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Introduction to Neutrosophic Hypernear-rings.

1 Introduction

Algebraic Hyperstructures are a natural extension/generalization of classical algebraic structures. This theory was introduced in 1934 by Marty. Since then, the theory and its applications to various aspect of sciences have been extensively studied by Corsini [8,9,10], Mittas [19,20], Stratigopoulos [23] and many other authors. For instance, Dasic in [11] studied the notion of hypernear-ring. He defined hypernear-rings, as the natural generalization of near-rings, endowed with quasicanonical hypergroups (R, +) with multiplication being distributive with respect to the hyperaddition on the left side, and such that (R, *) is a semigroup with bilaterally absorbing element. In [13], Gontineac called the hypernear-ring presented by Dasic as a zero symmetric hypernear-ring and he studied the concept of hypernear-ring in a more general case. Kyung et al. presented in [18] the notion of hyper R-sugroups of a hypernear-ring and they investigated some properties of hypernear-rings with respect to the hyper R-subgroups. For more comprehensive details on hyperstructures, the reader should see [12,18,21].

A well established branch of neutrosophic theory is the theory of neutrosophic algebraic structures. This aspect of neutrosophic theory was introduced in [24] by Kandasamy and Smarandache. They combined the elements of a given algebraic structure (X, *) with the indeterminate element I, and, the new structure (X(I), *) generated by X and I is called a neutrosophic algebraic structure. For more details about neutrosophic algebraic structures (see [6,14,22,25,26]). Recently, Agboola and Davvaz in [1,2,3,4] introduced the connections between neutrosophic set and the theory of algebraic hyperstructure. They studied neutrosophic BCI/BCK-Algebras, neutrosophic hypergroups, neutrosophic canonical hypergroups, and neutrosophic hyperrings. In this paper, we will investigate and present some interesting results arising from the study of hypernear-rings in a neutrosophic environment. This paper will add to the growing list of papers connecting algebraic hyperstructures and neutrosophic sets. More of such connections can be found in many recent publications some of which are [5,7,15,16,17].

2 Preliminaries

In this section, we will present some definitions and results that will be used later in the paper.

Definition 2.1. [18] A hypernear-ring is an algebraic structure (R, +, *) which satisfies the following axioms:

1. (R, +) is a quasi canonical hypergroup (not necessarily commutative), i.e., in (R, +) the following hold:

(a) x + (y + z) = (x + y) + z for all x, y, z [member of] R;

(b) There is 0 [member of] R such that x + 0 = 0 + x = x for all x [member of] R;

(c) For every x [member of] R there exists one and only one x' [member of] R such that 0 [member of] x + x', (we shall write -x for x' and we call it the opposite of x);

(d) z [member of] x + y implies y [member of] -x + z and x [member of] z - y.

If x [member of] R and A, B are subsets of R, then by A + B, A + x and x + B we mean

[mathematical expression not reproducible]

2. With respect to the multiplication, (R, *) is a semigroup having absorbing element 0, i.e., x * 0 = 0 for all x [member of] R. But, in general, 0x [not equal to] 0 for some x [member of] R.

3. The multiplication is distributive with respect to the hyperoperation "+" on the left side, i.e., x * (y + z)= x * y + x * z for all x, y, z [member of] R.

A hypernear-ring R is called zero symmetric if 0x = x0 = 0 for all x [member of] R.

Note that for all x, y [member of] R, we have -(-x) = x, 0 = -0, -(x + y) = -y - x and x(-y) = -xy.

Definition 2.2. [18] A two sided hyper R-subgroup of a hypernear-ring R is a subset H of R such that

1. (H, +) is a subhypergroup of (R, +),

(i) a, b [member of] H implies a + b [subset or equal to] H,

(ii) a [member of] H implies -a [member of] H,

2. RH [subset or equal to] H,

3. HR [subset or equal to] H.

If H satisfies (1) and (2), then it is called a left hyper R-subgroup of R. If H satisfies (1) and (3), then it is called a right hyper R-subgroup of R.

Definition 2.3. [6] Let (N, +, *) be any right nearring. The triple (N(I), +, *) is called a right neutrosophic nearring. For all x = (a, bI), y = (c, dI) [member of] N(I) with a, b, c, d [member of] N, we define:

1. x + y = (a, bI) + (c, dI) = (a + c, (b + d)I).

2. -x = -(a, bI) = (-a, -bI).

3. x * y = (a, bI).(c, dI) = (ac, (ad + bc + bd)I).

The zero element in (N, +) is represented by (0,0) in (N(I), +). Any element x [member of] N is represented by (x, 0) in N(I). I in N(I) is sometimes represented by (0,1) in N(I).

Definition 2.4. [6] Let (N(I), +, *) be a right neutrosophic nearring.

1. N(I) is called abelian, if (a, bI) + (c, dI) = (c, dI) + (a, bI) [for all] (a, bI), (c, dI) [member of] N(I).

2. N(I) is called commutative, if (a, bI) * (c, dI) = (c, dI) * (a, bI) [for all] (a, bI), (c, dI) [member of] N(I).

3. N(I) is said to be distributive, if N(I) = [N.sub.d](I), where

[N.sub.d](I) = {d [member of] N(I) : d(m + n) = dm + dn, [for all] m, n [member of] N(I)}.

4. N(I) is said to be zero-symmetric, if N(I) = [N.sub.0](I), where

[N.sub.0](I) = {n [member of] N(I) : n0 = 0}.

The following should be noted:

(i) N(I) is abelian only if (N, +) is abelian.

(ii) N(I) is commutative only if (N, *) is commutative.

(iii) N(I) is distributive only if N is distributive.

(iv) N(I) is zero-symmetric only if N is zero-symmetric

3 Development of neutrosophic hypernear-rings

In this section, we develop the concept of neutrosophic hypemear-rings and present some of their basic properties.

Definition 3.1. Let (R, +, *) be any hypernear-ring. The triple (R(I), +', [dot encircle]) is a neutrosophic hypernear-ring generated by R and I, where +' and [dot encircle] are hyperoperations.

For all [r.sub.1] = (u, vI), [r.sub.2] = (s, tI) [member of] R(I) with u, v, s, t [member of] R, we define :

1. [r.sub.1] +' r2 = (u, vI) +' (s, tI) = {(p, qI) : p [member of] u + s, q [member of] v + t} for all [r.sub.1], [r.sub.2] [member of] R,

2. [r.sub.1] [dot encircle] [r.sub.2] = (u, vI) [dot encircle] (s, tI) = (u [dot encircle] s, (u [dot encircle] t + v [dot encircle] s + v [dot encircle] t)I), the "[dot encircle]" and "+" on the right are respectively the ordinary multiplication and hyperaddition in R,

3. -[r.sub.1] = -(u, vI) = (-u, -vI).

We represent element x [member of] R by (x, 0I) [member of] R(I), and 0 [member of] (R, +) by (0, 0I) [member of] (R(I), +'). I [member of] R(I) may also be written as (0, I).

Lemma 3.2. Let (R(I), +', [dot encircle]) be any neutrosophic hypernear-ring. Let [r.sub.1] = (u, vI), [r.sub.2] = (s, tI) [member of] R(I) with u, v, s, t [member of] R. For all [r.sub.1], [r.sub.2] [member of] R(I) we have

1. -(-[r.sub.1]) = - (-a, -bI) = (-(-a) - (-b)I) = (a, bI),

2. -([r.sub.1] +' [r.sub.2]) = -[r.sub.1] - [r.sub.2],

3. -(0, 0I) = (0, 0I),

4. [r.sub.1] [dot encircle] (-[r.sub.2]) = -([r.sub.1] [dot encircle] [r.sub.2]).

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

Definition 3.3. Let (R(I), +', [dot encircle]) be a neutrosophic hypernear-ring. An element (x, yI) [member of] R(I) is said to be idempotentif [(x, yI).sup.2] = (x, yI).

Definition 3.4. Let (R(I), +', [dot encircle]) be a neutrosophic hypernear-ring.

1. R(I) is called zero-symmetric neutrosophic hypernear-ring, if R(I) = [R.sub.(0, 0I)](I), where

[R.sub.(0, 0I)](I) = {(x, yI) [member of] R(I) | (0, 0I) [dot encircle] (x, yI) = (0, 0I)}.

