2], [b.sub.2][I.sub.1], c[I.sub.2])), v = (([a'.sub.1], [b'.sub.1][I.sub.1], [c'.sub.1][I.sub.1]), ([a'.sub.2], [b.sub.2][I.sub.1], c'[I.sub.2])) [member of] V ([I.sub.1], [I.sub.2]) by : u + v = (([a.sub.1] + [a'.sub.1],([b.sub.1] + [b'.sub.1])[I.sub.1],([c.sub.1] + [c'.sub.1])[I.sub.2]),([a.sub.2] + [a'.sub.2],([b.sub.2] + [b'.sub.2])[I.sub.1],([c.sub.2] + [c'.sub.2])[I.sub.2])), [alpha] * u = {((x, y[I.sub.1], z[I.sub.2]),(p, q[I.sub.1], r[I.sub.2])) : x [member of] k * [a.sub.1], y [member of] k * [b.sub.1] [union] m * [a.sub.1] [union] m * [b.sub.1] [union] m * [c.sub.1] [union] n * [b.sub.1], z [member of] k * [c.sub.1] [union] n * [a.sub.1] [union] n * [c.sub.1], p [member of] k * [a.sub.2], q [member of] k * [b.sub.2] [union] m * [a.sub.2] [union] m * [b.sub.2] [union] m * [c.sub.2] [union] n * [b.sub.2], r [member of] k * [c.sub.2] [union] n * [a.sub.2] [union] n * [c.sub.2]}.

Let W([I.sub.1], [I.sub.2]) = K([I.sub.1], [I.sub.2]) * {(0, 0[I.sub.1], 0[I.sub.2])} [subset or equal to] V ([I.sub.1], [I.sub.2]). Then W([I.sub.1], [I.sub.2]) is a strong refined neutrosophic subhypervector space.

Proof. Since [theta] = ((0, 0[I.sub.1], 0[I.sub.2]),(0, 0[I.sub.1], 0[I.sub.2])) [member of] W([I.sub.1], [I.sub.2]). Then W([I.sub.1], [I.sub.2]) [not equal to] 0.

Now let [mathematical expression not reproducible].

Lastly, we can see from the definition of W([I.sub.1], [I.sub.2]) that W([I.sub.1], [I.sub.2]) contains a proper subset which is a hypervector space over K. To this end we can conclude that W([I.sub.1], [I.sub.2]) is a strong refined neutrosophic hypervector space.

Proposition 2.12. Let [W.sub.1][[I.sub.1], [I.sub.2]], [W.sub.2][[I.sub.1], [I.sub.2]], ..., Wn[[I.sub.1], [I.sub.2]] be refined neutrosophic subhypervector spaces of a strong refined neutrosophic hypervector space (V ([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) over a refined neutrosophic field K([I.sub.1], [I.sub.2]). Then [[intersection].sup.n.sub.i=1] [W.sub.i][[I.sub.1], [I.sub.2]] is a refined neutrosophic subhypervector space of V ([I.sub.1], [I.sub.2]).

Proof. Consider the collection of refined neutrosophic subhypervector space {[W.sub.i]([I.sub.1], [I.sub.2]) : i = 1, 2, ... n} of a strong refined neutrosophic hypervector space 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] = (k, 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].sup.n.sub.i=1] [W.sub.i]([I.sub.1], [I.sub.2]) then u, v [member of] [W.sub.i]([I.sub.1], [I.sub.2]) for all i = 1, 2, ... n.

Now for all scalars [alpha], [beta] [member of] K([I.sub.1], [I.sub.2]) we have that

[mathematical expression not reproducible].

Lastly, since [W.sub.i]([I.sub.1], [I.sub.2]) [for all] i = 1, 2, 3, ..., n contain proper subsets Wi which are hypervector space, [[intersection].sup.n.sub.i=1] [W.sub.i]([I.sub.1], [I.sub.2]) is a strong refined neutrosophic subhyperspace.

Proposition 2.13. Let W[[I.sub.1], [I.sub.2]] be a subset of a strong refined neutrosophic hypervector space (V ([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) over a refined neutrosophic field K([I.sub.1], [I.sub.2]). Then W[[I.sub.1], [I.sub.2]] is a refined neutrosophic subhypervector space of V ([I.sub.1], [I.sub.2]) if and only if for all u = (a, b[I.sub.1], c[I.sub.2]), v = (d, e[I.sub.1], [I.sub.2]) [member of] V ([I.sub.1], [I.sub.2]) and [alpha] = (k, m[I.sub.1], n[I.sub.2]) [member of] K([I.sub.1], [I.sub.2]), the following conditions hold:

1. W[[I.sub.1], [I.sub.2]] [not equal to] 0,

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

3. [alpha] * u [subset or equal to] W[[I.sub.1], [I.sub.2]],

4. W[[I.sub.1], [I.sub.2]] contains a proper subset which is a hypervector space over K.

Proposition 2.14. Let V ([I.sub.1], [I.sub.2]) be a strong refined neutrosophic hypervector space over 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 subhypervector spaces 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 subhypervector space if and only if [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]) [contains or equal to] [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 subhypervector spaces 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]) [contains or equal to] [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 subhypervector space 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 subhypervector space of V ([I.sub.1], [I.sub.2]), so [U.sub.1]([I.sub.1], [I.sub.2]) [union] [U.sub.2]([I.sub.1], [I.sub.2]) is a strong refined neutrosophic subhypervector space 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 subhypervector space 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]) [contains or equal to] [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 subhypervector space of V ([I.sub.1], [I.sub.2]) and suppose by contradiction that [U.sub.1]([I.sub.1], [I.sub.2]) [subset not equal to] [U.sub.2]([I.sub.1], [I.sub.2]) or [U.sub.1]([I.sub.1], [I.sub.2]) [contains not equal to] [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 subhypervector space, 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]) [contains or equal to] [U.sub.2]([I.sub.1], [I.sub.2]) as required.

Remark 2.15. If [W.sub.1][[I.sub.1], [I.sub.2]] and [W.sub.2][[I.sub.1], [I.sub.2]] are refined neutrosophic subhypervector spaces of a strong refined neutrosophic hypervector space V ([I.sub.1], [I.sub.2]) over a refined neutrosophic field K([I.sub.1], [I.sub.2]), then generally, [W.sub.1][[I.sub.1], [I.sub.2]] [union] [W.sub.2][[I.sub.1], [I.sub.2]] is not a refined neutrosophic subhypervector space of V ([I.sub.1], [I.sub.2]) except if [W.sub.1][[I.sub.1], [I.sub.2]] [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 2.16. Let [W.sub.1][[I.sub.1], [I.sub.2]] and [W.sub.2][[I.sub.1], [I.sub.2]] be two refined neutrosophic subhypervector spaces of a strong refined neutrosophic hypervector space (V ([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) over a refined neutrosophic field K([I.sub.1], [I.sub.2]). The sum of [W.sub.1][[I.sub.1], [I.sub.2]] and [W.sub.2][[I.sub.1], [I.sub.2]] denoted by [W.sub.1][[I.sub.1], [I.sub.2]] + [W.sub.2][[I.sub.1], [I.sub.2]] is defined by the set

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

If [W.sub.1][[I.sub.1], [I.sub.2]] [intersection] [W.sub.2][[I.sub.1], [I.sub.2]] = {[theta]}, then the sum of [W.sub.1][[I.sub.1], [I.sub.2]] and [W.sub.2][[I.sub.1], [I.sub.2]] is denoted by [W.sub.1][[I.sub.1], [I.sub.2]] [direct sum] [W.sub.2][[I.sub.1], [I.sub.2]] and it is called the direct sum of [W.sub.1][[I.sub.1], [I.sub.2]] and [W.sub.2][[I.sub.1], [I.sub.2]].

Proposition 2.17. Let [W.sub.1][[I.sub.1], [I.sub.2]] and [W.sub.2][[I.sub.1], [I.sub.2]] be two refined neutrosophic subhypervector spaces of a strong refined neutrosophic hypervector space (V ([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) over a refined neutrosophic field K([I.sub.1], [I.sub.2]).

1. [W.sub.1][[I.sub.1], [I.sub.2]] + [W.sub.2][[I.sub.1], [I.sub.2]] is a refined neutrosophic subhypervector space of V ([I.sub.1], [I.sub.2]).

2. [W.sub.1][[I.sub.1], [I.sub.2]] + [W.sub.2][[I.sub.1], [I.sub.2]] is the least refined neutrosophic subhypervector space of V ([I.sub.1], [I.sub.2]) containing [W.sub.1][[I.sub.1], [I.sub.2]] and [W.sub.2][[I.sub.1], [I.sub.2]].

Proof. 1. Since [theta] [member of] [W.sub.1][[I.sub.1], [I.sub.2]] and [theta] [member of] [W.sub.2][[I.sub.1], [I.sub.2]], {[theta] + [theta]} [subset or equal to] [W.sub.1][[I.sub.1], [I.sub.2]] + [W.sub.2][[I.sub.1], [I.sub.2]].

So,{[theta]} [subset or equal to] [W.sub.1][[I.sub.1], [I.sub.2]] + [W.sub.2][[I.sub.1], [I.sub.2]] [??] [theta] [member of] [W.sub.1][[I.sub.1], [I.sub.2]] + [W.sub.2][[I.sub.1], [I.sub.2]], therefore [W.sub.1][[I.sub.1], [I.sub.2]] + [W.sub.2][[I.sub.1], [I.sub.2]] is non-empty.

