# On Refined Neutrosophic Hypergroup.

1 Introduction and PreliminariesNeutrosophy 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 set [21] and respectively intuitionistic fuzzy logic [10]. 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+[, for more detailed information, the reader should see [15,17]. In 2013, Florentin Smarandache in [16] introduced 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] >. The birth of refinement of the neutrosophic components < T, I, F > has led 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]) where a, [b.sub.1], [b.sub.2], ..., [b.sub.n] are real or complex numbers. Using these refined neutrosophic numbers, the concept of refined neutrosophic set was introduced and this paved way for the development of refined neutrosophic algebraic structures. Agboola in [4] introduced the concept of refined neutrosophic structure and he studied refined neutrosophic groups in particular and presented their fundamental properties. Since then, several researchers in this field have studied this concept and a great deal of results have been published. In [1], Adeleke et al. presented results on refined neutrosophic rings, refined neutrosophic subrings and in [2], they presented results on refined neutrosophic ideals and refined neutrosophic homomorphisms. A comprehensive review of neutrosophy, neutrosophic triplet set and neutrosophic algebraic structures can be found in [12,18-20].

In [13], Marty, introduced the concept of hypergroups by considering the quotient of a group by its subgroup. And this was the birth of an interesting new branch of Mathematics known as "Algebraic hyperstructures" which is considered as a generalization of classical algebraic structures. In the classical algebraic structure, the composition of two elements is an element whereas in algebraic hyperstructure, the composition of two elements is a non-empty set. Since then, many different kinds of hyperstructures (hyperrings, hypermodules, hypervector spaces, ...) have been introduced and studied. Also, many theories of algebraic hyperstructures have been propounded as well as their applications to various areas of sciences and technology. For comprehensive details on hyperstructures, the reader should see [11,14]. The concept of neutrosophic hypergroup and their properties was introduced by Agboola and Davvaz in [7]. More connections between algebraic hyperstructures and neutrosophic set can be found in many recent publications, see [3,5,6,8,9]. The present paper is concerned with the development of connections between algebraic hyperstructures and neutrosphic algebraic structures and again concerned with studying the refinement of neutrosophic hypergroups in particular and present some of their basic properties.

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. [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.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.

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]), *) [right arrow] (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.k]) = [I.sub.k] 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 k = 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 [phi] : 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. [11] Let H be a non-empty set and o : H * 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 1.9. [11] 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 of H we have a o H = H o a = H. This condition is also called the reproduction axiom.

Definition 1.10. [11] A hypergroupoid (H, o) which is both a semihypergroup and a quasi- hypergroup is called a hypergroup.

Definition 1.11. [11] 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 1.12. [11] 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 exists 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 exists 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 1.13. [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, we define

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.

2 Formulation of Refined Neutrosophic Hypergroup

Definition 2.1. Let (H, *) be any hypergroup and let < H [union] ([I.sub.1], [I.sub.2]) > = {x = (a, b[I.sub.1], c[I.sub.2]) : a, b, c [member of] H}. The couple (H([I.sub.1], [I.sub.2]), *), is called a refined neutrosophic hypergroup generated by H, [I.sub.1] and [I.sub.2] under the hyperoperation *. The part a is called the non-neutrosophic part of x and the part b and c are called the neutrosophic parts of x.

If x = (a, b[I.sub.1], c[I.sub.2]) and y = (d, e[I.sub.1], f[I.sub.2]) are any two elements of H([I.sub.1], [I.sub.2]), where a, b, c, d [member of] H, we define

[mathematical expression not reproducible].

Note that a * d [subset or equal to] H, (a * e [union] b * d [union] b * e [union] b * f [union] c * e) [subset or equal to] H and (a * f [union] c * d [union] c * f) [subset or equal to] H.

Note 1. If the operation on H([I.sub.1], [I.sub.2]) is hyperaddition (+') then for all x = (a, b[I.sub.1], c[I.sub.2]) and y = (d, e[I.sub.1], f[I.sub.2]) elements of H([I.sub.1], [I.sub.2]), with a, b, c, d [member of] H, we define

x + 'y = (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]).

Here the addition on the right is the hyperaddition in H.

Proposition 2.2. Let (H, +) be a hypergroupoid, then, the refined neutrosophic hypergroup (H([I.sub.1], [I.sub.2]), +') is a hypergroup with identity element [theta] = (0, 0[I.sub.1], 0[I.sub.2]) iff (H, +) is a hypergroup with identity element 0.