2. R(I) is called a constant neutrosophic hypernear-ring, if R(I) = [R.sub.c](I) where

[R.sub.c](I) = {(x, yI) [member of] R(I) | (x, yI) [dot encircle] (p, qI) = (p, qI), [for all] (p, qI) [member of] R(I)}.

Proposition 3.5. Every neutrosophic hypernear-ring is a hypernear-ring.

Proof. Let (R(I), +', [dot encircle]) be a neutrosophic hypernear-ring.

1. We shall show that (R(I), +') is a quasi canonical hypergroup.

(a) Let (a, bI), (c, dI), (e, fI) [member of] R(I). Then

((a, bI)+' (c, dI))+' (e, fI) = {(x, yI): x [member of] a + c, y [member of] b + d} +' (e, fI)

= {(p, qI): p [member of] x + e, q [member of] y + f}

= {(p, qI) : p [member of] (a + c) + e, q [member of] (b + d) + f}

= {(p, qI): p [member of] a + (c + e), q [member of] b + (d + f)}

= {(p, qI): p [member of] a + u, q [member of] b + v}

= (a, bI) +' {(u, vI) : u [member of] c + e, v [member of] d + f}

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

(b) Let (0, 0I) [member of] R(I), then for all (a, bI) [member of] R(I) we have

(a, bI) +' (0, 0I) = {(x, yI) : x [member of] a + 0, y [member of] b + 0}

= {(x, y) : x [member of] a, y [member of] b}

= {(a, bI)}.

Following similar approach we can show that (0, 0I) + (a, bI) = {(a, bI)}. Hence there exists a neutral element in R(I).

(c) Let (a, bI), -(a, bI) [member of] R(I), then

(a, bI) +' (-(a, bI)) = (a, bI) +' (-a, -bI)

= {(x, yI) : x [member of] a + (-a), y [member of] b + (-b)}

= {(x, yI) : x [member of] (-a) + a, y [member of] (-b) + b}

= {(x, yI): x [member of] {0}, y [member of] {0}}

[??] (0, 0I) [member of] (a, bI)+' (-(a, bI)).

Hence -(a, bI) is the unique inverse of any (a, bI) [member of] R(I).

(d) Suppose that (x, yI) [member of] (a, bI) +' (c, dI) then

(x, yI) [member of] {(p, qI): P [member of] a + c, q [member of] b + d}

= {(p, qI): c [member of] -a + p, d [member of] -b + q}

= {(c, dI): c [member of] -a + p, d [member of] -b + q}

[??] (c, dI) [member of] -(a, bI) +' (x, yI).

Following the approach above we can also establish that (a, bI) [member of] (x, yI) - (c, dI).

2. Let (a, bI), (c, dI), (e, fI) [member of] R(I) then

(a) (a, bI) [dot encircle] (c, dI) = (p, qI) [member of] R(I), p = ac and q [member of] (ad + bc + bd).

(b) Let (a, bI), (c, dI), (e, fI) [member of] R(I) then

((a,bI) [dot encircle] (c, dI)) [dot encircle] (e, fI) = (ac, (ad + bc + bd)I) [dot encircle] (e, fI)

= ((ac)e, ((ac)f + (ad)e + (bc)e + (bd)e + (ad)f + (bc)f + (bd)f)I)

= (a(ce), (a(cf) + a(de) + b(ce) + b(de) + a(df) + b(cf) + b(df))I)

= (a(ce), (a(cf) + a(de) + a(df) + b(ce) + b(cf) + b(de) + b(df))I)

= (a, bI) [dot encircle] ((c, dI) [dot encircle] (e, fI)).

And (a, bI) [dot encircle] (0, 0I) = (a0, (a0 + b0 + b0)I) = (0, 0I).

3. Now, it remains to show the distributive of ([dot encircle]) with respect to (+') on the left side.

Let (a, bI), (c, dI), (e, dI) [member of] R(I) then

(a, bI) [dot encircle] ((c, dI) +' (e, fI)) = (a, bI) [dot encircle] {(x, yI): x [member of] c + e, y [member of] d + f}

= {(a, bI) [dot encircle] (x, yI) : x [member of] c + e, y [member of] d + f}

= (ax, (ax + ay + bx + by)I)

= (a(c + e), (a(c + e) + a(d + f) + b(c + e) + b(d + f))I)

= (ac + ae, q [member of] (ac + ae + ad + af + bc + be + bd + bf)I)

= (ac, (ac + ad + +bc + bd)I) +' (ae, [q.sub.2] [member of] (ae + af + be + bf)I)

= ((a, bI) [dot encircle] (c, dI)) +' ((a, bI) [dot encircle] (e, fI)).

Proposition 3.6. Let [{[R.sub.i](I)}.sup.n.sub.i] be a family of neutrosophic hypernear-rings. Then [[PI].sup.n.sub.i=1] [R.sub.i](I), +', [dot encircle]) is a neutrosophic hypernear-ring.

Proof. 1. Let ([a.sub.i], [b.sub.i]I), ([c.sub.i], [d.sub.i]I) [member of] [[PI].sup.n.sub.i=1] [R.sub.i](I), with [a.sub.i], [b.sub.i], [c.sub.i], [d.sub.i] [member of] [R.sub.i] for i = 1 ... n.

Following the approach in 1 of Proposition 3.5 we have that ([[PI].sup.n.sub.i=1] [R.sub.i](I), +') is a neutrosophic quasi canonical hypernear-ring.

2. Let ([a.sub.i], [b.sub.i]I), ([c.sub.i], [d.sub.i]I) [member of] [[PI].sup.n.sub.i=1] [R.sub.i](I), with [a.sub.i], [b.sub.i], [c.sub.i], [d.sub.i] [member of] [R.sub.i] for i = 1 ... n.

Following the approach in 2 of Proposition 3.5 we have that ([[PI].sup.n.sub.i=1] [R.sub.i](I), [dot encircle]) is a neutrosophic semihypergroup.

3. To show that ([dot encircle]) is distributive with respect to (+') on the left side, we follow the approach in 3 of Proposition 3.5

Hence we have that ([[PI].sup.n.sub.i=1] [R.sub.i](I), +', [dot encircle]) is a neutrosophic hypernear-ring.

Proposition 3.7. Let M(I) be neutrosophic hypernear-rings and N be a hypernear-ring. Then M(I) x N is a neutrosophic hypernear-ring.

Proof. The proof follows from Proposition 3.6 .

Example 3.8. Let (R(I), +) be a neutrosophic hypergroup and let [M.sup.R(I).sub.(0, 0I)] be defined by

[M.sup.R(I).sub.(0, 0I)] = {f : R(I) [right arrow] R(I)},

such that f ((0, 0I)) = (0, 0I). For all f, g [member of] [M.sup.R(I).sub.(0, 0I)] we define the hyperoperation f +' g of mappings as follows:

(f +' g)((x, yI)) = {h [member of] [M.sup.R(I).sub.(0, 0I)] | [for all] (x, yI) [member of] R(I), h((x, yI)) [member of] f((x, yI)) + g((x, yI))}.

f o g((x, yI)) = f (g((x, yI))).

Here the "+" on the right is the hyperoperation in (R, +) and o is a composition of functions. Then ([M.sup.R(I).sub.(0, 0I)], +', o) is a zero-symmetric neutrosophic hypernear-ring.

Let f, g [member of] [M.sup.R(I).sub.(0, 0I)] and (x, yI) [member of] R(I). Since f (x, yI) +' g(x, yI) [not equal to] 0, then there exists h [member of] [M.sup.R(I).sub.(0, 0I)] such that h((x, yI)) [member of] f ((x, yI)) +' g((x, yI)). Obviously, h((0, 0I)) [member of] f ((0, 0I)) +' g((0, 0I)) = {(0, 0I)}, i.e., h((0, 0I)) = (0, 0I).

Now we shall show that ([M.sup.R(I).sub.(0, 0I)], +') is a neutrosophic quasi canonical hypergroup .