Let u = (a, b[I.sub.1], c[I.sub.2]), v = (d, e[I.sub.1], f[I.sub.2]) [member of] [W.sub.1][[I.sub.1], [I.sub.2]] + [W.sub.2][[I.sub.1], [I.sub.2]], then [there exists] [u.sub.1] = ([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.2]), [u.sub.2] = ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2]) [member of] [W.sub.1][[I.sub.1], [I.sub.2]] and [v.sub.1] = ([d.sub.1], [e.sub.1][I.sub.1], [f.sub.1][I.sub.2]), [v.sub.2] = ([d.sub.2], [e.sub.2][I.sub.1], [f.sub.2][I.sub.2]) [member of] [W.sub.2][[I.sub.1], [I.sub.2]] such that u [member of] [u.sub.1] + [v.sub.1] and v [member of] [u.sub.2] + [v.sub.2].

Let [mathematical expression not reproducible].

Hence [alpha] * u + [beta] * v [subset or equal to] [W.sub.1][[I.sub.1], [I.sub.2]] + [W.sub.2][[I.sub.1], [I.sub.2]].

Now since [W.sub.1], [W.sub.2] are proper subsets of [W.sub.1][[I.sub.1], [I.sub.2]] and [W.sub.2][[I.sub.1], [I.sub.2]] respectively, with both [W.sub.1] and [W.sub.2] being hypervector space. Then [W.sub.1] + [W.sub.2] is a hypervector space which is properly contained in [W.sub.1][[I.sub.1], [I.sub.2]] + [W.sub.2][[I.sub.1], [I.sub.2]]. Then we can conclude that [W.sub.1][[I.sub.1], [I.sub.2]] + [W.sub.2][[I.sub.1], [I.sub.2]] is a refined neutrososphic subhypervector space.

2. Let W[[I.sub.1], [I.sub.2]] be refined neutrosophic subhypervector space of V [[I.sub.1], [I.sub.2]]such that [W.sub.1][[I.sub.1], [I.sub.2]] [subset or equal to] W[[I.sub.1], [I.sub.2]] and [W.sub.2][[I.sub.1], [I.sub.2]] [subset or equal to] W[[I.sub.1], [I.sub.2]].

Let u = (a, b[I.sub.1], c[I.sub.2]) [member of] [W.sub.1][[I.sub.1], [I.sub.2]] + [W.sub.2][[I.sub.1], [I.sub.2]], then [there exists][u.sub.1] = ([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.2]) [member of] [W.sub.1][[I.sub.1], [I.sub.2]] and [u.sub.2] = ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2]) [member of] [W.sub.2][[I.sub.1], [I.sub.2]] such that u [member of] [u.sub.1] + [u.sub.2].

Since [W.sub.1][[I.sub.1], [I.sub.2]] [subset or equal to] W[[I.sub.1], [I.sub.2]] and [W.sub.2][[I.sub.1], [I.sub.2]] [subset or equal to] W[[I.sub.1], [I.sub.2]], then [u.sub.1], [u.sub.2] [member of] W[[I.sub.1], [I.sub.2]].

Again since W[[I.sub.1], [I.sub.2]] is a refined neutrosophic subhypervector space of V [[I.sub.1], [I.sub.2]], then we have that [u.sub.1] + [u.sub.2] [subset or equal to] W[[I.sub.1], [I.sub.2]] [??] u [member of] W[[I.sub.1], [I.sub.2]].

Hence [W.sub.1][[I.sub.1], [I.sub.2]] + [W.sub.2][[I.sub.1], [I.sub.2]] [subset or equal to] W[[I.sub.1], [I.sub.2]] and the proof follows.

Remark 2.18. If V ([I.sub.1], [I.sub.2]) is a weak refined neutrosophic strongly left distributive hypervector space over a field K, then

1. W[[I.sub.1], [I.sub.2]] = [union]{k * u : k [member of] K} forms a weak refined neutrosophic subhypervector space of V ([I.sub.1], [I.sub.2]), where u = (a, b[I.sub.1], c[I.sub.2]) [member of] V ([I.sub.1], [I.sub.2]). This refined neutrosophic subhypervector space is said to be generated by the refined neutrosophic vector u and it is called a refined neutrosophic hyperline span by the refined neutrosophic vector u.

2. 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]), then the set W = [union]{[alpha] * u + [beta] * v, [alpha], [beta] [member of] K} is a weak refined neutrosophic subhypervector space of V ([I.sub.1], [I.sub.2]). This refined neutrosophic subhypervector space is called refined neutrosophic hyperlinear span of the refined neutrosophic vectors u and v.

Proposition 2.19. Let V ([I.sub.1], [I.sub.2]) be a weak refined neutrosophic strongly left distributive hypervector space over the field K and [u.sub.1], [u.sub.2], ..., [u.sub.n] [member of] V ([I.sub.1], [I.sub.2]), with [u.sub.i] = ([a.sub.i], [b.sub.i][I.sub.1], [c.sub.i][I.sub.2]) for i = 1, 2, 3 ... n. Then

1. W([I.sub.1], [I.sub.2]) = [union]{[[alpha].sub.1] * [u.sub.1] + [[alpha].sub.2] * [u.sub.2] + ... + [[alpha].sub.n] * [u.sub.n] : [[alpha].sub.1], [[alpha].sub.2], ..., [[alpha].sub.n] [member of] K} is a weak refined neutrosophic subhypervector space of V ([I.sub.1], [I.sub.2]).

2. W([I.sub.1], [I.sub.2]) is the smallest weak refined neutrosophic subhypervector space of V ([I.sub.1], [I.sub.2]) containing [u.sub.1], [u.sub.2], ..., [u.sub.n].

Proof. 1. The proof follows from similar approach as 1 of Proposition 2.17.

2. Suppose that M([I.sub.1], [I.sub.2]) is a weak refined neutrosophic subhypervector space of V ([I.sub.1], [I.sub.2]) containing [u.sub.1] = ([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.2]), [u.sub.2] = ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2]), ..., [u.sub.n] = ([a.sub.n], [b.sub.n][I.sub.1], [c.sub.n][I.sub.2]). Let t [member of] W([I.sub.1], [I.sub.2]), then there exist [[alpha].sub.1], [[alpha].sub.2], ..., [[alpha].sub.n] [member of] K such that

t [member of] [[alpha].sub.1] * ([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.2]) + [[alpha].sub.2] * ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2]) + ... + [[alpha].sub.n] * ([a.sub.n], [b.sub.n][I.sub.1], [c.sub.n][I.sub.2]).

Therefore t [member of] M([I.sub.1], [I.sub.2]) [??] W([I.sub.1], [I.sub.2]) [subset or equal to] M([I.sub.1], [I.sub.2]).

Hence W([I.sub.1], [I.sub.2]) is the smallest weak refined neutrosophic subhypervector space of V ([I.sub.1], [I.sub.2]) containing [u.sub.1], [u.sub.2], ..., [u.sub.n].

Proposition 2.20. Let V ([I.sub.1], [I.sub.2]) be a strong refined neutrosophic hypervector space over a refined neutrosophic field K([I.sub.1], [I.sub.2]), and let [u.sub.1] = ([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.2]), [u.sub.2] = ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2]), ..., [u.sub.n] = ([a.sub.n], [b.sub.n][I.sub.1], [c.sub.n][I.sub.2]) [member of] V ([I.sub.1], [I.sub.2]), [[alpha].sub.1] = ([k.sub.1], [m.sub.1][I.sub.1], [t.sub.1][I.sub.2]), [[alpha].sub.2] = ([k.sub.2], [m.sub.2][I.sub.1], [t.sub.2][I.sub.2]) ..., [[alpha].sub.n] = ([k.sub.n], [m.sub.n][I.sub.1], [t.sub.n][I.sub.2]).

Then:

1. W([I.sub.1], [I.sub.2]) = [union]{[[alpha].sub.1] * [u.sub.1] + [[alpha].sub.2] * [u.sub.2] + ... + [[alpha].sub.n] * [u.sub.n] : [[alpha].sub.1], [[alpha].sub.2], ..., [[alpha].sub.n] [member of] K([I.sub.1], [I.sub.2])} is a refined neutrosophic subhypervector space of V ([I.sub.1], [I.sub.2]).

2. W([I.sub.1], [I.sub.2]) is the smallest refined neutrosophic subhypervector space of V ([I.sub.1], [I.sub.2]) containing [u.sub.1], [u.sub.2], ..., [u.sub.n].

Proof: The proof follows from similar approach as that of Proposition 2.19.

Remark 2.21. The refined neutrosophic subhypervector space W([I.sub.1], [I.sub.2]) of the strong refined neutrosophic hypervector space V ([I.sub.1], [I.sub.2]) over a refined neutrosophic field K([I.sub.1], [I.sub.2]) of Proposition 2.20 is said to be generated by the refined neutrosophic vectors [u.sub.1], [u.sub.2], ..., [u.sub.n] and we write W([I.sub.1], [I.sub.2]) = span{[u.sub.1], [u.sub.2], ..., [u.sub.n]}.

Definition 2.22. Let (V ([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) be a strong refined neutrosophic hypervector space over a refined neutrosophic field K([I.sub.1], [I.sub.2]) and let B([I.sub.1], [I.sub.2]) = {[u.sub.1] = ([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.2]), [u.sub.2] = ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2]), ..., [u.sub.n] = ([a.sub.n], [b.sub.n][I.sub.1], [c.sub.n][I.sub.2])} be a subset of V ([I.sub.1], [I.sub.2]). B([I.sub.1], [I.sub.2]) is said to generate or span V ([I.sub.1], [I.sub.2]) if V ([I.sub.1], [I.sub.2]) = span(B([I.sub.1], [I.sub.2])).