Proof. Suppose (H, +)is a hypergroup and x = (a, b[I.sub.1], c[I.sub.2]), y = (d, e[I.sub.1], f[I.sub.2]), z = (g, h[I.sub.1], k[I.sub.2]) [member of] H([I.sub.1], [I.sub.2]). Then we show that (H([I.sub.1], [I.sub.2]), +') is a hypergroup. First, we shall show that (H([I.sub.1], [I.sub.2]), +') is a semihypergroup.

x +' (y +'z) = (a, b[I.sub.1], c[I.sub.2]) +' ((d, e[I.sub.1], f[I.sub.2]) +' (g, h[I.sub.1], k[I.sub.2]) = (a, b[I.sub.1], c[I.sub.2]) +' ((d + g,(e + h)[I.sub.1],(f + k)[I.sub.2])) = (a + (d + g),(b + (e + h))[I.sub.1],(c + (f + k))[I.sub.2]) = ((a + d) + g,((b + e) + h)[I.sub.1],((c + f) + k)[I.sub.2]) = (a + d,(b + e)[I.sub.1],(c + f)[I.sub.2]) +' (g, h[I.sub.1], k[I.sub.2]) = ((a, b[I.sub.1], c[I.sub.2]) +' (d.e[I.sub.1], f[I.sub.2])) +' (g, h[I.sub.1], k[I.sub.2]) = (x +0 y) +' z.

Secondly, we shall show that (H([I.sub.1], [I.sub.2]), +') is a quasihypergroup. That is, we want to show that x +' H([I.sub.1], [I.sub.2]) = H([I.sub.1], [I.sub.2]) +' x = H([I.sub.1], [I.sub.2]).

x +' H([I.sub.1], [I.sub.2]) = (a, b[I.sub.1], c[I.sub.2]) +' {(d, e[I.sub.1], f[I.sub.2]) : (d, e[I.sub.1], f[I.sub.2]) [member of] H([I.sub.1], [I.sub.2])} = (a + d,(b + e)[I.sub.1],(c + f)[I.sub.2]) [subset or equal to] H([I.sub.1], [I.sub.2]). [??] x +' H([I.sub.1], [I.sub.2]) [subset or equal to] H([I.sub.1], [I.sub.2]).

Now we show that H([I.sub.1], [I.sub.2]) [subset or equal to] x +' H([I.sub.1], [I.sub.2]), let z = (g, h[I.sub.1], k[I.sub.2]) [member of] H([I.sub.1], [I.sub.2]) with g, h, k [member of] H. There exist [a.sub.1], [a.sub.2], [a.sub.3] [member of] H such that g [member of] [a.sub.1] + H, h [member of] [a.sub.2] + H and k [member of] [a.sub.3] + H, since H is a hypergroup. Hence (g, h[I.sub.1], k[I.sub.2]) [member of] ([a.sub.1], [a.sub.2][I.sub.1], [a.sub.3][I.sub.2]) + H, which implies that H([I.sub.1], [I.sub.2]) [subset or equal to] x +' H([I.sub.1], [I.sub.2]). Accordingly, H([I.sub.1], [I.sub.2]) = x +' H([I.sub.1], [I.sub.2]). Similarly, we can show that H([I.sub.1], [I.sub.2]) = H([I.sub.1], [I.sub.2]) +' x. [??] We can conclude that (H([I.sub.1], [I.sub.2]), +') is a hypergroup. Conversely, suppose (H([I.sub.1], [I.sub.2]), +') is a hypergroup and x = (a, b[I.sub.1], c[I.sub.2]), y = (d, e[I.sub.1], f[I.sub.2]), z = (g, h[I.sub.1], k[I.sub.2]) [member of] H([I.sub.1], [I.sub.2]), with a, d, g, b = c = e = f = h = k = 0 [member of] H. Then we show that (H, +) is a hypergroup. Since H([I.sub.1], [I.sub.2]) is a hypergroup, x +' (y +' z) = (x +' y) +' z. But

[mathematical expression not reproducible].

Thus (H, +) is a semihypergroup.

Since H([I.sub.1], [I.sub.2]) is a quasihypergroup, for x = (a, b[I.sub.1], c[I.sub.2]) [member of] H([I.sub.1], [I.sub.2]) with a, b = c = 0 [member of] H we have that x +' H([I.sub.1], [I.sub.2]) = H([I.sub.1], [I.sub.2]) +' x = H([I.sub.1], [I.sub.2]). But

[mathematical expression not reproducible].

Since a [member of] H, a + H = H and H + a = H which implies that a + H = H + a = H. Hence, we can conclude that (H, +) is a hypergroup.

Proposition 2.3. Every refined neutrosophic hypergroup is a semihypergroup.

Proof. Let (H([I.sub.1], [I.sub.2]), *) be any refined neutrosophic hypergroup and let x = (a, b[I.sub.1], c[I.sub.2]), y = (d, e[I.sub.1], f[I.sub.2]), z = (g, h[I.sub.1], k[I.sub.2]) be arbitrary elements of H([I.sub.1], [I.sub.2]), where a, b, c, d, e, f, g, h, k [member of] H.