1. Let f, g, h [member of] [M.sup.R(I).sub.(0, 0I)] and (x, yI) [member of] R(I) then

(f +' g) +' h = {p | [for all] (x, yI) [member of] R(I), p((x, yI)) [member of] f ((x, yI)) + g((x, yI))} +' h

= {q | [for all] (x, yI) [member of] R(I), q((x, yI)) [member of] p((x, yI)) + h((x, yI))}

= {u | [for all] (x, yI) [member of] R(I), u((x, yI)) [member of] (f((x, yI)) + g((x, yI))) + h((x, yI))}

= {u | [for all] (x, yI) [member of] R(I) u((x, yI)) [member of] f((x, yI)) + g((x, yI)) + h((x, yI))}

= {u | [for all] (x, yI) [member of] R(I) u((x, yI)) [member of] f((x, yI)) + (g((x, yI)) + h((x, yI)))}

= {s | [for all] (x, yI) [member of] R(I), s((x, yI)) [member of] f ((x, yI)) + t((x, yI))}

= f +' {t | [for all] (x, yI) [member of] R(I), t((x, yI)) [member of] g((x, yI)) + h((x, yI))}

= f +' (g +' h).

2. Let [tau] [member of] [M.sup.R(I).sub.(0, 0I)] be defined by [tau]((x, yI)) = (0, 0I), then for all f [member of] [M.sup.R(I).sub.(0, 0I)] we have

(f +' [tau])((x, yI)) = {g | [for all] (x, yI) [member of] R(I), g [member of] f ((x, yI)) + [tau]((x, yI))}

= {g | [for all] (x, yI) [member of] R(I), g [member of] f ((x, yI)) + (0, 0I)}

= {f ((x, yI))}.

Similarly, it can be shown that ([tau] +' f)((x, yI)) = {f ((x, yI))}. Hence, there exists a neutral function [tau] [member of] [M.sup.R(I).sub.(0, 0I)].

3. Let f, -f [member of] [M.sup.R(I).sub.(0, 0I)] with (-f ((x, yI))) = -f ((x, yI)) then

(f +' (-f))((x, yI)) = {g | [for all] (x, yI) [member of] R(I), g [member of] f ((x, yI)) + (-f ((x, yI)))}

= {g | [for all] (x, yI) [member of] R(I), g [member of] f((x, yI)) - f ((x, yI))}

= {g | [for all] (x, yI) [member of] R(I), g [member of] {[tau]((x, yI))}

[therefore] [tau]((x, yI) [member of] f ((x, yI)) - f ((x, yI)) [??] (0, 0I) [member of] f ((x, yI)) - f ((x, yI)).

Hence -f is the unique inverse of any f [member of] [M.sup.R(I).sub.(0, 0I)].

4. Suppose that h [member of] f +' g. Then

h [member of] {p | [for all] (x, yI) [member of] R(I) p((x, yI)) [member of] f ((x, yI)) + g((x, yI))}

= {p | [for all] (x, yI) [member of] R(I) g((x, yI)) [member of] -f ((x, yI)) + p((x, yI))}

= {g | [for all] (x, yI) [member of] R(I) g((x, yI)) [member of] -f ((x, yI)) + p((x, yI))}.

Then we have that g [member of] (-f) +' h. Similarly, it can be shown that f [member of] h +' (-g). Hence h [member of] f +' g implies that f [member of] h +' (-g) and g [member of] (-f) +' h.

Accordingly, ([M.sup.R(I).sub.(0, 0I)], +') is a neutrosophic quasicanonical hypergroup.

It can easily be established that ([M.sup.R(I)], o) is a semihypergroup having t as a bilaterally absorbing element such that (f o [tau])(x, yI) = f ([tau](x, yI)) = f ((0, 01)) = (0, 0I) = [tau]((x, yI)) i.e., f o [tau] = f[tau] = [tau]. So, it remains to prove that the operation o is distributive with respect to the hyperoperation on the left side. Let f, g, h [member of] [M.sup.R(I).sub.(0, 0I)] then

f o (g +' h) = f o{t | [for all] (x, yI) [member of] R(I), t((x, yI)) [member of] g((x, yI)) + h((x, yI))}

= {p | [for all] (x, yI) [member of] R(I), p((x, yI)) [member of] ft((x, yI))}

= {p | [for all] (x, yI) [member of] R(I), p((x, yI)) [member of] fg((x, yI)) + fh((x, yI))}

= f o g((x, yI)) +' f o h((x, yI)).

Then it follows that ([M.sup.R(I).sub.(0, 0I)], o, +') is a zero-symmetric neutrosophic hypernear-ring.

Example 3.9. Let R(I) = {[x.sub.0] = (0, 0I), [x.sub.1] = (a, 0I), [x.sub.2] = (b, 0I), [x.sub.3] = (0, aI), [x.sub.4] = (0, bI), [y.sub.1] = (a, aI), [y.sub.2] = (a, bI), [y.sub.3] = (b, aI), [y.sub.4] = (b, bI)} be a neutrosophic set. Let [N.sub.x] = {[x.sub.0], [x.sub.1], [x.sub.2], [x.sub.3], [x.sub.4]} and [N.sub.y] = {[y.sub.1], [y.sub.2], [y.sub.3], [y.sub.4]}. Define hyperoperations +' and "[dot encircle]" on R(I) as in the table below.

Then (R(I), +', [dot encircle]) is neutrosophic hypemear-ring .

Proposition 3.10. Let R(I) be a neutrosophic hypernear-ring. R(I) is zero-symmetric only if R is zero-symmetric hypernear-ring.

Proof. Let R(I) be a neutrosophic hypernear-ring and let R be a zero symmetric hypernear-ring.

Then for all (x, yI) [member of] R(I) and for (0, 0I) [member of] R(I) we have

(0, 01) [dot encircle] (x, yI) = (0x, (0y + 0x + 0y)I)

= (0, 0I).

Hence, R(I) is zero-symmetric neutrosophic hypernear-ring.

Proposition 3.11. Let R(I) be a neutrosophic hypernear-ring and let R be a constant hypernear-ring. Then generally, R(I) is not a constant hypernear-ring.

Proof. Let R(I) be a neutrosophic hypernear-ring and let R be a constant hypernear-ring. Then, for all

(a, bI), (x, yI) [member of] R(I) we have,

(a, bI) [dot encircle] (x, yI) = (ax, (ay + bx + by)I)

= (x, (y + x + y)I) [therefore] R is constant.

[not equal to] (x, yI).

Remark 3.12. R(I) = {(x, yI) : x, y [member of] R} will be a constant hypernear-ring if x is the zero element in the constant hypernear-ring R. And, for each z [member of] R, z + z = z, and 0z = z.

To see this, pick any (0, bI), (0, aI) [member of] R(I). Since z + z = z and 0z = z for all z [member of] R we have

(0, bI) [dot encircle] (0, aI) = (0, (0a + b0 + ba)I)

= (0, (a + 0 + a)I)

= (0, (a + a)I)

= (0, aI).

Proposition 3.13. Every element in a constant neutrosophic hypernear-ring is idempotent.

Proof. The proof follows easily from definition of a constant neutrosophic hypernear-ring.

Definition 3.14. Let R(I) be a neutrososphic hypemear-ring and let N(I) be a nonempty subset of R(I). N(I) is called a neutrosophic subhypernear-ring if N(I) is a neutrosophic hypernear-ring and N(I) contains a proper subset which is a subhypernear-ring of R.

Proposition 3.15. Let R(I) be a neutrosophic hypernear-ring. The neutrosophic subset

[M(I).sub.(0,0I)] = {(x, yI) [member of] R(I) : (0, 0I) [dot encircle] (x, yI) = (0, 01)}

of R(I) is a zero-symmetric neutrosophic subhypernear-ring of R(I).

Proof. Let (a, bI), (c, dI) [member of] [M(I).sub.(0,0I)], then (0, 0I) [dot encircle] (a, bI) = (0, 0I) and (0, 0I) [dot encircle] (c, dI) = (0, 0I).

1. Since every element in [M(I).sub.(0,0I)] is of the form (a, bI), with a, b [member of] R. [M(I).sub.(0,0I)] can be written as ([M.sub.0], [M.sub.0](I)). Here [M.sub.0] is a zero-symmetry subhypernear-ring of R. Therefore, we can conclude that [M(I).sub.(0,0I)] contains a proper subset which is a zero-symmetric subhypernear-ring of R.

