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Neutrosophic Hyper BCK-Ideals.

1 Introduction

Algebraic hyperstructures represent a natural extension of classical algebraic structures and they were introduced in 1934 by the French mathematician F. Marty [17] when Marty defined hypergroups, began to analyze their properties, and applied them to groups and relational algebraic functions (See [17]). Since then, many papers and several books have been written on this topic. Hyperstructures have many applications to several sectors of both pure and applied sciences. (See [4, 5, 8, 11, 14, 19, 25]). In [16], Jun et al. applied the hyperstructures to BCK-algebras, and introduced the concept of a hyper BCK-algebra which is a generalization of a BCK-algebra. Since then, Jun et al. studied more notions and results in [12] and [15]. Also, several fuzzy versions of hyper BCK-algebras have been considered in [10] and [13]. The neutrosophic set, which is developed by Smarandache ([20], [21] and [22]), is a more general platform that extends the notions of classic set, (intuitionistic) fuzzy set and interval valued (intuitionistic) fuzzy set. Borzooei et al. [6] studied neutrosophic deductive filters on BL-algebras. Zhang et al. [26] applied the notion of neutrosophic set to pseudo-BCI algebras, and discussed neutrosophic regular filters and fuzzy regular filters. Neutrosophic set theory is applied to varios part and received attentions from many researches were proceed to develop, improve and expand the neutrosophic theory ([1], [2], [3], [7], [9], [18], [23] and [24]).

Our purpose is to introduce the notions of neutrosophic (strong, weak, s-weak) hyper BCK-ideal, and reflexive neutrosophic hyper BCK-ideal. We consider their relations and related properties. We discuss characterizations of neutrosophic (weak) hyper BCK-ideal. We give conditions for a neutrosophic set to be a (reflexive) neutrosophic hyper BCK-ideal and a neutrosophic strong hyper BCK-ideal. We are interested in finding some provisions for a neutrosophic strong hyper BCK-ideal to be a reflexive neutrosophic hyper BCK-ideal. We discuss conditions for a neutrosophic weak hyper BCK-ideal to be a neutrosophic s-weak hyper BCK-ideal.

2 Preliminaries

In this section, we give the basic definitions of hyper BCK-ideals and neutrosophic set.

For a nonempty set H a function [omicron] : H x H [right arrow] P*(H) is called a hyper operation on H. If A,B [??] H, then A [omicron] B = j{a [omicron] b | a [member of] A, b [member of] B}.

A nonempty set H with a hyper operation "[omicron]" and a constant 0 is called a hyper BCK-algebra (See [16]), if it satisfies the following conditions: for any x,y,z [member of] H,

(HBCK1) (x [omicron] z) [omicron] (y [omicron] z) [much less than] x [omicron] y,

(HBCK2) (x [omicron] y) [omicron] z = (x [omicron] z) [omicron] y,

(HBCK3) x [omicron] H [much less than] {x},

(HBCK4) x [much less than] y and y [much less than] x imply x = y,

where x [much less than] y is defined by 0 [member of] x [omicron] y. Also for any A,B [??] H, A [much less than] B is defined by [for all]a [member of] A, [there exists]b [member of] B such that a [much less than] b.

Lemma 2.1. ([16]) In a hyper BCK-algebra H, the condition (HBCK3) is equivalent to the following condition:

([for all]x,y [??] H)(x [omicron] y [??] {x}). (2.1)

Lemma 2.2. ([16]) Let H be a hyper BCK-algebra. Then

(i) x [omicron] 0 [much less than] {x}, 0 [omicron] x [much less than] {0} and 0 [omicron] 0 [much less than] {0}, for all x [member of] H

(ii) (A [omicron] B) [omicron] C = (A [omicron] C) [omicron] B, A [omicron] B < A and 0 [omicron] A [much less than] {0}, for any nonempty subsets A, B and C of H.

Lemma 2.3. ([16]) In any hyper BCK-algebra H, we have:

0 [omicron] 0 = {0}, 0 [much less than] x, x [much less than] x and A [much less than] A, (2.2)

A [??] B implies A [much less than] B, (2.3)

0 [omicron] x = {0} and 0 [omicron] A = {0}, (2.4)

A [??] {0} implies A = {0}, (2.5)

x [member of] x [omicron] 0, (2.6)

for all x, y, z [member of] H and for all nonempty subsets A, B and C of H.

Let I [??] H be such that 0 [member of] I. Then I is said to be (See [16] and [15])

* hyper BCK-ideal of H if

([for all]x, y [member of] H) (x [omicron] y [much less than] A, y [member of] A [??] x [member of] A). (2.7)

* weak hyper BCK-ideal of H if

([for all]x, y [member of] H)(x [omicron] y « A, y [member of] A [??] x [member of] A). (2.8)

* strong hyper BCK-ideal of H if

([for all]x, y [member of] H)((x [omicron] y) [intersection] A [not equal to] [empty set], y [member of] A [??] x [member of] A). (2.9)

A subset I of a hyper BCK-algebra H is said to be reflexive if (x [omicron] x) [??] I for all x [member of] H.