Then,

[mathematical expression not reproducible].

[subset or equal to] H([I.sub.1], [I.sub.2]).

Hence, (H([I.sub.1], [I.sub.2]), *) is a hypergroupoid. Next

[mathematical expression not reproducible].

Accordingly, (H([I.sub.1], [I.sub.2]), *) is a semihypergroup.

Proposition 2.4. A refined neutrosophic hypergroup is not always a quasihypergroup.

Proof. To see this, consider a refined neutrosophic hypergroup, say (H([I.sub.1], [I.sub.2]), *), where (0, 0[I.sub.1], 0[I.sub.2]) [not member of] H([I.sub.1], [I.sub.2]). Then for x = (a, b[I.sub.1], c[I.sub.2]) [member of] H([I.sub.1], [I.sub.2]) we have that

[mathematical expression not reproducible].

We can see that x * H([I.sub.1], [I.sub.2]) = H([I.sub.1], [I.sub.2]) * x [not equal to] H([I.sub.1], [I.sub.2]). This implies that reproduction axioms fails to hold in this case.

We note that the reproduction axioms fails to hold in some refined neutrosophic hypergroup, hence there exist some neutrosophic hypergroups that are not hypergroup. This observation is recorded in the next proposition.

Proposition 2.5. Let (H([I.sub.1], [I.sub.2]), *) be a refined neutrosophic hypergroup, then

1. (H([I.sub.1], [I.sub.2]), *) in general is not a hypergroup;

2. (H([I.sub.1], [I.sub.2]), *) always contain a hypergroup.

Proof. 1. From Proposition 2.4 above, we can see that the reproduction axiom is not always satisfied. Then the proof follows.

2. It follows from the definition of a neutrosophic hypergroup.

Example 2.6. Let H = {a, b} be a set with the hyperoperation defined as follows

a * a = a, a * b = b * a = b and b * b = {a, b}.

Let

H([I.sub.1], [I.sub.2]) = {a, b, [[alpha].sub.1] = (a, a[I.sub.1], a[I.sub.2]), [[alpha].sub.2] = (a, a[I.sub.1], b[I.sub.2]), [[alpha].sub.3] = (a, b[I.sub.1], a[I.sub.2]), [[alpha].sub.4] = (a, b[I.sub.1], b[I.sub.2]), [[beta].sub.1] = (b, b[I.sub.1], b[I.sub.2]), [[beta].sub.2] = (b, b[I.sub.1], a[I.sub.2]), [[beta].sub.3] = (b, a[I.sub.1], b[I.sub.2]), [[beta].sub.4] = (b, a[I.sub.1], a[I.sub.2])} be a refined neutrosophic set and let *' be a hyperoperation on H([I.sub.1], [I.sub.2]) defined in the table below.

Take [alpha] = {[[alpha].sub.1] = (a, a[I.sub.1], a[I.sub.2]), [[alpha].sub.2] = (a, a[I.sub.1], b[I.sub.2]), [[alpha].sub.3] = (a, b[I.sub.1], a[I.sub.2]), [[alpha].sub.4] = (a, b[I.sub.1], b[I.sub.2])} and [beta] = {[[beta].sub.1] = (b, b[I.sub.1], b[I.sub.2]), [[beta].sub.2] = (b, b[I.sub.1], a[I.sub.2]), [[beta].sub.3] = (b, a[I.sub.1], b[I.sub.2]), [[beta].sub.4] = (b, a[I.sub.1], a[I.sub.2])}.

It is clear from the table that (H([I.sub.1], [I.sub.2]), *) is a refined neutrosophic hypergroup since it contains a proper subset {a, b} which is a hypergroup under *.

Example 2.7. Let H = {a, b, c} and define "*" on H as follows

Let [mathematical expression not reproducible] be a refined neutrosophic set and let *' be a hyperoperation on H([I.sub.1], [I.sub.2]), then using the definition of * in Table 2, (H([I.sub.1], [I.sub.2]), *') is a refined neutrosophic hypergroup.

Example 2.8. Let H([I.sub.1], [I.sub.2]) = {e, a, b, c,([I.sub.1], [I.sub.2]), (a[I.sub.1], a[I.sub.2]), (b[I.sub.1], b[I.sub.2]), (c[I.sub.1], c[I.sub.2])} be a refined neutrosophic semi group where [a.sup.2] = [b.sup.2] = [c.sup.2] = e, ab = ba = c and ac = ca = b and let P([I.sub.1], [I.sub.2]) = {e, a,(a[I.sub.1], a[I.sub.2])} be a refined neutrosophic subset of H([I.sub.1], [I.sub.2]). Then for all x, y [member of] H([I.sub.1], [I.sub.2]) define

xoy = xP([I.sub.1], [I.sub.2])y.