2. We shall show that ([M(I).sub.(0,0I)], +') is a zero-symmetric neutrosophic subhypergroup.

(0, 0I) [dot encircle] [(a, bI) +' (c, dI)] = (0, 0I) [dot encircle] {(p, qI) : p [member of] a + c, q [member of] b + d}

= (0, 0I) [dot encircle] (p, qI)

= (0p, (0q + 0p + 0q)I)

= (0, 0I).

This shows that (a, bI) + (c, dI) [subset or equal to] [M(I).sub.(0,0I)].

And, for all (a, bI) [member of] [M(I).sub.(0,0I)],

(0, 0I) [dot encircle] (-(a, bI)) = -((0, 0I) [dot encircle] (a, bI)) by Lemma 3.2

= -(0a, (0b + 0a + 0b)I)

= -(0, 0I) = (0, 0I) by Lemma 3. 2 .

This shows that -(a, bI) [member of] [M(I).sub.(0,0I)].

Thus, ([M(I).sub.(0,0I)], +') is a zero-symmetric neutrosophic subhypergroup.

3. We shall show that ([M(I).sub.(0,0I)], [dot encircle]) is a zero-symmetric neutrosophic subsemihypergroup.

(0, 0I) [dot encircle] [(a, bI) [dot encircle] (c, dI)] = (0, 0I) * [(ac, (ad + bc + bd)I)]

= (0(ac), (0(ad) + 0(bc) + 0(bd) + 0(ac) + 0(ad) + 0(bc) + 0(bd))I)

= ((0a)c, ((0a)d + (0b)c + (0b)d + (0a)c + (0a)d + (0b)c + (0b)d)I)

= (0c, (0d + 0c + 0d + 0c + 0d + 0c + 0d)I)

= (0, 0I).

This shows that (a, bI) [dot encircle] (c, dI) [member of] [M(I).sub.(0,0I)].

Thus ([M(I).sub.(0,0I)], [dot encircle]) is a zero-symmetric neutrosophic subsemihypergroup.

Hence, we can conclude that ([M(I).sub.(0,0I)], +', [dot encircle]) is a zero-symmetric neutrosophic subhypernear-ring of R(I).

Definition 3.16. Let (A, +, *) beany hypernear-ring and let (M(I), +') be neutrosophic hypergroup. Suppose that

[psi] : A * M(I) [right arrow] M(I)

is an action of A on M(I) defined by a * (x, yI) = (ax, ayI), for all a [member of] A and x, y [member of] M. M(I) is called a neutrosophic A-hypergroup, for all a, b [member of] A and (x, yI) [member of] M(I), the following conditions hold:

1. (a + b)(x, yI) = a(x, yI) +' b(x, yI).

2. (a * b) * (x, yI) = a(b(x, yI)).

3. a * I = aI.

4. 0 * (x, yI) = (0, 0I) and a * (0, 0I) = (0, 0I) for all (x, yI) [member of] M(I) and a [member of] A.

If we replace A with a neutrosophic hypernear-ring A(I), then M(I) becomes a neutrosophic A(I)-hypergroup.

Proposition 3.17. Every neutrosophic A-hypergroup is an A-hypergroup.

Proof. Suppose that M(I) is a neutrosophic A-hypergroup. Then (M(I), +') is a hypergroup. The required result follows.

Definition 3.18. Let S(I) be a nonempty subset of a neutrosophic A-hypergroup M(I). S(I) is said to be a two-sided neutrosophic A-subhypergroup of M(I) if

1. (S(I), +') is a neutrosophic subhypergroup of (M(I), +'),

2. AS(I) [subset or equal to] S(I) and

3. S(I)A [subset or equal to] S(I).

S(I) is said to be left neutrosophic A-subhypergroup if 1 and 2 are met. And (I) is called right neutrosophic A-subhypergroup if properties 1 and 3 are satisfied.

Example 3.19. Let M(I) be a neutrosophic A-hypergroup and (x, yI) [member of] M(I), then the set

A(x, yI) = {a(x, yI) : a [member of] A}

is a left neutrosophic A-subhypergroup of M(I).

To see this, let u, v [member of] A(x, yI). Then there exist [a.sub.1], [a.sub.2] [member of] A such that u = [a.sub.1](x, yI) and v = [a.sub.2](x, yI) so that

u +' v = [a.sub.1](x, yI) +' [a.sub.2](x, yI) = ([a.sub.1]x, [a.sub.1]yI) +' ([a.sub.2]x, [a.sub.2]yI)

= {(p, qI): p [member of] [a.sub.1]x + [a.sub.2]x, q [member of] [a.sub.1]y + [a.sub.2]y}

= {(p, qI): P [member of] ([a.sub.1] + [a.sub.2])x, q [member of] ([a.sub.1] + [a.sub.2])y}

= (([a.sub.1] + [a.sub.2])x, ([a.sub.1] + [a.sub.2])yI)

= ([a.sub.1] + [a.sub.2])(x, yI)

[subset or equal to] A(x, yI) [therefore] [a.sub.1] + [a.sub.2] [subset or equal to] A.

Also, for any u [member of] A(x, yI), there exists [a.sub.1] [member of] A such that u = [a.sub.1](x, yI).

Then -u = -([a.sub.1](x, yI)) = -[a.sub.1](x, yI), this implies that -u [member of] A(x, yI), since -[a.sub.1] [member of] A.

Since, A(x, yI) can be written as (Ax, Ay(I)), A(x, yI) contains a proper subset which is a subhypergroup. Hence, A(x, yI) is a neutrosophic subhypergroup.

Now, it remains to show that AA(x, yI) [subset or equal to] A(x, yI). Let a [member of] A and u = [a.sub.1](x, yI) [member of] A(x, yI).

We have that

au = a([a.sub.1](x, yI)) = (a[a.sub.1])(x, yI) [member of] A(x, yI), since a[a.sub.1] [member of] A.

Hence, the set A(x, yI) = {a(x, yI) : a [member of] A} is a left neutrosophic A-subhypergroup of M(I).

Proposition 3.20. Let B(I) and D(I) be two left neutrosophic A-subhypergroups ofa neutrosophic A-hypergroup M(I) and let [[B.sub.i](I).sub.i[member of][LAMBDA]] be a family of left neutrosophic A-subhypergroups of an A-hypergroup M(I). Then,

1. B(I) + D(I) is a left neutrosophic A-subhypergroup of M(I).

2. [[intersection].sub.i[member of][LAMBDA]] [B.sub.i](I) is also a left neutrosophic A-subhypergroups of M(I).

Proof. 1. We can easily show that B(I) + D(I) is a neutrosophic subhypergroup of M(I).

Now it remain to show that A(B(I) + D(I)) [subset or equal to] B(I) + D(I).

Since B(I) and D(I) are left neutrosophic A-subhypergroup of M(I), AB(I) [subset or equal to] B(I) and AD(I) [subset or equal to] D(I).

So, for a [member of] A, ([b.sub.1], [b.sub.2]I) [member of] B(I) and ([d.sub.1], [d.sub.2]I) [member of]

D(I) we have A(B(I) + D(I)) = a(([b.sub.1], [b.sub.2]I) + ([d.sub.1], [d.sub.2]I))

= a{(p, qI) : p [member of] [b.sub.1] + [d.sub.1], q [member of] [b.sub.2] + [d.sub.2]}

= {(ap, aqI) : ap [member of] a[b.sub.1] + a[d.sub.1], aq [member of] a[b.sub.2] + a[d.sub.2]}

= {(u, vI) : u [member of] a[b.sub.1] + a[d.sub.1], v [member of] a[b.sub.2] + a[d.sub.2]}

= (u, vI) [member of] AB(I)+ AD(I)

[subset or equal to] B(I) + D(I).

Hence, B(I) + D(I) is a left neutrosophic A-subhypergroup.

2. Let [b.sub.1] = (x, yI), [b.sub.2] = (u, vI) [member of] [[intersection].sub.i[member of][LAMBDA]] [B.sub.i](I) and a [member of] A. Then for all i [member of] [LAMBDA], [b.sub.1], [b.sub.2] [member of] [B.sub.i](I).