Example 2.23. Let V ([I.sub.1], [I.sub.2]) = [R.sup.3]([I.sub.1], [I.sub.2]) be a strong refined neutrosophic hypervector space over a neutrosophic field R([I.sub.1], [I.sub.2]) and let B([I.sub.1], [I.sub.2]) = {[u.sub.1] = ((1, 0[I.sub.1], 0[I.sub.2]),(0, 0[I.sub.1], 0[I.sub.2]),(0, 0[I.sub.1], 0[I.sub.2])), [u.sub.2] = ((0, 0[I.sub.1], 0[I.sub.2]),(1, 0[I.sub.1], 0[I.sub.2]),(0, 0[I.sub.1], 0[I.sub.2])), [u.sub.3] = ((0, 0[I.sub.1], 0[I.sub.2]),(0, 0[I.sub.1], 0[I.sub.2]),(1, 0[I.sub.1], 0[I.sub.2]))}. Then B([I.sub.1], [I.sub.2]) spans V ([I.sub.1], [I.sub.2]).

Example 2.24. Let V ([I.sub.1], [I.sub.2]) = [R.sup.2]([I.sub.1], [I.sub.2]) be a weak refined neutrosophic hypervector space over a field R and let B([I.sub.1], [I.sub.2]) = {[u.sub.1] = ((1, 0[I.sub.1], 0[I.sub.2]),(0, 0[I.sub.1], 0[I.sub.2])), [u.sub.2] = ((0, 0[I.sub.1], 0[I.sub.2]),(1, 0[I.sub.1], 0[I.sub.2])), [u.sub.3] = ((0, [I.sub.1], 0[I.sub.2]),(0, 0[I.sub.1], 0[I.sub.2])), [u.sub.4] = ((0, 0[I.sub.1], 0[I.sub.2]), (0, [I.sub.1], 0[I.sub.2])), [u.sub.5] = ((0, 0[I.sub.1], [I.sub.2]), (0, 0[I.sub.1], 0[I.sub.2])), [u.sub.6] = ((0, 0[I.sub.1], 0[I.sub.2]), (0, 0[I.sub.1], [I.sub.2]))}. Then B([I.sub.1], [I.sub.2]) spans V ([I.sub.1], [I.sub.2]).

Definition 2.25. Let W[[I.sub.1], [I.sub.2]] and X[[I.sub.1], [I.sub.2]] be two refined neutrosophic subhypervector spaces of a strong refined neutrosophic hypervector space (V ([I.sub.1], [I.sub.2]), +, *, K([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]) is said to be the direct sum of W[[I.sub.1], [I.sub.2]] and X[[I.sub.1], [I.sub.2]] written V ([I.sub.1], [I.sub.2]) = W[[I.sub.1], [I.sub.2]] [direct sum] X[[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 = w + x where w [member of] W[[I.sub.1], [I.sub.2]] and x [member of] X[[I.sub.1], [I.sub.2]].

Proposition 2.26. Let W[[I.sub.1], [I.sub.2]] and X[[I.sub.1], [I.sub.2]] be two refined neutrosophic subhypervector spaces of a strong refined neutrosophic hypervector space (V ([I.sub.1], [I.sub.2]), +, *, K([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]) = W[[I.sub.1], [I.sub.2]] [direct sum] X[[I.sub.1], [I.sub.2]] if and only if the following conditions hold:

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

2. W[[I.sub.1], [I.sub.2]] [intersection] X[[I.sub.1], [I.sub.2]] = {[theta]}.

Proof. Same as in classical case.

Definition 2.27. Let (V ([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) be a strong refined neutrosophic hypervector space over a refined neutrosophic field K([I.sub.1], [I.sub.2]). The refined neutrosophic vector u = (a, b[I.sub.1], c[I.sub.2]) [member of] V ([I.sub.1], [I.sub.2]) is said to be a linear combination of the refined neutrosophic vectors [u.sub.1] = ([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.2]), [u.sub.2] = ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.1]), ..., [u.sub.n] = ([a.sub.n], [b.sub.n][I.sub.1], [c.sub.n][I.sub.2]) [member of] V ([I.sub.1], [I.sub.2]) if there exist refined neutrosophic scalars [[alpha].sub.1] = ([k.sub.1], [m.sub.1][I.sub.1], [t.sub.1][I.sub.2]), [[alpha].sub.2] = ([k.sub.2], [m.sub.2][I.sub.1], [t.sub.2][I.sub.2]), ..., [[alpha].sub.n] = ([k.sub.n], [m.sub.n][I.sub.1], [t.sub.n][I.sub.2]) [member of] K([I.sub.1], [I.sub.2]) such that

u [member of] [[alpha].sub.1] * [u.sub.1] + [[alpha].sub.2] * [u.sub.2] + ... + [[alpha].sub.n] * [u.sub.n].

Definition 2.28. Let (V ([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) be a strong refined neutrosophic hypervector space over a refined neutrosophic field K([I.sub.1], [I.sub.2]) and let B([I.sub.1], [I.sub.2]) = {[u.sub.1] = ([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.2]), [u.sub.2] = ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2]), ..., [u.sub.n] = ([a.sub.n], [b.sub.n][I.sub.1], [c.sub.n][I.sub.2])} be a subset of V ([I.sub.1], [I.sub.2]).

1. B([I.sub.1], [I.sub.2]) is called a linearly dependent set if there exist refined neutrosophic scalars [[alpha].sub.1] = ([k.sub.1], [m.sub.1][I.sub.1], [t.sub.1][I.sub.2]), [[alpha].sub.2] = ([k.sub.2], [m.sub.2][I.sub.1], [t.sub.2][I.sub.2]), ..., [[alpha].sub.n] = ([k.sub.n], [m.sub.n][I.sub.1], [t.sub.n][I.sub.2]) (not all zero) such that

[theta] [member of] [[alpha].sub.1] * [u.sub.1] + [[alpha].sub.2] * [u.sub.2] + ... + [[alpha].sub.n] * [u.sub.n].

2. B([I.sub.1], [I.sub.2]) is called a linearly independent set if

[theta] [member of] [[alpha].sub.1] * [u.sub.1] + [[alpha].sub.2] * [u.sub.2] + ... + [[alpha].sub.n] * [u.sub.n]

implies that [[alpha].sub.1] = [[alpha].sub.2] = ... = [[alpha].sub.n] = (0, 0[I.sub.1], 0[I.sub.2]).

Proposition 2.29. Let (V ([I.sub.1], [I.sub.2]), +, *, K) be a weak refined neutrosophic hypervector space over a field K. Any singleton set of non-null refined neutrosophic vector of the weak refined neutrosophic hypervector space V ([I.sub.1], [I.sub.2]) is linearly independent.

Proof. Suppose that [theta] [not equal to] v = (a, b[I.sub.1], c[I.sub.2]) [member of] V ([I.sub.1], [I.sub.2]). Let [theta] [member of] k * v and suppose that [theta] [not equal to] k [member of] K. Then [k.sup.-1] [member of] K and therefore, [k.sup.-1] * [theta] [subset or equal to] [k.sup.-1] * (k * v) so that

[theta] [member of] ([k.sup.-1]k) * v = 1 * v = {(x, y[I.sub.1], z[I.sub.2]) : x [member of] 1 * a, y [member of] 1 * b, z [member of] 1 * c} = {(x, y[I.sub.1], z[I.sub.2]) : x [member of] {a}, y [member of] {b}, z [member of] {c}} = {(a, b[I.sub.1], c[I.sub.2])} = {v}.

This shows that v = [theta] which is a contradiction. Hence, k = [theta] and thus, the singleton {v} is a linearly independent set.

Proposition 2.30. Let (V ([I.sub.1], [I.sub.2]), +, *, K) be a weak refined neutrosophic hypervector space over a field K. Any set of refined neutrosophic vectors of the weak refined neutrosophic hypervector space V ([I.sub.1], [I.sub.2]) containing the null refined neutrosophic vector is always linearly dependent.

Proposition 2.31. Let (V ([I.sub.1], [I.sub.2]), +, *, K) be a weak refined neutrosophic hypervector space over a field K and let B([I.sub.1], [I.sub.2]) = {[u.sub.1] = ([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.2]), [u.sub.2] = ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2]), ..., [u.sub.n] = ([a.sub.n], [b.sub.n][I.sub.1], [c.sub.n][I.sub.2])} be a subset of V ([I.sub.1], [I.sub.2]). Then B([I.sub.1], [I.sub.2]) is a linearly independent set if and only if at least one element of B([I.sub.1], [I.sub.2]) can be expressed as a linear combination of the remaining elements of B([I.sub.1], [I.sub.2]).

Proof : This can be easily established.

Proposition 2.32. Let (V ([I.sub.1], [I.sub.2]), +, *, K) be a weak refined neutrosophic hypervector space over a field K and let

B([I.sub.1], [I.sub.2]) = {[u.sub.1] = ([a.sub.1], [b.sub.1]I, [c.sub.1][I.sub.1]), [u.sub.2] = ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2]), ..., [u.sub.n] = ([a.sub.n], [b.sub.n][I.sub.1], c[I.sub.2])}

be a subset of V ([I.sub.1], [I.sub.2]). Then B([I.sub.1], [I.sub.2]) is a linearly dependent set if and only if at least one element of B([I.sub.1], [I.sub.2]) can be expressed as a linear combination of the remaining elements of B([I.sub.1], [I.sub.2]).