Then (H([I.sub.1], [I.sub.2])), o) is a refined neutrosophic hypergroup.

Example 2.9. Let V ([I.sub.1], [I.sub.2]) be a weak refined neutrosophic vector space over a field K. Then for all 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]) define

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

Then (V ([I.sub.1], [I.sub.2]), o) is a hypergroup.

To see this we proceed as follows :

Firstly we show that (V ([I.sub.1], [I.sub.2]), o) is a hypergroupoid.

So, for 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]), o) and k [member of] K we have that

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

Next we show (V ([I.sub.1], [I.sub.2]), o) is a semi-hypergroup, i.e., o is associative.

Let x = (a, b[I.sub.1], c[I.sub.2]), y = (d, e[I.sub.1], f[I.sub.2]) and z = (g, h[I.sub.1], j[I.sub.2]) [member of] V ([I.sub.1], [I.sub.2]) then we want to show that x o (y o z) = (x o y) o z.

Consider [mathematical expression not reproducible].

Next, we show that o satisfies the reproduction axiom.

Let x = (a, b[I.sub.1], c[I.sub.2]) [member of] V ([I.sub.1], [I.sub.2]) with a, b, c [member of] V then [mathematical expression not reproducible]. Therefore V ([I.sub.1], [I.sub.2]) o (a, b[I.sub.1], c[I.sub.2]) = (a, b[I.sub.1], c[I.sub.2]) o V ([I.sub.1], [I.sub.2]) = V ([I.sub.1], [I.sub.2]).

Example 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]). For all 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]) and k = (p, q[I.sub.1], r[I.sub.2]) [member of] K([I.sub.1], [I.sub.2]) define

[mathematical expression not reproducible].

Then V ([I.sub.1], [I.sub.2], o) is refined neutrosophic hypergroup.

Proposition 2.11. Let (H([I.sub.1], [I.sub.2]), *1) and (K([I.sub.1], [I.sub.2]), *2) be any two refined neutrosophic hypergroups. Then, (H([I.sub.1], [I.sub.2]) * K([I.sub.1], [I.sub.2]), *) is a refined neutrosophic hypergroup, where

([x.sub.1], [x.sub.2]) * ([y.sub.1], [y.sub.2]) = {(x, y) : x [member of] [x.sub.1] *1 [y.sub.1], y [member of] [x.sub.2] *2 [y.sub.2], [for all]([x.sub.1], [x.sub.2]), ([y.sub.1], [y.sub.2]) [member of] H([I.sub.1], [I.sub.2]) * K([I.sub.1], [I.sub.2])}.

Proof. Let ([x.sub.1], [x.sub.2]), ([y.sub.1], [y.sub.2]) [member of] H([I.sub.1], [I.sub.2]) * K([I.sub.1], [I.sub.2]), where x = (a, b[I.sub.1], c[I.sub.2]) and y = (d, e[I.sub.1], f[I.sub.2]) then

[mathematical expression not reproducible].

Then (H([I.sub.1], [I.sub.2])) * K([I.sub.1], [I.sub.2]), *) is a refine neutrosophic hypergroupoid.

Let, ([x.sub.1], [x.sub.2]),([y.sub.1], [y.sub.2]),(z1, z2) [member of] H([I.sub.1], [I.sub.2]) * K([I.sub.1], [I.sub.2]), where x = (a, b[I.sub.1], c[I.sub.2]), y = (d, e[I.sub.1], f[I.sub.2]) and z = (g, h[I.sub.1]j[I.sub.2]) then [mathematical expression not reproducible].

Hence, (H([I.sub.1], [I.sub.2]) * K([I.sub.1], [I.sub.2]), *) is a refined neutrosophic semi-hypergroup. Lastly, let ([x.sub.1], [x.sub.2]) [member of] H([I.sub.1], [I.sub.2]) * K([I.sub.1], [I.sub.2]) then [mathematical expression not reproducible] since H([I.sub.1], [I.sub.2]) and K([I.sub.1], [I.sub.2]) are hypergroups. = H([I.sub.1], [I.sub.2]) * K([I.sub.1], [I.sub.2]). Hence (H([I.sub.1], [I.sub.2])) * K([I.sub.1], [I.sub.2]), *) is a refined neutrosophic quasi hypergroup. Then we can conclude that (H([I.sub.1], [I.sub.2])) * K([I.sub.1], [I.sub.2]), *) is a refined neutrososphic hypergroup.

Proposition 2.12. Let (H([I.sub.1], [I.sub.2]), *) be a refined neutrosophic hypergroup and let (K, o) be a hypergroup. Then, (H([I.sub.1], [I.sub.2]) * K, *') is a refined neutrosophic hypergroup, where

([h.sub.1], [k.sub.1]) *' ([h.sub.2], [k.sub.2]) = {(h, k) : h [member of] [h.sub.1] *' [h.sub.2], k [member of] [k.sub.1] o [k.sub.2], [for all]([h.sub.1], [k.sub.1]), ([h.sub.2], [k.sub.2]) [member of] N(H) * K}.