Since each [B.sub.i](I) is a left neutrosophic A-subhyperspace, for all i [member of] [LAMBDA], [b.sub.1] - [b.sub.2] = [b.sub.1] + (-[b.sub.2]) [subset or equal to] [B.sub.i](I), [B.sub.i](I) contains a proper subset which is a subhypergroup and A[b.sub.1] [subset or equal to] [B.sub.i](I).

Thus,

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[mathematical expression not reproducible] contains [mathematical expression not reproducible] which is a subhyperspace, and [mathematical expression not reproducible].

Proposition 3.21. Let M(I) be a constant neutrosophic M(I)-hypergroup. Then

1. any neutrosophic subhypergroup of M(I) is a left neutrosophic M(I)-subhypergroup of M(I).

2. M(I) is the only right neutrosophic M(I)-subhypergroup of M(I).

Proof. 1. We know from Remark 3.12 that

M(I) = {(0, mI) : m [member of] M}.

Now, let N(I) be any neutrosophic subhypergroup of M(I).

Let (0, xI) [member of] N(I) and (0, mI) [member of] M(I) be arbitrary. Then

(0, mI) * (0, xI) = (0 * 0, 0 * x + m * 0 + mxI) = (0, (x + 0 + x)I) = (0, xI) [member of] N(I). [therefore] M(I)N(I) [subset or equal to] N(I).

Hence, N(I) a left neutrosophic M(I) -subhypergroup of M(I).

2. First, (M(I), +') is a neutrosophic subhypergroup of (M(I), +').

It remains to show that M(I) is the only neutrosophic subhypergroup of (M(I), +') satisfying axiom 3 of Definition 3.18, i.e., M(I)M(I) [subset or equal to] M(I).

To see this, suppose we can find another neutrosophic subhypergroup U(I) of M(I) such that

U(I)M(I) [subset or equal to] U(I). Then we have (0, 0I)M(I) = M(I) [subset or equal to] U(I), since M(I) is constant.

So we must have that M(I) = U(I). The proof is complete.

Definition 3.22. A neutrosophic subhypergroup A(I) of a neutrosophic hypergroup (R(I), +') is said to be normal if for all (x, yI) [member of] R(I), we have (x, yI) + A(I) - (x, yI) [subset or equal to] A(I).

Definition 3.23. Let (R(I), +', *) be a neutrosophic hypernear-ring and let A(I) be a normal neutrosophic subhypergroup of (R(I), +).

1. A(I) is called a left neutrosophic hyperideal of R(I), if for all (a, bI) [member of] A(I), (x, yI) [member of] R(I), we have (x, yI)(a, bI) [member of] A(I).

2. A(I) is called a right neutrosophic hyperideal of R(I), if for all (x, yI), (u, vI) [member of] R(I), we have ((x, yI) +' A(I))(u, vI) - (x, yI)(u, vI) [subset or equal to] A(I).

3. A(I) is called a neutrosophic hyperideal of R(I), if it is both a left and right neutrosophic hyperideal of R(I).

Definition 3.24. Let (M(I), +') be a neutrosophic A-hypergroup over a hypernear-ring A. A normal neutrosophic subhypergroup B(I) of M(I) is called a neutrosophic hyperideal of M(I) if for all ([b.sub.1], [b.sub.2]I) [member of] B(I), (x, yI) [member of] M(I) and a [member of] A we have

a * ((x, yI) +' ([b.sub.1], [b.sub.2]I)) - a * (x, yI) [subset or equal to] B(I).

Proposition 3.25. Let M(I) be a neutrosophic hypernear-ring. If U(I) and V(I) are any two neutrosophic hyperideals of M(I) and [[U.sub.i](I).sub.i[member of][LAMBDA]] is a family of neutrosophic hyperideals of M(I), then

1. U(I) + V(I) = {a|a [member of] u + v for some u [member of] U(I), v [member of] V(I)} is a neutrosophic hyperideal of M(I).

2. U(I)V(I) = {a|a [member of] [[summation].sup.n.sub.i=1] [u.sub.i][v.sub.i] for some [u.sub.i] [member of] U(I), [v.sub.i] [member of] V(I)} is a neutrosophic hyperideal of M(I).

3. [[intersection].sub.i[member of][LAMBDA]] [U.sub.i](I) is a neutrosophic hyperideal of M(I).

Proof. The proof is the same as the proof in classical case.

Proposition 3.26. Let (M(I), +) be a neutrosophic A-hypergroup over a hypernear-ring A. Let B(I) be a neutrosophic hyperideal of M(I) and D(I) be a left neutrosophic A-subhypergroup of M(I).

Then D(I) + B(I) is a left neutrosophic A-subhypergroup of M(I).

Proof. We want to show that A(D(I) + B(I)) C D(I) + B(I).

Since B(I) is a neutrosophic hyperideal of M(I) and D(I) is a left neutrosophic A-subhypergroup of M(I), for ([b.sub.1], [b.sub.2]I) [member of] B(I), (x, yI) [member of] D(I) and a [member of] A we have

a * ((x, yI) + ([b.sub.1], [b.sub.2]I)) - a * (x, yI) [subset or equal to] B(I).

So, for ([b.sub.3], [b.sub.4]I) [member of] B(I),

a * ((x, yI) + ([b.sub.1], [b.sub.2]I)) = ([b.sub.3], [b.sub.4]I) + a * (x, yI)

= (ax, (ay)I) + ([b.sub.3], [b.sub.4]I)

[subset or equal to] D(I) + B(I).

Hence, D(I) + B(I) is a left neutrosophic A-subhypergroup of M(I).

Proposition 3.27. Let M(I) be a neutrosophic A(I)-hypergroup over a neutrosophic hypernear-ring A(I). If B(I) and D(I) are any two neutrosophic hyperideals of M(I), then

(B(I) : D(I)) = {([a.sub.1], [a.sub.2]I) [member of] A(I) : ([a.sub.1], [a.sub.2]I)D(I) [subset or equal to] B(I)}

is a neutrosophic hyperideal of M(I).

Proof. Let ([b.sub.1], [b.sub.2]I) [member of] (B(I) : D(I)), ([d.sub.1], [d.sub.2]I) [member of] D(I), ([a.sub.1], [a.sub.2]I) [member of] A(I) and ([m.sub.1], [m.sub.2]I) [member of] M(I) be arbitrary elements. Then ([b.sub.1], [b.sub.2]I)([d.sub.1], [d.sub.2]I) = ([b.sub.1], [b.sub.2]I) and it implies that [b.sub.1][d.sub.1] = [b.sub.1], [b.sub.1][d.sub.2] + [b.sub.2][d.sub.1] + [b.sub.2][d.sub.2] = [b.sub.2]. We want to show that (B(I) : D(I)) is a neutrosophic hyperideal of M(I).

To see this, we only need to show that

([a.sub.1], [a.sub.2]I) * (([m.sub.1], [m.sub.2]I) + ([b.sub.1], [b.sub.2]I)) - ([a.sub.1], [a.sub.2]I)([m.sub.1], [m.sub.2]I) [subset or equal to] (B(I) : D(I)).