Proof : Suppose that B([I.sub.1], [I.sub.2]) is a linearly dependent set. Then there exist scalars [k.sub.1], [k.sub.2], ..., [k.sub.n] not all zero in K such that [theta] [member of] [k.sub.1] * [u.sub.1] + [k.sub.2] * [u.sub.2] + ... + [k.sub.n] * [u.sub.n].

Suppose that [k.sub.1] [not equal to] 0, then [k.sup.-1.sub.1] [member of] K and therefore

[mathematical expression not reproducible]

so that

[theta] [member of] 1 * [u.sub.1] + {u}

where u = (a, b[I.sub.1], c[I.sub.2]) [member of] ([k.sup.-1.sub.1][k.sub.2]) * [u.sub.2] + ... + ([k.sup.-1.sub.n] [k.sub.n]) * [u.sub.n].

Thus [theta] [member of] {(a + [a.sub.1], (b + [b.sub.1])[I.sub.1], (c + [c.sub.1])[I.sub.2])} from which we obtain [u.sub.1] = ([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.2]) = -u = -(a, b[I.sub.1], c[I.sub.2]) so that

[mathematical expression not reproducible].

This shows that [u.sub.1] [member of] span{[u.sub.2], [u.sub.3], ..., [u.sub.n]}.

Conversely, suppose that [u.sub.1] [member of] span{[u.sub.2], [u.sub.3], ..., [u.sub.n]} and suppose that 0 [not equal to] -1 [member of] K. Then there exist [k.sub.2], [k.sub.3], ..., [k.sub.n] [member of] K such that

[u.sub.1] [member of] [k.sub.2] * [u.sub.2] + [k.sub.3] * [u.sub.3] + ... + [k.sub.n] * [u.sub.n]

and we have

[u.sub.1] + (-[u.sub.1]) [member of] (-1) * [u.sub.1] + [k.sub.2] * [u.sub.2] + [k.sub.3] * [u.sub.3] + * + [k.sub.n] * [u.sub.n].

From which

[theta] [member of] (-1) * [u.sub.1] + [k.sub.2] * [u.sub.2] + [k.sub.3] * [u.sub.3] + ... + [k.sub.n] * [u.sub.n].

Since -1 [not equal to] 0 [member of] K, it follows that B([I.sub.1], [I.sub.2]) is a linearly dependent set.

Proposition 2.33. Let (V ([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) be a strong refined neutrosophic hypervector space over a refined neutrosophic field K([I.sub.1], [I.sub.2]) and let [B.sub.1]([I.sub.1], [I.sub.2]) and [B.sub.2]([I.sub.1], [I.sub.2]) be subsets of V ([I.sub.1], [I.sub.2]) such that [B.sub.1]([I.sub.1], [I.sub.2]) [subset or equal to] [B.sub.2]([I.sub.1], [I.sub.2]). If [B.sub.1]([I.sub.1], [I.sub.2]) is linearly dependent, then [B.sub.2]([I.sub.1], [I.sub.2]) is linearly dependent.

Proposition 2.34. Let (V ([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) be a strong refined neutrosophic hypervector space over a refined neutrosophic field K([I.sub.1], [I.sub.2]) and let [B.sub.1]([I.sub.1], [I.sub.2]) and [B.sub.2]([I.sub.1], [I.sub.2]) be subsets of V ([I.sub.1], [I.sub.2]) such that [B.sub.1]([I.sub.1], [I.sub.2]) [subset or equal to] [B.sub.2]([I.sub.1], [I.sub.2]). If [B.sub.1]([I.sub.1], [I.sub.2]) is linearly independent, then [B.sub.2]([I.sub.1], [I.sub.2]) is linearly independent.

Definition 2.35. Let (V ([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) be a strong refined neutrosophic hypervector space over a refined neutrosophic field K([I.sub.1], [I.sub.2]) and let B([I.sub.1], [I.sub.2]) = {[u.sub.1] = ([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.1]), [u.sub.2] = ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2]), ...} be a subset of V ([I.sub.1], [I.sub.2]). B([I.sub.1], [I.sub.2]) is said to be a basis for V ([I.sub.1], [I.sub.2]) if the following conditions hold:

1. B([I.sub.1], [I.sub.2]) is a linearly independent set

2. V ([I.sub.1], [I.sub.2]) = span(B([I.sub.1], [I.sub.2])).

If B([I.sub.1], [I.sub.2]) is finite and its cardinality is n, then V ([I.sub.1], [I.sub.2]) is called an n-dimensional strong refined neutrosophic hypervector space and we write [dim.sub.s](V ([I.sub.1], [I.sub.2])) = n. If B([I.sub.1], [I.sub.2]) is not finite, then V ([I.sub.1], [I.sub.2]) is called an infinite-dimensional strong refined neutrosophic hypervector space.

Definition 2.36. Let (V ([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) be a weak refined neutrosophic hypervector space over a field K and let B([I.sub.1], [I.sub.2]) = {[u.sub.1] = ([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.1]), [u.sub.2] = ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2]), ...} be a subset of V ([I.sub.1], [I.sub.2]). B([I.sub.1], [I.sub.2]) is said to be a basis for V ([I.sub.1], [I.sub.2]) if the following conditions hold:

1. B([I.sub.1], [I.sub.2]) is a linearly independent set

2. V ([I.sub.1], [I.sub.2]) = span(B([I.sub.1], [I.sub.2])).

If B([I.sub.1], [I.sub.2]) is finite and its cardinality is n, then V ([I.sub.1], [I.sub.2]) is called an n-dimensional weak refined neutrosophic hypervector space and we write [dim.sub.w](V ([I.sub.1], [I.sub.2])) = n. If B([I.sub.1], [I.sub.2]) is not finite, then V ([I.sub.1], [I.sub.2]) is called an infinite-dimensional weak refined neutrosophic hypervector space.

Example 2.37. In Example 2.23 B([I.sub.1], [I.sub.2]) is a basis for V ([I.sub.1], [I.sub.2]) and dims(V ([I.sub.1], [I.sub.2])) = 3.

Example 2.38. In Example 2.24 B([I.sub.1], [I.sub.2]) is a basis for V ([I.sub.1], [I.sub.2]) and dimW (V ([I.sub.1], [I.sub.2])) = 6.

Proposition 2.39. Let (V ([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) be a strong refined neutrosophic hypervector space over a refined neutrosophic field K([I.sub.1], [I.sub.2]) and let B([I.sub.1], [I.sub.2]) = {[u.sub.1] = ([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.2]), [u.sub.2] = ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2]), ..., [u.sub.n] = ([a.sub.n], [b.sub.n][I.sub.1], [c.sub.n][I.sub.2])} be a subset of V ([I.sub.1], [I.sub.2]). Then B([I.sub.1], [I.sub.2]) is a basis for V ([I.sub.1], [I.sub.2]) if and only if each refined neutrosophic vector u = (a, b[I.sub.1], c[I.sub.2]) [member of] V ([I.sub.1], [I.sub.2]) can be expressed uniquely as a linear combination of the elements of B([I.sub.1], [I.sub.2]).

Proof. Suppose that each refined neutrosophic vector u = (a, bI, c[I.sub.2]) [member of] V ([I.sub.1], [I.sub.2]) can be expressed uniquely as a linear combination of the elements of B([I.sub.1], [I.sub.2]). Then u [member of] span(B([I.sub.1], [I.sub.2])) = V ([I.sub.1], [I.sub.2]). Since such a representation is unique, it follows that B([I.sub.1], [I.sub.2]) is a linearly independent set and since u [member of] V ([I.sub.1], [I.sub.2]) is arbitrary, it follows that B([I.sub.1], [I.sub.2]) is a basis for V ([I.sub.1], [I.sub.2]). Conversely, suppose that B([I.sub.1], [I.sub.2]) is a basis for V ([I.sub.1], [I.sub.2]), then V ([I.sub.1], [I.sub.2]) = span(B([I.sub.1], [I.sub.2])) and B([I.sub.1], [I.sub.2]) is linearly independent. Now it remains to show that u = (a, b[I.sub.1], c[I.sub.2]) [member of] V ([I.sub.1], [I.sub.2]) can be expressed uniquely as a linear combination of the elements of B([I.sub.1], [I.sub.2]).

To this end, for [[alpha].sub.1] = ([k.sub.1], [m.sub.1][I.sub.1], [p.sub.1][I.sub.2]), [[alpha].sub.2] = ([k.sub.2], [m.sub.2][I.sub.1], [p.sub.2][I.sub.2]), ..., [[alpha].sub.n] = ([k.sub.n], [m.sub.n][I.sub.1], [p.sub.n][I.sub.2]), [[beta].sub.1] = ([r.sub.1], [s.sub.1][I.sub.1], [t.sub.1][I.sub.2]), [[beta].sub.2] = ([r.sub.2], [s.sub.2][I.sub.1], [t.sub.2][I.sub.2]), ..., [[beta].sub.n] = ([r.sub.n], [s.sub.n][I.sub.1], [t.sub.n][I.sub.2]) [member of] K([I.sub.1], [I.sub.2]), let us express u in two ways as follows:

u [member of] [[alpha].sub.1] * [u.sub.1] + [[alpha].sub.2] * [u.sub.2] + ... + [[alpha].sub.n] * [u.sub.n], (1)

u [member of] [[beta].sub.1] * [u.sub.1] + [[beta].sub.2] * [u.sub.2] + ...+ [[beta].sub.n] * [u.sub.n]. (2)

From equation(2), we have

[mathematical expression not reproducible]. (3).