Proof : It follows from similar approach to the proof of Proposition 2.11 .

Proposition 2.13. Let (H([I.sub.1], [I.sub.2]), *) be a refined neutrosophic hypergroup, then for all elements of H([I.sub.1], [I.sub.2]) no two elements combine to give empty set.

Proof. Let (a, b[I.sub.1], c[I.sub.2]),(x, y[I.sub.1], z[I.sub.2]) [member of] H([I.sub.1], [I.sub.2]). Suppose (a, b[I.sub.1], [I.sub.2]) * (x, y[I.sub.1], z[I.sub.2]) = 0. Then since H([I.sub.1], [I.sub.2]) is a neutrosophic hypergroup, by reproduction axiom we have

H([I.sub.1], [I.sub.2]) = (a, b[I.sub.1], c[I.sub.2]) * H([I.sub.1], [I.sub.2]) = (a, b[I.sub.1], c[I.sub.2]) * ((x, y[I.sub.1], [I.sub.2]) * H([I.sub.1], [I.sub.2])) = ((a, b[I.sub.1], c[I.sub.2]) * (x, y[I.sub.1], [I.sub.2])) * H([I.sub.1], [I.sub.2]) = 0 * H([I.sub.1], [I.sub.2]) = 0.

This is absurd, hence there exist no two elements of H([I.sub.1], [I.sub.2]) that combine to give empty set.

Definition 2.14. Let H([I.sub.1], [I.sub.2]) be a refined neutrosophic hypergroup and let K[[I.sub.1], [I.sub.2]] be a proper subset of H([I.sub.1], [I.sub.2]). Then K[[I.sub.1], [I.sub.2]] is said to be a refined neutrosophic semi-subhypergroup of H([I.sub.1], [I.sub.2]) if x * y [subset or equal to] K[[I.sub.1], [I.sub.2]] for all x, y [member of] K[[I.sub.1], [I.sub.2]].

Definition 2.15. Let H([I.sub.1], [I.sub.2]) be a refined neutrosophic hypergroup and let K[[I.sub.1], [I.sub.2]] be a proper subset of H([I.sub.1], [I.sub.2]). Then

1. K[[I.sub.1], [I.sub.2]] is said to be a refined neutrosophic subhypergroup of H([I.sub.1], [I.sub.2]) if K[[I.sub.1], [I.sub.2]] is a refined neutrosophic hypergroup, that is, K[[I.sub.1], [I.sub.2]] must contain a proper subset which is a hypergroup.

2. K[[I.sub.1], [I.sub.2]] is said to be a refined pseudo neutrosophic subhypergroup of H([I.sub.1], [I.sub.2]) if K[[I.sub.1], [I.sub.2]] is a refined neutrosophic hypergroup which contains no proper subset which is a hypergroup.

Note 2. A refined neutrosophic hypergroup is a much more complicated structure than the structure of a refined neutrosophic group. In a refined neutrosophic group, the intersection of any two refined neutrosophic subgroups is a refined neutrosophic subgroup, this is not usually so in the case of a refined neutrosophic hypergroups, since the reproductive axioms fails to hold in this case. This has led to the consideration of different kinds of refined neutrosophic subhypergroups, which are; Closed, Ultraclosed and Conjugable.

Proposition 2.16. Let M([I.sub.1], [I.sub.2]) and N([I.sub.1], [I.sub.2]) be any refined neutrosophic subhypergroups of a refined neutrososphic hypergroup H([I.sub.1], [I.sub.2]), then M([I.sub.1], [I.sub.2]) [intersection] N([I.sub.1], [I.sub.2]) is a refined neutrosophic semi-subhypergroup.

Proof. M([I.sub.1], [I.sub.2])[intersection]N([I.sub.1], [I.sub.2]) [not equal to] 0, since M([I.sub.1], [I.sub.2]) and N([I.sub.1], [I.sub.2]) are non-empty subhypergroups of H([I.sub.1], [I.sub.2]). Now, let x = ([a.sub.1], [b.sub.1][I.sub.1], [c.sub.1][I.sub.2]), y = ([a.sub.2], [b.sub.2][I.sub.1], [c.sub.2][I.sub.2]) [member of] M([I.sub.1], [I.sub.2]) [intersection] N([I.sub.1], [I.sub.2]).