Now,

[([a.sub.1], [a.sub.2]I) * (([m.sub.1], [m.sub.2]1) + ([b.sub.1], [b.sub.2]I)) - ([a.sub.1], [a.sub.2]I) ([m.sub.1], [m.sub.2]I)]([d.sub.1], [d.sub.2]I)

= [([a.sub.1], [a.sub.2]I)([m.sub.1] + [b.sub.1], ([m.sub.2] + [b.sub.2])I) - ([a.sub.i], [a.sub.2]I)([m.sub.1], [m.sub.2]i)]([d.sub.1], [d.sub.2]I)

= [([a.sub.1][m.sub.1] + [a.sub.1][b.sub.1], ([a.sub.1][m.sub.2] + [a.sub.1][b.sub.2] + [a.sub.2][m.sub.1] + [a.sub.2][b.sub.1] + [a.sub.2][m.sub.2] + [a.sub.2][b.sub.2])I) - ([a.sub.1][m.sub.1], ([a.sub.1][m.sub.2] + [a.sub.2][m.sub.1] + [a.sub.2][m.sub.2]I)]([d.sub.1], [d.sub.2]I)

= [[a.sub.1][b.sub.1], ([a.sub.1][b.sub.2] + [a.sub.2][b.sub.1] + [a.sub.2][b.sub.2])I}([d.sub.1], [d.sub.2]I)

= (([a.sub.1][b.sub.1])[d.sub.1], (([a.sub.1][b.sub.1])[d.sub.2] + ([a.sub.1][b.sub.2])[d.sub.1] + ([a.sub.1][b.sub.2])[d.sub.2] + ([a.sub.2][b.sub.1])[d.sub.1] + ([a.sub.2][b.sub.1])[d.sub.2] + ([a.sub.2][b.sub.2])[d.sub.1] + ([a.sub.2][b.sub.2])[d.sub.2])I)

= ([a.sub.1]([b.sub.1][d.sub.1]), ([a.sub.1]([b.sub.1][d.sub.2]) + [a.sub.1]([b.sub.2][d.sub.1]) + [a.sub.1]([b.sub.2][d.sub.2]) + [a.sub.2]([b.sub.1][d.sub.1]) + [a.sub.2] ([b.sub.1][d.sub.2]) + [a.sub.2]([b.sub.2][d.sub.1]) + [a.sub.2]([b.sub.2][d.sub.2]))I)

= ([a.sub.1][b.sub.1], ([a.sub.1]([b.sub.1][d.sub.2] + [b.sub.2][d.sub.1] + [b.sub.2][d.sub.2]) + [a.sub.2] ([b.sub.1][d.sub.2] + [b.sub.2][d.sub.1] + [b.sub.2][d.sub.2]) + [a.sub.2][b.sub.1])I)

= ([a.sub.1][b.sub.1], ([a.sub.1][b.sub.2] + [a.sub.2][b.sub.2] + [a.sub.2][b.sub.1])I)

= ([a.sub.1], [a.sub.2]I)([b.sub.1], [b.sub.2]I).

Hence, ([a.sub.1], [a.sub.2]I) * (([m.sub.1], [m.sub.2]I) + ([b.sub.1], [b.sub.2]I)) - ([a.sub.1], [a.sub.2]I)([m.sub.1], [m.sub.2]I) [subset or equal to] (B(I) : D(I)).

Definition 3.28. Let M(I) be a neutrosophic hypernear-ring and let (x, yI) [member of] M(I). The set

Ann((x, yI)) = {(m, nI) [member of] M(I) : (x, yI)(m, nI) = (0, 0I)}

is called the right annihilator of (x, yI).

Proposition 3.29. Let M(I) be a zero-symmetric neutrosophic hypernear-ring. For any (x, yI) [member of] M(I), Ann((x, yI)) is a right neutrosophic M(I)-subhypergroup of M(I).

Proof. Ann((x, yI)) [not equal to] 0. Since there exists (0, 0I) [member of] M(I) such that

(x, yI)(0, 0I) = (x0, (x0 + y0 + y0)I) = (0, 0I).

Let (a, bI), (c, dI) [member of] Ann((x, yI)), then (x, yI)(a, bI) = (0, 0I) and (x, yI)(c, dI) = (0, 0I) from which we have xa = 0, xb + ya + yb = 0, xc = 0 and xd + yc + yd = 0.

So,

(x, yI)[(a,bI) + (c,dI)] = (x, yI){(p,qI): p [member of] a + c,q [member of] b + d}

= (x, yI)(p,qI)

= (xp, (xq + yp + yq)I)

= (0, 0I).

This implies that (a, bI) + (c, dI) [subset or equal to] Ann((x, yI)).

Also,

(x, yI)(-(a, bI)) = -((x, yI)(a,bI)) From Lemma 3.2

= -(0, 0I)

= (0, 0I).

Lastly, we will show that Ann((x, yI))M(I) [subset or equal to] Ann((x, yI)). To this end, let (u, vI) [member of] M(I) and (a, bI) [member of] Ann((x, yI)) so that

(x, yI)[(a, bI)(u, vI)] = (x, yI)(au, (av + bu + bv)I)

= (x(au), (x(av) + x(bu) + x(bv) + y (au) + y (av) + y (bu) + y(bv))I)

= ((xa)u, ((xa)v + (xb)u + (xb)v + (ya)u + (ya)v + (yb)u + (yb)v)I)

= ((xa)u, (((xb + ya + yb)u) + ((xa + xb + ya + yb)v))I)

= (0, 0I).

So, (a, bI)(u, vI) [member of] Ann((x, yI)) from which it follows that Ann((x, yI))M(I) [subset or equal to] Ann((x, yI)).

Hence, Ann((x, yI)) is a right neutrosophic M(I)-subhypergroup of M(I).

Definition 3.30. Let W(I) be a neutrosophic hyperideal of a neutrosophic hypernear-ring (M(I), +', *).

The quotient M(I)/W(I) is defined by the set {[m] = m + W(I) : m [member of] M(I)}.

Proposition 3.31. Lei M(I)/W(I) = {[m] = m + W(I) : m [member of] V(I)}.

For every [m] = ([m.sub.1], [m.sub.2]I) + W(I), [n] = ([n.sub.1], [n.sub.2]I) + W(I) [member of] M(I)/W(I) we define:

[m] [direct sum] [n] = ([m.sub.1], [m.sub.2]I) + W(I) [direct sum] ([n.sub.1], [n.sub.2]I) + W(I) = (([m.sub.1] +' [n.sub.1]), ([m.sub.2] +' [n.sub.2])I) + W(I)

and

[m] [dot encircle] [n] = [m * n] = (mini, ([m.sub.1][n.sub.2] + [m.sub.2][n.sub.1] + [m.sub.2][n.sub.2])I) + W(I).

(M(I)/W(I), [direct sum], [dot encircle]) is a neutrosophic hypernear-ring called neutrosophic quotient hypernear-ring.

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

Definition 3.32. Let M(I) and N(I) be any two neutrosophic hypernear-rings.

[alpha] : M(I) [right arrow] N(I)

is called a neutrosophic hypernear-ring homomorphism, if the following conditions hold:

1. [alpha] is a hypernear-ring homomorphism,

2. [alpha](I) = I.

Note: If M(I) and N(I) are any two neutrosophic A-hypergroups. Then [alpha] is called a neutrosophic A-hypergroup homomorphism if [alpha] is a A-hypergroup homomorphism and [alpha](I) = I.

Definition 3.33. Let a be a neutrosophic homomorphism from M(I) into N(I) then

1. Ker[alpha] = {(x, yI) [member of] M(I) : [alpha]((x, yI)) = (0, 0I)} and

2. Im[alpha] = {(a, bI) [member of] N(I) : (a, bI) = [alpha]((x, yI)), (x, yI) [member of] M(I)}.

Proposition 3.34. Let A(I) and B(I) be two neutrosophic A-hypergroup over a zero-symmetric hypernear-ring A. Let [alpha] : A(I) [right arrow] B(I) be a neutrosophic A-hypergroup homomorphism, then

1. Ker[alpha] is not a neutrosophic hyperideal of A(I).

2. Ker[alpha] is a two-sided A-subhypergroup of A.

3. Im[alpha] is a left A--neutrosophic subhypegroup of B(I).

Proof. 1. Since [alpha] is a neutrosophic A-hypergroup homomorphism, we know that [alpha](I) = I.

Then ([a.sub.1], [a.sub.2]I) [member of] A(I) with [a.sub.1], [a.sub.2] [member of] A will be in the Ker[alpha] if and only if [a.sub.2] = 0. This implies that

ker[alpha] = {([a.sub.1], 0I) [member of] A(I)}

which is just a subhypergroup of A. Hence, ker [alpha] is not a neutrosophic hyperideal of A(I).

2. From 1, we have that ker[alpha] = {([a.sub.1], 0I) [member of] A(I)} is a subhypergroup of A. So, it remains to show that A(Ker[alpha]) [subset or equal to] Ker[alpha] and (Ker[alpha])A [subset or equal to] Ker[alpha]

To see this, let b [member of] A and ([a.sub.1], 0I) [member of] Ker[alpha] be arbitrary then

b[alpha](([a.sub.1], 0I)) = [alpha]((b[a.sub.1], (b0)I)) = [alpha]((b[a.sub.1], 0I)) = (0, 0I) [member of] Ker[alpha],

and

[alpha](([a.sub.1], 0I))b = [alpha](([a.sub.1]b, (0b)I)) = [alpha](([a.sub.1]b, 0I)) = (0, 0I) [member of] Ker[alpha].