From equations (1) and (3), we have

[mathematical expression not reproducible].

Since B([I.sub.1], [I.sub.2]) is linearly independent, it follows that

[[alpha].sub.1] - [[beta].sub.1] = [[alpha].sub.2] - [[beta].sub.2] = ... = [[alpha].sub.n] - [[beta].sub.n] = (0, 0[I.sub.1], 0[I.sub.2])

and therefore,

[[alpha].sub.1] = [[beta].sub.1], [[alpha].sub.2] = [[beta].sub.2], ..., [[alpha].sub.n] = [[beta].sub.n].

This shows that u has been expressed uniquely as a linear combination of the elements of B([I.sub.1], [I.sub.2]). The proof is complete.

Proposition 2.40. Let (V ([I.sub.1], [I.sub.2]), +, *, K) be a weak refined neutrosophic hypervector space over a field K and let [B.sub.1]([I.sub.1], [I.sub.2]) = {[u.sub.1] = ([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.2]), [u.sub.2] = ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2]), ..., [u.sub.n] = ([a.sub.n], [b.sub.n][I.sub.1], c[I.sub.2])} be a linearly independent subset of V ([I.sub.1], [I.sub.2]). If u [member of] V ([I.sub.1], [I.sub.2])\[B.sub.1]([I.sub.1], [I.sub.2]) = V ([I.sub.1], [I.sub.2]) [intersection] (B[([I.sub.1], [I.sub.2])).sup.c] is arbitrary, then [B.sub.2]([I.sub.1], [I.sub.2]) = {[u.sub.1] = ([a.sub.1], [b.sub.1]I, [c.sub.1][I.sub.2]), [u.sub.2] = ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2]), ..., [u.sub.n] = ([a.sub.n], [b.sub.n][I.sub.1], [c.sub.n][I.sub.2]), u} is a linearly dependent set if and only if u [member of] span(([B.sub.1]([I.sub.1], [I.sub.2])).

Proof. Suppose that [B.sub.2]([I.sub.1], [I.sub.2]) is a linearly dependent set. Then there exist scalars [k.sub.1], [k.sub.2], ..., [k.sub.n], k not all zero such that

[theta] [member of] [k.sub.1] * [u.sub.1] + [k.sub.2] * [u.sub.2] + ... + [k.sub.n] * [u.sub.n] + k * u. (4)

Suppose that k = 0, then there exist at least one of the [k'.sub.i]s say [k.sub.1] [not equal to] 0 and equation (4) becomes

[theta] [member of] [k.sub.1] * [u.sub.1] + [k.sub.2] * [u.sub.2] + ... + [k.sub.n] ... [u.sub.n] (5)

from which it follows that the set

[B.sub.1]([I.sub.1], [I.sub.2]) = {[u.sub.1] = ([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.2]), [u.sub.2] = ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2]), ..., [u.sub.n] = ([a.sub.n], [b.sub.n][I.sub.1], c[I.sub.2])} is linearly dependent. This contradicts the hypothesis that [B.sub.1]([I.sub.1], [I.sub.2]) is linearly independent. Hence k [not equal to] 0 and therefore [k.sup.-1] [member of] K. From equation (4), we have

[mathematical expression not reproducible].

Conversely, suppose that u [member of] span([B.sub.1]([I.sub.1], [I.sub.2])). Then there exist [k.sub.1], [k.sub.2], ..., [k.sub.n] [member of] K such that

[mathematical expression not reproducible].

Since u [not member of] [B.sub.1]([I.sub.1], [I.sub.2]) and [B.sub.1]([I.sub.1], [I.sub.2]) is linearly independent, it follows that {[u.sub.1], [u.sub.2], ..., [u.sub.n], u} = [B.sub.2]([I.sub.1], [I.sub.2]) is a linearly dependent set. The proof is complete.

Definition 2.41. Let W[[I.sub.1], [I.sub.2]] be a refined neutrosophic subhypervector space of a strong refined neutrosophic hypervector space (V ([I.sub.1], [I.sub.2]), +, *, K([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] = v + W[[I.sub.1], [I.sub.2]] : v [member of] V ([I.sub.1], [I.sub.2])}.

Proposition 2.42. Let V ([I.sub.1], [I.sub.2])/W[[I.sub.1], [I.sub.2]] = {[v] = v + W[[I.sub.1], [I.sub.2]] : v [member of] V ([I.sub.1], [I.sub.2])}. If for every [u], [v] [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]) we define:

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

and

[alpha] [??] [u] = [[alpha] * u] = {[x] : x [member of] [alpha] * u}.

(V ([I.sub.1], [I.sub.2])/W[[I.sub.1], [I.sub.2]], [direct sum], [??], K([I.sub.1], [I.sub.2])) is a strong refined neutrosophic hypervector space over a refined neutrosophic field K([I.sub.1], [I.sub.2]) called a strong refined neutrosophic quotient hypervector space.

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

Proposition 2.43. Let W[[I.sub.1], [I.sub.2]] be a refined neutrosophic subhypervector space of a strong refined neutrosophic hypervector space V ([I.sub.1], [I.sub.2]) over a refined neutrosophic field K([I.sub.1], [I.sub.2]), let (V ([I.sub.1], [I.sub.2])/W[[I.sub.1], [1.sub.2]]) be as defined in Proposition 2.42, then the following hold:

1. W[[I.sub.1], [I.sub.2]] is finite dimensional and [dim.sub.s]W[[I.sub.1], [I.sub.2]] [less than or equal to] [dim.sub.s]V ([I.sub.1], [I.sub.2]).

2. [dim.sub.s](V ([I.sub.1], [I.sub.2])/W[[I.sub.1], 12]) = [dim.sub.s]V ([I.sub.1], [I.sub.2]) - [dim.sub.s]W[[I.sub.1], [I.sub.2]].

Proof:

1. Let [B.sub.1]([I.sub.1], [I.sub.2]) be the basis for W[[I.sub.1], [I.sub.2]] and let [B.sub.2]([I.sub.1], [I.sub.2]) be a basis for V ([I.sub.1], [I.sub.2]). Since W[[I.sub.1], [I.sub.2]] [subset or equal to] V ([I.sub.1], [I.sub.2]) then [B.sub.1]([I.sub.1], [I.sub.2]) is contained in [B.sub.2]([I.sub.1], [I.sub.2]). Therefore [B.sub.1]([I.sub.1], [I.sub.2]) is a linearly independent subset of V ([I.sub.1], [I.sub.2]). Then we have that [absolute value of ([B.sub.1]([I.sub.1], [I.sub.2]))] [less than or equal to] [absolute value of ([B.sub.2]([I.sub.1], [I.sub.2]))]. Now, since [absolute value of ([B.sub.1]([I.sub.1], [I.sub.2]))] [less than or equal to] [absolute value of ([B.sub.2]([I.sub.1], [I.sub.2]))] and V ([I.sub.1], [I.sub.2]) is finite dimensional we can conclude that W([I.sub.1], [I.sub.2]) is finite dimensional and

[dim.sub.s]W([I.sub.1], [I.sub.2]) = [absolute value of ([B.sub.1]([I.sub.1], [I.sub.2]))] [less than or equal to] [absolute value of ([B.sub.2]([I.sub.1], [I.sub.2]))] = [dim.sub.s]V ([I.sub.1], [I.sub.2]).

2. Let {[u.sub.1], [u.sub.2], ..., [u.sub.m]} be a basis of W[[I.sub.1], [I.sub.2]]. Then this can be filled out to a basis, {[u.sub.1], [u.sub.2] ..., [u.sub.m], [v.sub.1], [v.sub.2], ..., [v.sub.n]} of V ([I.sub.1], [I.sub.2]), where m+n = [dim.sub.s]V ([I.sub.1], [I.sub.2]) and m = [dim.sub.s]W[[I.sub.1], [I.sub.2]]. Let [[v.sub.1]], [[v.sub.2]], ..., [[v.sub.n]] be the images in V ([I.sub.1], [I.sub.2])/W[[I.sub.1], [I.sub.2]], of [v.sub.1], [v.sub.2], ..., [v.sub.n]. Since any vector v [member of] V ([I.sub.1], [I.sub.2]) is in a linear combination of [u.sub.1], [u.sub.2], ..., [u.sub.m], [v.sub.1], [v.sub.2], ..., [v.sub.n], we have that

v [member of] [[alpha].sub.1] * [u.sub.1] + [[alpha].sub.2] * [u.sub.2] + ... + [[alpha].sub.m] * [u.sub.m] + [[beta].sub.1] * [v.sub.1] + [[beta].sub.2] * [v.sub.2] + ... + [[beta].sub.n] * [v.sub.n],

then

[mathematical expression not reproducible].