Then ([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]) [member of] M([I.sub.1], [I.sub.2]) and ([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]) [member of] N([I.sub.1], [I.sub.2]). Since, M([I.sub.1], [I.sub.2]) and N([I.sub.1], [I.sub.2]) are refined neutrosophic subhypergroup, we have that ([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]) [subset or equal to] M([I.sub.1], [I.sub.2]) and ([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]) [subset or equal to] N([I.sub.1], [I.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]) [subset or equal to] M([I.sub.1], [I.sub.2]) [intersection] N([I.sub.1], [I.sub.2]). Hence, M([I.sub.1], [I.sub.2]) [intersection] N([I.sub.1], [I.sub.2]) is a refined neutrosophic semi-subhypergroup.

Proposition 2.17. Let M([I.sub.1], [I.sub.2]) and N([I.sub.1], [I.sub.2]) be any refined neutrosophic semi-subhypergroups of a refined neutrososphic commutative hypergroup H([I.sub.1], [I.sub.2]), then the set

M([I.sub.1], [I.sub.2])N([I.sub.1], [I.sub.2]) = {xy : x [member of] M([I.sub.1], [I.sub.2]), y [member of] N([I.sub.1], [I.sub.2])}

is a refined neutrosophic semi-subhypergroup of H([I.sub.1], [I.sub.2]).

Definition 2.18. Let K([I.sub.1], [I.sub.2]) be a refined neutrosophic subhypergroup of a refined neutrosophic hypergroup (H([I.sub.1], [I.sub.2]), *). Then,

1. K([I.sub.1], [I.sub.2]) is said to be closed on the left (right) if for all [k.sub.1], [k.sub.2] [member of] K([I.sub.1], [I.sub.2]), x [member of] H([I.sub.1], [I.sub.2]) we have [k.sub.2] [member of] x * [k.sub.1]([k.sub.2] [member of] [k.sub.1] * x) implies that x [member of] K([I.sub.1], [I.sub.2]);

2. K([I.sub.1], [I.sub.2]) is said to be ultraclosed on the left (right) if for all x [member of] H([I.sub.1], [I.sub.2]) we have x * K([I.sub.1], [I.sub.2]) [intersection] x * (H([I.sub.1], [I.sub.2])\K([I.sub.1], [I.sub.2])) = 0 (K([I.sub.1], [I.sub.2]) * x [intersection] (H([I.sub.1], [I.sub.2])\K([I.sub.1], [I.sub.2])) * x = 0);

3. K([I.sub.1], [I.sub.2]) is said to be left (right) conjugable if K([I.sub.1], [I.sub.2]) is left (right) closed and if for all x [member of] H([I.sub.1], [I.sub.2]), there exists h [member of] H([I.sub.1], [I.sub.2]) such that x * h [subset or equal to] K([I.sub.1], [I.sub.2]) (h * x [subset or equal to] K([I.sub.1], [I.sub.2]));

4. K([I.sub.1], [I.sub.2]) is said to be (closed, ultraclosed, conjugable) if it is left and right (closed, ultraclosed, conjugable).

Proposition 2.19. Let K[[I.sub.1], [I.sub.2]] be a refined neutrosophic subhypergroup of H([I.sub.1], [I.sub.2]), A[[I.sub.1], [I.sub.2]] [subset or equal to] K[[I.sub.1], [I.sub.2]] and B[[I.sub.1], [I.sub.2]] [subset or equal to] H([I.sub.1], [I.sub.2]), then

1. A[[I.sub.1], [I.sub.2]](B[[I.sub.1], [I.sub.2]] [intersection] K[[I.sub.1], [I.sub.2]]) [subset or equal to] A[[I.sub.1], [I.sub.2]]B[[I.sub.1], [I.sub.2]] [intersection] K[[I.sub.1], [I.sub.2]] and

2. (B[[I.sub.1], [I.sub.2]] [intersection] K[[I.sub.1], [I.sub.2]])A[[I.sub.1], [I.sub.2]] [subset or equal to] B[[I.sub.1], [I.sub.2]]A[[I.sub.1], [I.sub.2]] [intersection] K[[I.sub.1], [I.sub.2]].

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

Proposition 2.20. 1. If K[[I.sub.1], [I.sub.2]] is a left closed refined neutrososphic subhypergroup in H[[I.sub.1], [I.sub.2]], A[[I.sub.1], [I.sub.2]] [subset or equal to] K[[I.sub.1], [I.sub.2]] and B[[I.sub.1], [I.sub.2]] [subset or equal to] H[[I.sub.1], [I.sub.2]], then (B[[I.sub.1], [I.sub.2]] [intersection] K[[I.sub.1], [I.sub.2]])/A[[I.sub.1], [I.sub.2]] = (B[[I.sub.1], [I.sub.2]]/A[[I.sub.1], [I.sub.2]]) [intersection] K[[I.sub.1], [I.sub.2]].