It implies that A(Ker[alpha]) [subset or equal to] Ker[alpha] and (Ker[alpha]) A [subset or equal to] Ker[alpha]. Hence ker [alpha] is a two-sided A-subhypergroup of A.

3. By definition Im[alpha] = {(a, bI) [member of] B(I) : (a, bI) = [alpha]((x, yI)), (x, yI) [member of] A(I)}. It is clear that Im[alpha] is a neutrosophic subhypergroup of B(I). So, it remains to show that A(Im[alpha]) [subset or equal to] (Im[alpha]). Now, let (a, bI) [member of] Im[alpha] and [a.sub.1] [member of] A be arbitrary, where (a, bI) = [alpha]((x, yI)) and (x, yI) [member of] A(I), then

[a.sub.1](a, bI) = [a.sub.1][alpha]((x, yI)) = [alpha](([a.sub.1]x), ([a.sub.1]y)I).

Since A(I) is a A-hypergroup, then (([a.sub.1]x), ([a.sub.1]y)I) [member of] A(I). So, we can say that [alpha]((([a.sub.1]x), ([a.sub.1]y)I)) [member of] Im[alpha]. And it implies that A(Im[alpha]) [subset or equal to] Im[alpha]. Hence, Im[alpha] is a left neutrosophic A-subhypergroup of B(I).

Proposition 3.35. Let A(I) and B(I) be any two neutrosophic hypernear-ring and let [alpha] : A(I) [right arrow] B(I) be a neutrosophic hypernear-ring homomorphism, then

1. Ker[alpha] is not a neutrosophic hyperideal of A(I).

2. Ker[alpha] is a subhypernear-ring of A.

3. Im[alpha] is neutrosophic subhypernear-ring of B(I).

Proof. 1. The proof follows similar approach as proof 1 of Proposition 3.34.

2. Ker[alpha] = {([a.sub.1], 0) [member of] A(I)}. It is easy to show that ker[alpha] is a subhypernear-ring of A.

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

4 Conclusion

We investigated and presented some of the interesting results arising from the study of hypernear-rings in the neutrosophic environment. The concept of neutrosophic A-hypergroup of a hypernear-ring A, neutrosophic A(I)-hypergroup of a neutrosophic hypernear-ring A(I) and their respective neutrosophic substructures were presented. It was shown that a constant neutrosophic hypernear-ring in general is not a constant hypernearrings. We hope to study more advanced properties of neutrosophic hypernear-ring in our future work.

Doi: 10.5281/zenodo.4001123

5 Acknowledgment

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

References

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

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

[4] Agboola, A.A.A., Davvaz, B. and Smarandache, F. "Neutrosophic quadruple algebraic hyperstructures", Ann. Fuzzy Math.Inform. Vol. 14, pp. 29-42, 2017.

[5] Agboola, A.A.A, Ibrahim, M.A., Adeleke, E.O, Akinleye, S.A., "On Refined Neutrosophic Algebraic Hyperstructures I", International Journal of Neutrosophic Science (UNS), Vol. 5(1), pp. 29-37, 2020.

[6] Akinleye, S.A., Adeleke, E.O. and Agboola, A.A.A. "Introduction to neutrosophic near-rings", Ann. Fuzzy Math.Inform. Vol. 12, pp. 397-409, 2016.

[7] Al-Tahan, M. and Davvaz, B., "Refined neutrosophic quadruple (po-)hypergroups and their fundamental group ", Neutrosophic Sets and Systems, Vol. 27, pp. 138-153, 2019.

[8] Corsini. P., "Hypergroupes regulier et hypermodules", Vol 20, pp. 121-135, 1975.

[9] Corsini. P., "Prolegomena of Hypergroup Theory", Aviani Editore, Tricemo, Italian, 1993.

[10] Corsini. P. and Loereanu, V. "Application of Hyperstructure Theory", Vol. 5 Advances in Mathematics. Kluwer Academics Publishers, Dordrecht, The Netherlands, 2003.

[11] Dasic, V., "Hypernear-rings, Algebraic Hyperstructures and Applications", Proceedings of the Fourth International Congress Xanthi, Greece, pp. 75-85, World Sci. Publ., Teaneck, NJ, 1991.

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

[13] Gontineac, V.M., "On hypernear-rings and H-hypergroups", Proc. Fifth Int. Congress on Algebraic Hyperstrutures and Application (AHA 1993), Hadronic Press, Inc., USA, pp. 171-179, 1994.

[14] Ibrahim, M.A., Agboola, A.A.A, Adeleke, E.O, Akinleye, S.A., "Introduction to Neutrosophic Subtraction Algebra and Neutrosophic Subtraction Semigroup", International Journal of Neutrosophic Science (UNS), Vol. 2(2), pp. 47-62, 2020.

[15] Ibrahim, M.A., Agboola, A.A.A, Adeleke, E.O, Akinleye, S.A., "On Neutrosophic Quadruple Hypervector Spaces", International Journal of Neutrosophic Science (IJNS), Vol. 4(1), pp. 20-35, 2020.

[16] Ibrahim, M.A., Agboola, A.A.A, Badmus, B.S, Akinleye, S.A., "On Refined Neutrosophic Hypervector Spaces", International Journal of Neutrosophic Science (IJNS), Vol. 8(1), pp. 50-71, 2020.

[17] Ibrahim, M.A., Agboola, A.A.A, Badmus, B.S, Akinleye, S.A., "On Refined Neutrosophic Hypergroup", to appear in International Journal of Neutrosophic Science (IJNS).

[18] Kyung Ho Kim, Davvaz, B. and Eun Hwan Roh, "On Hyper R-Subgroups of Hypernear-rings", Scientiae Mathematicae Japonicae Online, pp. 649-656, 2007.

[19] Mittas, J., "Hypergroupes canoniques", Mathematica Balkanica, Vol. 2, pp. 165-179, 1972.

[20] Mittas, J., "Hyperanneaux et certaines de leurs proprietes", Vol. 269, pp. A623-a626, 1969.

[21] Massouros, C., "Some properties of certain Subhypergroups", RATIO MATHEMATICA Vol. 25, pp. 67-76, 2013.

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

[23] Stratisgopoulos, D., "Certaines classes d'hypercorps et d'hyperanneaux, in Hypergroups, Other Multivalued Structures and Their Applications", University of Udine, Udine, Italy, pp. 105-110, 1985.

[24] 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

[25] 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.