Thus [[v.sub.1]], [[v.sub.2]] ..., [[v.sub.n]] span V ([I.sub.1], [I.sub.2])/W[[I.sub.1], [I.sub.2]]. We claim that they are linearly independent, for if

[theta] [member of] [[lambda].sub.1] * [[v.sub.1]] [direct sum] [[lambda].sub.2] * [[v.sub.2]] [direct sum] ... [direct sum] [[lambda].sub.n] * [[v.sub.n]]

then

[theta] [member of] [[lambda].sub.1] * [[v.sub.1]] [direct sum] [[lambda].sub.2] * [[v.sub.2]] [direct sum] ... [direct sum] [[lambda].sub.n] * [[v.sub.n]] [direct sum] W[[I.sub.1], [I.sub.2]] [theta] [subset or equal to] [[lambda].sub.1] * [[v.sub.1]] [direct sum] [[lambda].sub.2] * [[v.sub.2]] [direct sum] ... [direct sum] [[lambda].sub.n] * [[v.sub.n]] [direct sum] [[gamma].sub.1] * [[u.sub.1]] [direct sum] [[gamma].sub.2] * [[u.sub.2]] [direct sum] ... [direct sum] [[gamma].sub.m] * [[u.sub.m]]

which by the linear independence of the set {[u.sub.1], [u.sub.2] ..., [u.sub.m], [v.sub.1], [v.sub.2] ..., [v.sub.n]} forces [[lambda].sub.1] = [[lambda].sub.2] = ... = [[lambda].sub.n] = [[gamma].sub.1] = [[gamma].sub.2] = ... = [[gamma].sub.m] = 0.

This shows that V ([I.sub.1], [I.sub.2])/W[[I.sub.1], [I.sub.2]] has a basis of n elements, and

[dim.sub.s](V ([I.sub.1], [I.sub.2])/W[[I.sub.1], [I.sub.2]]) = n = (n + m) - m = [dim.sub.s]V ([I.sub.1], [I.sub.2]) - [dim.sub.s]W[[I.sub.1], [I.sub.2]].

Proposition 2.44. Let [W.sub.1]([I.sub.1], [I.sub.2]) and [W.sub.2]([I.sub.1], [I.sub.2]) be finite dimensional weak refined neutrosophic subhypervector spaces of a weak refined neutrosophic vector space V ([I.sub.1], [I.sub.2]) over a field K. Then [W.sub.1]([I.sub.1], [I.sub.2]) + [W.sub.2]([I.sub.1], [I.sub.2]) is a finite dimensional refined neutrosophic subhypervector space of V ([I.sub.1], [I.sub.2]) and

[dim.sub.w]([W.sub.1]([I.sub.1], [I.sub.2]) + [W.sub.2]([I.sub.1], [I.sub.2])) = [dim.sub.w]([W.sub.1]([I.sub.1], [I.sub.2])) + [dim.sub.w]([W.sub.2]([I.sub.1], [I.sub.2])) - [dim.sub.w] ([W.sub.1]([I.sub.1], [I.sub.2])[intersection][W.sub.2]([I.sub.1], [I.sub.2])).

If V ([I.sub.1], [I.sub.2]) = [W.sub.1]([I.sub.1], [I.sub.2]) [direct sum] [W.sub.2]([I.sub.1], [I.sub.2]) then

[dim.sub.w]([W.sub.1]([I.sub.1], [I.sub.2]) + [W.sub.2]([I.sub.1], [I.sub.2])) = [dim.sub.w]([W.sub.1]([I.sub.1], [I.sub.2])) + [dim.sub.w]([W.sub.2]([I.sub.1], [I.sub.2])).

Proof: We know that [W.sub.1]([I.sub.1], [I.sub.2])[intersection][W.sub.2]([I.sub.1], [I.sub.2])is a refined neutrosophic subhypervector space of both [W.sub.1]([I.sub.1], [I.sub.2]) and [W.sub.2]([I.sub.1], [I.sub.2]). So [W.sub.1]([I.sub.1], [I.sub.2])[intersection][W.sub.2]([I.sub.1], [I.sub.2]) is a finite dimensional refined neutrosophic subhypervector space of V ([I.sub.1], [I.sub.2]).

Suppose that [dim.sub.w]([W.sub.1]([I.sub.1], [I.sub.2]) [intersection] [W.sub.2]([I.sub.1], [I.sub.2])) = k, [dim.sub.w]([W.sub.1]([I.sub.1], [I.sub.2])) = m and [dim.sub.w]([W.sub.2]([I.sub.1], [I.sub.2])) = n then we have that k [less than or equal to] m and k [less than or equal to] n.

Now, let {[u.sub.1], [u.sub.2], ..., [u.sub.k]} be a basis of [W.sub.1]([I.sub.1], [I.sub.2]) [intersection] [W.sub.2]([I.sub.1], [I.sub.2]). Then we have that {[u.sub.1], [u.sub.2], ..., [u.sub.k]} is a linearly independent set of refined neutrosophic vectors in [W.sub.1][[I.sub.1], [I.sub.2]] and [W.sub.2][[I.sub.1], [I.sub.2]] with k [less than or equal to] m and k [less than or equal to] n, then it follows that either {[u.sub.1], [u.sub.2], ..., [u.sub.k]} is a basis of [W.sub.1][[I.sub.1], [I.sub.2]] and [W.sub.2][[I.sub.1], [I.sub.2]] or it can be extended to a basis for [W.sub.1][[I.sub.1], [I.sub.2]] and [W.sub.2][[I.sub.1], [I.sub.2]].

Let {[u.sub.1], [u.sub.2], ..., [u.sub.k], [v.sub.1], [v.sub.2], ..., [v.sub.m- k]} be a basis for [W.sub.1][[I.sub.1], [I.sub.2]], and let {[u.sub.1], [u.sub.2], ..., [u.sub.k], [w.sub.1], [w.sub.2], ..., [w.sub.n-k]} be a basis of [W.sub.1][[I.sub.1], [I.sub.2]].

Then the refined neutrosophic subhypervector space [W.sub.1][[I.sub.1], [I.sub.2]] + [W.sub.2][[I.sub.1], [I.sub.2]] is spanned by the refined neutrosophic vectors {[u.sub.1], [u.sub.2], ..., [u.sub.k], [v.sub.1], [v.sub.2], ... [v.sub.m-k], [w.sub.1], [w.sub.2], ..., [w.sub.n-k]} and these refined neutrosophic vectors form an independent set. For suppose

[theta] [member of] [k.summation over (i=1)] [[alpha].sub.i][u.sub.i] + [m.summation over (j=1)][[beta].sub.j][v.sub.j] + [n.summation over (r=1)] [[gamma].sub.r][w.sub.r].

Then

[mathematical expression not reproducible]

which shows that [[summation].sup.n.sub.r=1][[gamma].sub.r][w.sub.r] belongs to [W.sub.1][[I.sub.1], [I.sub.2]]. As [[summation].sup.n.sub.r=1][[gamma].sup.r][w.sub.r] also belongs to [W.sub.2][[I.sub.1], [I.sub.2]], it follows that

[n.summation over (r=1)][[gamma].sup.r][w.sub.r] = [k.summation over (i=1)][[lambda].sub.i][u.sub.i]

for certain scalars [[lambda].sub.1], [[lambda].sub.2], ..., [[lambda].sub.k].

Because the set {[u.sub.1], [u.sub.2], ..., [u.sub.k], [w.sub.1], [w.sub.2], ..., [w.sub.n-k]} is independent, each of the scalars [[gamma].sub.r] = 0. Thus

[theta] [member of] [k.summation over (i=1)][[alpha].sub.i][u.sub.i] + [m.summation over (j=1)][[beta].sub.j][v.sub.j]

and since {[u.sub.1], [u.sub.2], ..., [u.sub.k], [v.sub.1], [v.sub.2], ..., [v.sub.m-k]} is also an independent set, each [[alpha].sub.i] = 0 and each [[beta].sub.j] = 0. Thus, {[u.sub.1], [u.sub.2], ..., [u.sub.k], [v.sub.1], [v.sub.2], ..., [v.sub.m-k], [w.sub.1], [w.sub.2], ..., [w.sub.n-k]} is a basis for [W.sub.1][[I.sub.1], [I.sub.2]] + [W.sub.2][[I.sub.1], [I.sub.2]].

Finally, [dim.sub.w]([W.sub.1]([I.sub.1], [I.sub.2]) + [W.sub.2]([I.sub.1], [I.sub.2])) = k + m - k + n - k = m + n - k = [dim.sub.w]([W.sub.1]([I.sub.1], [I.sub.2])) + dimw([W.sub.2]([I.sub.1], [I.sub.2])) - [dim.sub.w]([W.sub.1]([I.sub.1], [I.sub.2]) [intersection] [W.sub.2]([I.sub.1], [I.sub.2])).

Definition 2.45. Let (V ([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) and W([I.sub.1], [I.sub.2]), +', *', K([I.sub.1], [I.sub.2])) be two strong refined neutrosophic hypervector spaces over a neutrosophic field K([I.sub.1], [I.sub.2]).

A mapping [phi] : V ([I.sub.1], [I.sub.2]) [right arrow] W([I.sub.1], [I.sub.2]) is called a strong refined neutrosophic hypervector space homomorphism if the following conditions hold:

1. [phi] is a strong hypervector space homomorphism.

2. [phi](0, [I.sub.1], [I.sub.2]) = (0, [I.sub.1], [I.sub.2]).

If in addition [phi] is a bijection, we say that V ([I.sub.1], [I.sub.2]) is isomorphic to W([I.sub.1], [I.sub.2]) and we write V ([I.sub.1], [I.sub.2]) [congruent to] W([I.sub.1], [I.sub.2]).