2. If K[[I.sub.1], [I.sub.2]] is a right closed subhypergroup in H[[I.sub.1], [I.sub.2]], A[[I.sub.1], [I.sub.2]] [subset or equal to] K[[I.sub.1], [I.sub.2]] and B[[I.sub.1], [I.sub.2]] [subset or equal to] H[[I.sub.1], [I.sub.2]], then (B[[I.sub.1], [I.sub.2]] [intersection] K[[I.sub.1], [I.sub.2]])\A[[I.sub.1], [I.sub.2]] = B[[I.sub.1], [I.sub.2]]\A[[I.sub.1], [I.sub.2]] [intersection] K[[I.sub.1], [I.sub.2]].

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

Proposition 2.21. Let K[[I.sub.1], [I.sub.2]], M[[I.sub.1], [I.sub.2]] be two refined neutrosophic subhypergroups of a refined neutrosophic hypergroup H[[I.sub.1], [I.sub.2]] and suppose that K[[I.sub.1], [I.sub.2]] is left (or right) closed in H[[I.sub.1], [I.sub.2]]. Then K[[I.sub.1], [I.sub.2]] [intersection] M[[I.sub.1], [I.sub.2]] is left (or right) closed in M[[I.sub.1], [I.sub.2]].

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

Proposition 2.22. Let (H([I.sub.1], [I.sub.2])), *) be a refined neutrosophic hypergroup and let [rho] be an equivalence relation on H([I.sub.1], [I.sub.2]).

1. If [rho] is regular, then H([I.sub.1], [I.sub.2])/[rho] is a refined neutrosophic hypergroup.

2. If [rho] is strongly regular, then H([I.sub.1], [I.sub.2])/[rho] is a refined neutrosophic group.

The proposition will be proved with the example provided below.

Example 2.23. If (G([I.sub.1], [I.sub.2]), +) is a refined neutrosophic abelian hypergroup, [rho] is an equivalence relation in G([I.sub.1], [I.sub.2]), which has classes [bar.x] = {x, -x}, then for all [bar.x], [bar.y] of G([I.sub.1], [I.sub.2])/[rho], we define

[bar.x]o[bar.y] = {[bar.x + y], [bar.x - y]}.

Then (G([I.sub.1], [I.sub.2])/[rho], o) is a refined neutrosophic hypergroup.

Proof. Let [bar.x], [bar.y] [member of] G([I.sub.1], [I.sub.2])/[rho], where [bar.x] = [bar.(a, b[I.sub.1], c[I.sub.2])], and [bar.y] = [bar.(d, e[I.sub.1], f[I.sub.2])] then [bar.x]o[bar.y] = [bar.(a, b[I.sub.1], c[I.sub.2])] o [bar.(d, e[I.sub.1], f[I.sub.2])] = [bar.{(a + d,(b + e)[I.sub.1],(c + f)[I.sub.2])], [bar.(a - d,(b - e)[I.sub.1],(c - f)[I.sub.2])}] = [bar.{x + y, x - y}] [member of] G([I.sub.1], [I.sub.2])/[rho].

Then (G([I.sub.1], [I.sub.2])/[rho], o) is a refined neutrosophic hypergroupoid.

Next we show that o satisfies the associative law. Let [bar.x], [bar.y], [bar.z] [member of] G([I.sub.1], [I.sub.2])/[rho], where [bar.x] = [bar.(a, b[I.sub.1], c[I.sub.2])], [bar.y] = [bar.(d, e[I.sub.1], f[I.sub.2])] and [bar.z] = [bar.(g, h[I.sub.1], j[I.sub.2])] then

[mathematical expression not reproducible].

Now we show that o satisfies the reproduction axiom. Let [bar.x] [member of] G([I.sub.1], [I.sub.2])/[rho] then

[mathematical expression not reproducible].

Hence we say that (G([I.sub.1], [I.sub.2])/[rho], o) is a refined neutrosophic hypergroup.

Definition 2.24. Let ([H.sub.1]([I.sub.1], [I.sub.2]), *1) and ([H.sub.2]([I.sub.1], [I.sub.2])), *2) be any two refined neutrosophic hypergroups and let f : [H.sub.1]([I.sub.1], [I.sub.2]) [right arrow] [H.sub.2])([I.sub.1], [I.sub.2]) be a map. Then

1. f is called a refined neutrosophic homomorphism if:

(a) for all x, y of [H.sub.1]([I.sub.1], [I.sub.2]), f(x *1 y) [subset or equal to] f(x) *2 f(y),

(b) f([I.sub.k]) = [I.sub.k] for k = 1, 2.

2. f is called a good refined neutrosophic homomorphism if:

(a) for all x, y of [H.sub.1]([I.sub.1], [I.sub.2]), f(x *1 y) = f(x) *2 f(y),

(b) f([I.sub.k]) = [I.sub.k] for k = 1, 2.

3. f is called a refined neutrosophic isomorphism if f is a refined neutrosophic homomorphism and [f.sup.-1] is also a refined neutrosophic homomorphism.