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

(1) M.A. Ibrahim and (2) A.A.A. Agboola

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

muritalaibrahim40@gmail.com (1) and agboolaaaa@funaab.edu.ng (2)
Table 1: (i) Cayley table for the hyper operation +' and (ii) Cayley
table for the hyper operation "[dot encircle]"
(i)
+'               [x.sub.0]     [x.sub.1]     [x.sub.2]     [x.sub.3]
[x.sub.0]       {[x.sub.0]}   {[x.sub.1]}   {[x.sub.2]}   {[x.sub.3]}
[x.sub.1]       {[x.sub.1]}   {[x.sub.0],   {[x.sub.1]    {[y.sub.1]}
                               [x.sub.1]    [x.sub.2]}
                              [x.sub.2]}
[x.sub.2]       {[x.sub.2]}   {[x.sub.1],   {[x.sub.0],   {[y.sub.3]}
                              [x.sub.2]}     [x.sub.1]
                                            [x.sub.2]}
[x.sub.3]       {[x.sub.3]}   {[y.sub.1]}   {[y.sub.3]}   {[x.sub.0],
                                                           [x.sub.3]
                                                          [x.sub.4]}
[x.sub.4]       {[x.sub.4]}   {[y.sub.2]}   {[y.sub.4]}   {[x.sub.3],
                                                          [x.sub.4]}
[y.sub.1]       {[y.sub.1]}   {[x.sub.3],   {[y.sub.1],   {[x.sub.1],
                              [y.sub.1],    [y.sub.3]}    [y.sub.1],
                              [y.sub.3]}                  [y.sub.2]}
[y.sub.2]       {[y.sub.2]}   {[x.sub.4],   {[y.sub.2],   {[y.sub.1],
                              [y.sub.2],    [y.sub.4]}    [y.sub.2]}
                              [y.sub.4]}
[y.sub.3]       {[y.sub.3]}   {[y.sub.1],   {[x.sub.3],   {[x.sub.2],
                              [y.sub.3]}    [y.sub.1],    [y.sub.3],
                                            [y.sub.3]}    [y.sub.4]}
[y.sub.4]       {[y.sub.4]}   {[y.sub.2],   {[x.sub.4],   {[y.sub.3],
                              [y.sub.4]}    [y.sub.2],    [y.sub.4]}
                                            [y.sub.4]}
+'               [x.sub.4]     [y.sub.1]     [y.sub.2]     [y.sub.3]
[x.sub.0]       {[x.sub.4]}   {[y.sub.1]}   {[y.sub.2]}   {[y.sub.3]}
[x.sub.1]       {[y.sub.2]}   {[x.sub.3],   {[x.sub.4],   {[y.sub.1],
                              [y.sub.1],    [y.sub.2],    [y.sub.3]}
                              [y.sub.3]}    [y.sub.4]}
[x.sub.2]       {[y.sub.4]}   {[y.sub.1],   {[y.sub.2],   {[x.sub.3],
                              [y.sub.3]}    [y.sub.4]}    [y.sub.1],
                                                          [y.sub.3]}
[x.sub.3]       {[x.sub.3],   {[x.sub.1],   {[y.sub.1],   {[x.sub.2],
                [x.sub.4]}    [y.sub.1],    [y.sub.3]}    [y.sub.3],
                              [y.sub.2]}                  [y.sub.4]}
[x.sub.4]       {[x.sub.0],   {[y.sub.1],   {[x.sub.1],   {[y.sub.3],
                 [x.sub.3]    [y.sub.2]}    [y.sub.1],    [y.sub.4]}
                [x.sub.4]}                  [y.sub.2]}
[y.sub.1]       {[y.sub.1],      R(I)       {[x.sub.3],   {[x.sub.1],
                [y.sub.2]}                  [x.sub.4],    [x.sub.2],
                                            [N.sub.y]}    [N.sub.y]}
[y.sub.2]       {[x.sub.1],   {[x.sub.3],      R(I)       {[N.sub.y]}
                [y.sub.1],    [x.sub.4],
                [y.sub.2]}    [N.sub.y]}
[y.sub.3]       {[y.sub.3],   {[x.sub.1],   {[N.sub.y]}      R(I)
                [y.sub.4]}    [x.sub.2],
                              [N.sub.y]}
[y.sub.4]       {[x.sub.2],   {[N.sub.y]}   {[x.sub.1],   {[x.sub.2],
                [y.sub.3],                  [x.sub.2],    [x.sub.4],
                [y.sub.4]}                  [N.sub.y]}    [N.sub.y]}
+'               [y.sub.4]
[x.sub.0]       {[y.sub.4]}
[x.sub.1]       {[y.sub.2],
                [y.sub.4]}
[x.sub.2]       {[x.sub.4],
                [y.sub.2],
                [y.sub.4]}
[x.sub.3]       {[y.sub.3],
                [y.sub.4]}
[x.sub.4]       {[x.sub.2],
                [y.sub.3],
                [y.sub.4]}
[y.sub.1]       {[N.sub.y]}
[y.sub.2]       {[x.sub.1],
                [x.sub.2],
                [N.sub.y]}
[y.sub.3]       {[x.sub.3],
                [x.sub.4],
                [N.sub.y]}
[y.sub.4]          R(I)
(ii)
[dot encircle]   [x.sub.0]     [x.sub.1]     [x.sub.2]     [x.sub.3]
[x.sub.0]        [x.sub.0]     [x.sub.0]     [x.sub.0]     [x.sub.0]
[x.sub.1]        [x.sub.0]     [x.sub.1]     [x.sub.2]     [x.sub.3]
[x.sub.2]        [x.sub.0]     [x.sub.2]     [x.sub.1]     [x.sub.4]
[x.sub.3]        [x.sub.0]     [x.sub.3]     [x.sub.4]     [x.sub.3]
[x.sub.4]        [x.sub.0]     [x.sub.4]     [x.sub.3]     [x.sub.4]
[y.sub.1]        [x.sub.0]     [y.sub.1]     [y.sub.4]    {[x.sub.0],
                                                          [x.sub.3],
                                                          [x.sub.4]}
[y.sub.2]        [x.sub.0]     [y.sub.2]     [y.sub.3]    {[x.sub.3],
                                                          [x.sub.4]}
[y.sub.3]        [x.sub.0]     [y.sub.3]     [y.sub.2]    {[x.sub.3],
                                                          [x.sub.4]}
[y.sub.4]        [x.sub.0]     [y.sub.4]     [y.sub.1]    {[x.sub.0],
                                                          [x.sub.3],
                                                          [x.sub.4]}
[dot encircle]    [x.sub.4]     [y.sub.1]     [y.sub.2]     [y.sub.3]
[x.sub.0]         [x.sub.0]     [x.sub.0]     [x.sub.0]     [x.sub.0]
[x.sub.1]         [x.sub.4]     [y.sub.1]     [y.sub.2]     [y.sub.3]
[x.sub.2]         [x.sub.3]     [y.sub.4]     [y.sub.3]     [y.sub.2]
[x.sub.3]         [x.sub.4]    {[x.sub.0],   {[x.sub.3],   {[x.sub.3],
                               [x.sub.3],    [x.sub.4]}    [x.sub.4]}
                               [x.sub.4]}
[x.sub.4]         [x.sub.3]    {[x.sub.0],   {[x.sub.3],   {[x.sub.3],
                               [x.sub.3],    [x.sub.4]}    [x.sub.4]}
                               [x.sub.4]}
[y.sub.1]        {[x.sub.0],   {[x.sub.1],   {[x.sub.1],   {[x.sub.2],
                 [x.sub.3],    [y.sub.1],    [y.sub.1],    [y.sub.3],
                 [x.sub.4]}    [y.sub.2]}    [y.sub.2]}    [y.sub.4]}
[y.sub.2]        {[x.sub.3],   {[x.sub.1],   {[x.sub.1],   {[x.sub.2],
                 [x.sub.4]}    [y.sub.1],    [y.sub.1],    [y.sub.3],
                               [y.sub.2]}    [y.sub.2]}    [y.sub.4]}
[y.sub.3]        {[x.sub.3],   {[x.sub.2],   {[x.sub.2],   {[x.sub.1],
                 [x.sub.4]}    [y.sub.3],    [y.sub.3],    [y.sub.1],
                               [y.sub.4]}    [y.sub.4]}    [y.sub.2]}
[y.sub.4]        {[x.sub.0],   {[x.sub.2],   {[x.sub.2],   {[x.sub.1],
                 [x.sub.3],    [y.sub.3],    [y.sub.3],    [y.sub.1],
                 [x.sub.4]}    [y.sub.4]}    [y.sub.4]}    [y.sub.2]}
[dot encircle]    [y.sub.4]
[x.sub.0]         [x.sub.0]
[x.sub.1]         [y.sub.4]
[x.sub.2]         [y.sub.1]
[x.sub.3]        {[x.sub.0],
                 [x.sub.3],
                 [x.sub.4]}
[x.sub.4]        {[x.sub.0],
                 [x.sub.3],
                 [x.sub.4]}
[y.sub.1]        {[x.sub.2],
                 [y.sub.3],
                 [y.sub.4]}
[y.sub.2]        {[x.sub.2],
                 [y.sub.3],
                 [y.sub.4]}
[y.sub.3]        {[x.sub.1],
                 [y.sub.1],
                 [y.sub.2]}
[y.sub.4]        {[x.sub.1],
                 [y.sub.1],
                 [y.sub.2]}
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Author:Ibrahim, M.A.; Agboola, A.A.A.
Publication:International Journal of Neutrosophic Science
Date:Oct 1, 2020
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