Proposition 2.46. Let (V ([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) and (W([I.sub.1], [I.sub.2]), +, *, K([I.sub.1], [I.sub.2])) be two strong refined neutrosophic hypervector spaces over a refined neutrosophic field K([I.sub.1], [I.sub.2]) and let [phi] : V ([I.sub.1], [I.sub.2]) [right arrow] W([I.sub.1], [I.sub.2]) be a bijective strong refined neutrosophic hypervector space homomorphism.

If B([I.sub.1], [I.sub.2]) = {[u.sub.1] = ([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.2]), [u.sub.2] = ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2]), ..., [u.sub.n] = ([a.sub.n], [b.sub.n][I.sub.1], [c.sub.n][I.sub.n])} is a basis for V ([I.sub.1], [I.sub.2]), then B'([I.sub.1], [I.sub.2]) = [phi](B([I.sub.1], [I.sub.2])) = {[phi]([u.sub.1]), [phi]([u.sub.2]), ..., [phi]([u.sub.n])} is a basis for W([I.sub.1], [I.sub.2]).

Proof. Suppose that B([I.sub.1], [I.sub.2]) is a basis for V ([I.sub.1], [I.sub.2]). Then for an arbitrary u = (a, b[I.sub.1], c[I.sub.2]) [member of] V ([I.sub.1], [I.sub.2]), there exist refined neutrosophic scalars [[alpha].sub.1] = (k1, m1[I.sub.1], t1[I.sub.2]), [[alpha].sub.2] = (k2, m2[I.sub.1], t2[I.sub.2]), ..., [[alpha].sub.n] = (kn, mn[I.sub.1], tn[I.sub.2]) [member of] K([I.sub.1], [I.sub.2]) such that

u [member of] [[alpha].sub.1] * [u.sub.1] + [[alpha].sub.2] * [u.sub.2] + ... + [[alpha].sub.n] * [u.sub.n]

[??] [phi](u) [member of] [phi]([[alpha].sub.1] * [u.sub.1] + [[alpha].sub.2] * [u.sub.2] + ... + [[alpha].sub.n] * un)

= [[alpha].sub.1] *' [phi]([u.sub.1]) +0 [[alpha].sub.2] *' [phi]([u.sub.2]) +' ... +' [[alpha].sub.n] *' [phi]([u.sub.n]).

Since [phi] is surjective, it follows that [phi](u), [phi]([u.sub.1]), [phi]([u.sub.2]), ..., [phi]([u.sub.n]) [member of] W([I.sub.1], [I.sub.2]) and therefore [phi](u) [member of] span(B'([I.sub.1], [I.sub.2])). To complete the proof, we must show that B'([I.sub.1], [I.sub.2]) is linearly independent. To this end, suppose that

[phi]([theta]) [member of] [[beta].sub.1] *' [phi]([u.sub.1]) +' [[beta].sub.2] *' [phi]([u.sub.2]) +' ... +' [[beta].sub.n] *' [phi]([u.sub.n])

where [beta]1 = ([p.sub.1], [q.sub.1][I.sub.1], [s.sub.1][I.sub.2]), [[beta].sub.2] = ([p.sub.2], [q.sub.2][I.sub.1], [s.sub.2][I.sub.2]), ..., [[beta].sub.n] = ([p.sub.n], [q.sub.n][I.sub.1], [s.sub.n][I.sub.2]) [member of] K([I.sub.1], [I.sub.2]), then

[phi]([theta]) [member of] [phi]([[beta].sub.1] * [u.sub.1]) +' [phi]([[beta].sub.2] * [u.sub.2]) +' * +' [phi]([[beta].sub.n] * [u.sub.n]) = [phi]([[beta].sub.1] * [u.sub.1] + [[beta].sub.2] * [u.sub.2] + ... + [[beta].sub.n] * [u.sub.n]).

Since [phi] is injective, we must have

[theta] [member of] [[beta].sub.1] * [u.sub.1] + [[beta].sub.2] * [u.sub.2] + ... + [[beta].sub.n] * [u.sub.n].

Also, since B([I.sub.1], [I.sub.2]) is linearly independent, we must have

[[beta].sub.1] = [[beta].sub.2] = ... = [[beta].sub.n] = (0, 0[I.sub.1], 0[I.sub.2]).

Hence B'([I.sub.1], [I.sub.2]) = {[phi]([u.sub.1]), [phi]([u.sub.2]), ..., [phi]([u.sub.n])} is linearly independent and therefore a basis for W([I.sub.1], [I.sub.2]).

Remark 2.47. Suppose we wish to transform a refined neutrosophic hypervector space into a neutrosophic hypervector space, an interesting question to ask will be, can we find a mapping that will help us achieve this? The answer to this is Yes. The mapping [phi] : V ([I.sub.1], [I.sub.2]) [right arrow] V (I) defined by

[phi]((x, y[I.sub.1], z[I.sub.2])) = (x,(y + z)I) [for all] x, y, z [member of] V

will make such transformation possible. This mapping is a non-neutrosophic one. This make sense since every refined neutrosophic hypervector space and neutrosophic hypervector spaces are hypervector spaces.

Proposition 2.48. Let (V ([I.sub.1], [I.sub.2]), +, *) be a weak refined neutrosophic vector space over a field K and let V (I) be a weak neutrosophic vector space over K. The mapping [phi] : V ([I.sub.1], [I.sub.2]) [right arrow] V (I) defined by

[phi]((x, y[I.sub.1], z[I.sub.2])) = (x,(y + z)I) [for all] x, y, z [member of] V

is a good linear transformation.

Proof. [phi] is well defined. Suppose (x, y[I.sub.1], z[I.sub.2]) = (x'y'[I.sub.1], z'[I.sub.2]) then we that x = x', y = y' and z' = z'. So,

[phi]((x, y[I.sub.1], z[I.sub.2])) = (x,(y + z)I) = x' + (y' + z')I = [phi](x', y'[I.sub.1], z'[I.sub.2]).

Now, suppose (x, y[I.sub.1], z[I.sub.2]),(x', y'[I.sub.1], 1z'[I.sub.2]) [member of] V ([I.sub.1], [I.sub.2]) then

[phi]((x, y[I.sub.1], z[I.sub.2]) + (x', y'[I.sub.1], z'[I.sub.2])) = [phi]((x + x'), (y + y')[I.sub.1], (z + z')[I.sub.2]) = (x + x'), (y + y' + z + z')I = (x + x'), ((y + z) + (y' + z'))I = (x + x'), ((y + z)I + (y' + z')I) = (x,(y + z)I) + (x', (y' + z')I) = [phi](x, y[I.sub.1], z[I.sub.2]) + [phi](x, y[I.sub.1], z[I.sub.2]).

[phi](k o (x, y[I.sub.1], z[I.sub.2])) = [phi]{(a, b[I.sub.1], c[I.sub.2]) : a [member of] k o x, b [member of] k o y, c [member of] k o z} = {[phi](a, b[I.sub.1], c[I.sub.2]) : a [member of] k o x, b [member of] k o y, c [member of] k o z} = {(u, vI) : u [member of] a, v [member of] b + c} = {(u, vI) : u [member of] k o x, v [member of] k o y + k o z} = {(u, vI) : u [member of] k o x, v [member of] k o (y + z)} = k o (x,(y + z)I) = k o [phi](x, y[I.sub.1], z[I.sub.2]).

Hence [phi] is a good linear transformation.

Proposition 2.49. Let Lk(V ([I.sub.1], [I.sub.2]), V (I)) be the set of good linear transformation from a weak refined neutrosophic vector space V ([I.sub.1], [I.sub.2]) over a field K into a weak neutrosophic vector space V (I) over a field K. Define addition and scalar multiplication as below;

([phi] + [psi])(x, y[I.sub.1], z[I.sub.2]) = [phi]((x, y[I.sub.1], z[I.sub.2])) + [psi]((x, y[I.sub.1], z[I.sub.2]))

and for k [member of] K

(k[phi])((x, y[I.sub.1], z[I.sub.2])) = k[phi](x, y[I.sub.1], z[I.sub.2]).

Then, it can be shown that ([L.sub.k](V ([I.sub.1], [I.sub.2]), V (I)), +, *) is a weak neutrosophic strongly distributive hypervector space.

Definition 2.50. Let [phi] : V ([I.sub.1], [I.sub.2]) [right arrow] V (I) be a good linear transformation, then ker [phi] = {(x, y[I.sub.1], z[I.sub.2]) : [phi]((x, y[I.sub.1], z[I.sub.2])) = (0, 0I)} = {(x, y[I.sub.1], z[I.sub.2]) : (x,(y + z)I) = (0, 0I)} = {(0, y[I.sub.1],(-y)[I.sub.2])}.

Proposition 2.51. Let [phi] : V ([I.sub.1], [I.sub.2]) [right arrow] V (I) be a good linear transformation.

1. ker [phi] is a subhyperspace of V ([I.sub.1], [I.sub.2]).

2. If W[[I.sub.1], [I.sub.2]] is a refined neutrosophic subhyperspace of V ([I.sub.1], [I.sub.2]), then the image of W[[I.sub.1], [I.sub.2]], [phi](W[[I.sub.1], [I.sub.2]]) is a neutrosophic subhyperspace of V (I).

3 Conclusion

This paper studied refinement of neutrosophic hypervector space, linear dependence, independence, bases and dimension of refined neutrosophic hypervector spaces and presented some of their basic properties. Also, the paper established the existence of a good linear transformation between a weak refined neutrosophic hypervector space V ([I.sub.1], [I.sub.2]) and a weak neutrosophic hypervector space V (I). We hope to present and study more properties of refined neutrosophic Hypervector spaces in our future papers.