4. f is called a 2-refined neutrosophic homomorphism if for all x, y of [H.sub.1]([I.sub.1], [I.sub.2]), [f.sup.-1](f(x) *2 f(y)) = [f.sup.-1](f(x *1 y)).

5. f is called an almost strong refined neutrosophic homomorphism if for all x, y of [H.sub.1]([I.sub.1], [I.sub.2]), [f.sup.-1](f(x) *2 f(y)) = [f.sup.-1](f(x)) *1 [f.sup.-1](f(y)).

Proposition 2.25. Let (H([I.sub.1], [I.sub.2])), *) be a refined neutrosophic hypergroup and let [rho] be a regular equivalence relation on H([I.sub.1], [I.sub.2]). Then, the map [phi] : H([I.sub.1], [I.sub.2]) [right arrow] H([I.sub.1], [I.sub.2])/[rho] defined by [phi](x) = [bar.x] is not a refined neutrosophic homomorphism (good refined neutrosophic homomorphism).

Proof. It is clear since I [member of] H([I.sub.1], [I.sub.2]) but [phi]([I.sub.k]) [not equal to] [I.sub.k].

Note 3. Suppose we wish to establish any relationship between the refined neutrosophic hypergroups and the parent neutrosophic hypergroups, or any other neutrosophic hypergroup. Then, our task will be to find a mapping [phi] say, such that

[phi] : H([I.sub.1], [I.sub.2]) [right arrow] H(I).

For all (x, y[I.sub.1], z[I.sub.2]) [member of] H([I.sub.1], [I.sub.2]) define [phi] by

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

In what follows we present some of the basic properties of such mapping.

Proposition 2.26. Let (H([I.sub.1], [I.sub.2]), +') be a refined neutrosophic hypergroup and let (H(I), +) be a neutrosophic hypergroup. The mapping [phi] defined in 1 above is a good homomorphism.

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] H([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]).

Hence [phi] is a good homomorphism.

Definition 2.27. Let (H([I.sub.1], [I.sub.2]), +') be a refined neutrosophic hypergroup with identity element (0, 0[I.sub.1], 0[I.sub.2]) and (H([I.sub.1], [I.sub.2]), +) be a neutrosophic hypergroup with identity element (0, 0I). Let [phi] : H([I.sub.1], [I.sub.2]) [right arrow] H(I) be a good homomorphism, 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 2.28. Let [phi] : H([I.sub.1], [I.sub.2]) [right arrow] H(I) be a good homomorphism.

1. ker [phi] is a semi-subhypergroup of H([I.sub.1], [I.sub.2]).

2. Im[phi] is a subhypergroup of H(I).

Proof. 1. Let (a, b[I.sub.1], c[I.sub.2]),(x, y[I.sub.1], z[I.sub.2]) [member of] ker [phi], then

[phi]((a, b[I.sub.1], c[I.sub.2]) +' (x, y[I.sub.1], z[I.sub.2])) = [phi]((a, b[I.sub.1], c[I.sub.2])) + [phi]((x, y[I.sub.1], z[I.sub.2])) = (0, 0I) + (0, 0I) = (0, 0I) [??] (a, b[I.sub.1], c[I.sub.1]) +' (x, y[I.sub.1], z[I.sub.2]) [subset or equal to] ker[phi].

Hence, ker[phi] is a semi-subhypergroup.

2. Let (a, b[I.sub.1], c[I.sub.2]) [member of] H([I.sub.1], [I.sub.2]), then

[mathematical expression not reproducible].

Following similar approach we can show that [phi](H([I.sub.1], [I.sub.2])) + [phi]((a, b[I.sub.1], c[I.sub.2])) = [phi](H([I.sub.1], [I.sub.2])). Thus, Im[phi] is a subhypergroup of H(I).

3 Conclusion

In this paper, we have studied the refinement of neutrosophic hyperstructures. In particular, we have studied refined neutrosophic hypergroups and presented several results and examples. Also, we have established the existence of a good homomorphism between a refined neutrosophic hypergroup H([I.sub.1], [I.sub.2]) and a neutrosophic hypergroup H(I). We hope to present and study more advance properties of refined neutrosophic Hypergroups in our future papers.

Doi : 10.5281/zenodo.3958093

4 Acknowledgment

The Authors wish to thank the anonymous reviewers for their valuable and useful comments which have led to the improvement of the paper.

References

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Received: April 04, 2020 Accepted: July 08, 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)

Caption: Table 1: Cayley table for the binary operation "*'"

Table 2: Cayley table for the binary operation "*" * a b c a a b c b b {a,b} {b,c} c c {b,c} {a, b, c}

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Title Annotation: | ISSUE 2 |
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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: | Sep 1, 2020 |

Words: | 5671 |

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