DOI: 10.5281/zenodo.3900146

4 Acknowledgment

The Authors wish to thank the anonymous reviewers for their valuable comments and suggestions which have been used for the improvement of the paper.

References

[1] Adeleke, E.O, Agboola, A.A.A and Smarandache, F. Refined Neutrosophic Rings I, International Journal of Neutrosophic Science (IJNS), Vol. 2(2), pp. 77-81, 2020.

[2] Adeleke, E.O, Agboola, A.A.A and Smarandache, F. Refined Neutrosophic Rings II, International Journal of Neutrosophic Science (IJNS), Vol. 2(2), pp. 89-94, 2020.

[3] Agboola, A.A.A., Ibrahim, A.M. and Adeleke, E.O, Elementary Examination of NeutroAlgebras and AntiAlgebras Viz-a-Viz the Classical Number Systems, Vol. 4, pp. 16-19, 2020.

[4] Agboola, A.A.A. On Refined Neutrosophic Algebraic Structures, Neutrosophic Sets and Systems 10, pp 99-101, 2015.

[5] Agboola, A.A.A. and Akinleye, S.A., Neutrosophic Hypervector Spaces, ROMAI Journal, Vol. 11, pp. 1-16, 2015.

[6] Agboola, A.A.A. and Akinleye, S.A., Neutrosophic Vector Spaces, Neutrosophic Sets and Systems 4, pp 9-18, 2014.

[7] Agboola, A.A.A. and Davvaz, B., Introduction to Neutrosophic Hypergroups, Romai J.,Vol. 9(2), pp. 1-10, 2013.

[8] Agboola, A.A.A and Davvaz, B., On Neutrosophic Canonical Hypergroups and Neutrosophic Hyperrings, Neutrosophic Sets and Systems. Vol. 2, pp. 34-41, 2014.

[9] Asokkumar, A., Hyperlattice formed by the idempotents of a hyperring, Tamkang J. Math., Vol. 3(38), pp. 209-215, 2007.

[10] Asokkumar, A. and Veelrajan, M., Characterizations of regular hyperrings, Italian J. Pure and App. Math., Vol. 22, pp. 115-124, 2007.

[11] Asokkumar, A. and Veelrajan, M., Hyperrings of matrices over a regular hyperring, Italian J. Pure and App. Math. Vol. 23, pp. 113-120, 2008.

[12] Ameri. R. and Dehghan, O.R., On Dimension of Hypervector Spaces, European Journal of Pure and Applied Mathematics, Vol. 1(2), pp. 32-50, 2008.

[13] Atanassov, K., Intuitionistic fuzzy sets, Fuzzy Sets and Systems, Vol. 20, pp. 87-96, 1986.

[14] Bera, T. and Mahapatra, N.K., Introduction to neutrosophic soft groups, Neutrosophic Sets and Systems, Vol. 13, pp. 118-127, 2016, doi.org/10.5281/zenodo.570845.

[15] Bera, T. and Mahapatra, N.K., On neutrosophic normal soft groups, International Journal of Applied and Computational Mathematics., Vol. 3, pp. 3047-3066, 2017. DOI 10.1007/s40819-016- 0284-2.

[16] Bera, T. and Mahapatra, N.K., On neutrosophic soft rings, OPSEARCH, Vol. 54, pp. 143-167, 2017. DOI 10.1007/ s12597-016-0273-6.

[17] Bera, T and Mahapatra, N. K., On neutrosophic soft linear spaces, Fuzzy Information and Engineering, Vol. 9, pp 299-324, 2017.

[18] Bera, T and Mahapatra, N. K., On neutrosophic soft field, IJMTT, Vol. 56(7), pp. 472-494, 2018.

[19] Bonansinga, P., Quasicanonical hypergroups, (Italian), Atti Soc. Peloritana Sci. Fis. Mat. Natur., Vol. 27, pp. 9-17, 1981.

[20] Bonansinga, P., Weakly quasicanonical hypergroups, (Italian), Atti Sem. Mat. Fis. Univ. Modena, Vol. 30, pp. 286-298, 1981.

[21] Corsini, P., Prolegomena of Hypergroup Theory, Second edition, Aviani editor, (1993).

[22] Corsini, P., Finite canonical hypergroups with partial scalar identities, (Italian), Rend. Circ. Mat. Palermo (2), Vol. 36, pp. 205-219, 1987.

[23] Davvaz, B. and Leoreanu-Fotea, V., Hyperring Theory and Applications, International Academic Press, Palm Harber, USA, 2007.

[24] De Salvo, M., Hyperrings and hyperfields, Annales Scientifiques de l'Universite de Clermont-Ferrand II, Vol. 22, pp. 89-107, 1984.

[25] Marty, F. Role de la notion dhypergroupe dans letude des groupes non abeliens, Comptes Renclus Acad. Sci. Paris Math, Vol. 201, pp 636-638, 1935.

[26] Krasner, M., A class of hyperrings and hyperfields, International Journal of Mathematics and Mathematical Science, Vol 6(2), pp. 307-312, 1983.

[27] Mittas, J., Hypergroupes canoniques, Math. Balkanica, Vol. 2, pp. 165-179, 1972.

[28] Muthusamy Velrajan and Arjunan Asokkumar, Note on Isomophism Theorems of Hyperrings, Int. J. Math. Math. Sc., Hindawi Publishing Corporation, pp. 1-12, 2010.

[29] Nakassis, A., Recent results in hyperring and hyperfield theory, Internat. J. Math.Math. Sci., Vol. 11(2), pp. 209-220, 1988.

[30] Sanjay Roy and Samanta, T.K., A Note on Hypervector Spaces, Discussiones Mathematicae -General Algebra and Applications Vol. 31(1), pp. 75-99, 2011. < http : //eudml.org/doc/276586 >.

[31] Scafati Tallini, M., Characterization of Remarkable Hypervector spaces, Proc. of 8th int. Congress on Algebraic Hyperstructures and Applications, Samotraki, Greece, Sept. 1-9-2002, Spanidis Press, Xanthi, Greece, ISBN 960-87499-5-6, pp. 231-237, 2003.

[32] Serafimidis, K., Sur les hypergroups canoniques ordonne's et strictement ordonne's,(French) [Ordered and strictly ordered canonical hypergroups], Rend. Mat., Vol. 7(6), pp. 231-238. 1986.

[33] Serafimidis, K., Konstantinidou, M. and Mittas, J., Sur les hypergroups canoniques strictement re'ticule's, (French) [On strictly lattice-ordered canonical hypergroups], Riv. Mat. Pura Appl., Vol. 2, pp 21-35, 1987.

[34] Spartalis, S., A class of hyperrings, Riv. Mat. Pura Appl., Vol. 4, pp. 55- 64, 1989.

[35] Smarandache, F., A Unifying Field in Logics: Neutrosophic Logic, Neutrosophy, Neutrosophic Set, Neutrosophic Probability, American Research Press, Rehoboth, 2003.

[36] Smarandache, F., (T,I,F)- Neutrosophic Structures, Neutrosophic Sets and Systems, Vol. 8, pp. 3-10, 2015.

[37] Vasantha Kandasamy W.B and Smarandache, F., Basic Neutrosophic Algebraic Structures and Their Applications to Fuzzy and Neutrosophic Models, Hexis, Church Rock, (2004), http://fs.gallup.unm.edu/eBook-otherformats.htm

[38] Vasantha Kandasamy, W.B. and Florentin Smarandache, Some Neutrosophic Algebraic Structures and Neutrosophic N-Algebraic Structures, Hexis, Phoenix, Arizona, (2006), http://www.gallup.unm.edu/ smarandache/eBooks-otherformats.htm.

[39] Vasantha Kandasamy, W.B., Neutrosophic Rings, Hexis, Phoenix, Arizona, (2006) http://www.gallup.unm.edu/ smarandache/eBooks-otherformats.htm.

[40] Wadei Al-Omeri and Smarandache, F., New Neutrosophic Set via Neutrosophic Topological Spaces. Excerpt from Neutrosophic Operation Research Vol I, Pons Editions: Brussels, Belgium, pp. 189-209, 2017.

[41] Wadei Al-Omeri, Neutrosophic crisp Sets Via Neutrosophic crisp Topological Spaces, Neutrosophic Set and Systems Vol 13, pp 96- 104, 2016.

[42] Wadei Al-Omeri and Saeid Jafari, On Generalized Closed Sets and Generalized Pre-Closed Sets in Neutrosophic Topological Spaces, Mathematics, Vol 7, pp 1- 12, 2019. Doi: doi.org/10.3390/math/7010001.

[43] Zadeh, L.A., Fuzzy Sets, Information and Control, Vol. 8, pp. 338-353, 1965.

Received: March 28, 2020 Accepted: June 14, 2020

(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), sa akinleye@yahoo.com (4)
COPYRIGHT 2020 Neutrosophic Sets and Systems
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2021 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Title Annotation:ISSUE 1
Author:Ibrahim, M.A.; Agboola, A.A.A.; Badmus, B.S.; Akinleye, S.A.
Publication:International Journal of Neutrosophic Science
Article Type:Report
Date:Aug 1, 2020
Words:12129
Previous Article:Neutrosophic Ideal layers & Some Generalizations for GIS Topological Rules.
Next Article:Plithogenic n- Super Hypergraph in Novel Multi-Attribute Decision Making.
Topics:

Terms of use | Privacy policy | Copyright © 2022 Farlex, Inc. | Feedback | For webmasters |