Let H be a non-empty set. A neutrosophic set (NS) in H (See [21]) is a structure of the form:

A := {<x; [A.sub.T] (x), [A.sub.I](x)> | x [member of] H}

where [A.sub.T] : H [right arrow] [0,1] is a truth membership function, [A.sub.I] : H [right arrow] [0,1] is an indeterminate membership function, and [A.sub.F] : H [right arrow] [0,1] is a false membership function. For abbreviation, we continue to write A = ([A.sub.T], [A.sub.I], [A.sub.F]) for the neutrosophic set

A := {<x; [A.sub.T](x),[A.sub.I](x),[A.sub.F](x)> | x [member of] H}.

Given a neutrosophic set A = ([A.sub.T], [A.sub.I], [A.sub.F]) in a hyper BCK-algebra H and a subset S of H, by *[A.sub.T], *[A.sub.T], *[A.sub.I], *[A.sub.I], *[A.sub.F] and *[A.sub.F] we mean

[mathematical expression not reproducible]

respectively.

Notation. From now on, in this paper, we assume that H is a hyper BCK-algebra.

3 Neutrosophic hyper BCK-ideals

In this section, we introduced the notions of neutrosophic (strong, weak, s-weak) hyper BCK-ideal. reflexive neutrosophic hyper BCK-ideal and discuss their properties.

Definition 3.1. Let A = ([A.sub.T],[A.sub.I],[A.sub.F]) be a neutrosophic set in H. Then A is said to be a neutrosophic hyper BCK-ideal of H if it satisfies the following assertions for all x,y [member of] H,

[mathematical expression not reproducible] (3.1)

[mathematical expression not reproducible] (3.2)

Example 3.2. Let H = {0,a,b} be a hyper BCK-algebra. The hyper operation "[omicron]" on H described by Table 1.

We define a neutrosophic set A = ([A.sub.T], [A.sub.I], [A.sub.F]) on H by Table 2.

It is easy to check that A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic hyper BCK-ideal of H. Proposition 3.3. For any neutrosophic hyper BCK-ideal A = ([A.sub.T], [A.sub.I], [A.sub.F]) of H, the following assertions are valid.

(1) A = ([A.sub.T], [A.sub.I], [A.sub.F]) satisfies

[mathematical expression not reproducible] (3.3)

(2) If A = ([A.sub.T], [A.sub.I], [A.sub.F]) satisfies

[mathematical expression not reproducible] (3.4)

then the following assertion is valid.

[mathematical expression not reproducible] (3.5)

Proof. By (2.2) and (3.1) we have

[A.sub.T](0) [greater than or equal to] [A.sub.T](x), [A.sub.I](0) [greater than or equal to] [A.sub.I](x) and [A.sub.F](0) [less than or equal to] [A.sub.F](x).

Assume that A = ([A.sub.T], [A.sub.I], [A.sub.F]) satisfies the condition (3.4). For all x,y [member of] H, there exists [a.sub.0], [b.sub.0], [c.sub.0] [member of] x [omicron] y such that

[A.sub.T] ([a.sub.0]) = *[A.sub.T] (x [omicron] y), [A.sub.I] ([b.sub.0]) = *[A.sub.I] (x [omicron] y) and [A.sub.F] ([c.sub.0]) = *[A.sub.F] (x [omicron] y).

Now condition (3.2) implies that

[mathematical expression not reproducible]

This completes the proof.

We define the following sets:

[mathematical expression not reproducible]

where A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic set in H and [[epsilon].sub.T], [[epsilon].sub.I], [[epsilon].sub.F] [member of] [0,1].

Lemma 3.4 ([12]). Let A be a subset of H. If I is a hyper BCK-ideal of H such that A [much less than] I, then A is contained in I.

Theorem 3.5. A neutrosophic set A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic hyper BCK-ideal of H if and only if the nonempty sets U([A.sub.T], [[epsilon].sub.I]), U([A.sub.I], [[epsilon].sub.I]) and L([A.sub.F], [[epsilon].sub.F]) are hyper BCK-ideals of H for all [[epsilon].sub.T], [[epsilon].sub.I], [[epsilon].sub.F] [member of] [0, 1].

Proof. Assume that A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic hyper BCK-ideal of H and suppose that U([A.sub.T], [[epsilon].sub.T]), U([A.sub.I], [[epsilon].sub.I]) and L([A.sub.F], [[epsilon].sub.F]) are nonempty for all [[epsilon].sub.T], [[epsilon].sub.I], [[epsilon].sub.F] [member of] [0, 1]. It is easy to see that 0 [member of] U([A.sub.T],[[epsilon].sub.T]), 0 [member of] U([A.sub.I],[[epsilon].sub.I]) and 0 [member of] L([A.sub.F],[[epsilon].sub.F]). Let x,y [member of] H be such that x [??] y [much less than] U([A.sub.T],[[epsilon].sub.T]) and y [member of] U([A.sub.T],[[epsilon].sub.T]). Then [A.sub.T](y) [greater than or equal to] [[epsilon].sub.T] and for any a [member of] x [??] y there exists [a.sub.0] [member of] U([A.sub.T],[[epsilon].sub.T]) such that a [much less than] [a.sub.0]. We conclude from (3.1) that [A.sub.T](a) [greater than or equal to] [A.sub.T]([A.sub.0]) > [[epsilon].sub.T] for all a [member of] x [??] y. Hence *[A.sub.T](x [??] y) [greater than or equal to] [[epsilon].sub.T], and so

[A.sub.T] (x) [greater than or equal to] min {*[A.sub.T] (x [??] y), [A.sub.T] (y)} [greater than or equal to] [[epsilon].sub.T],

that is, x [member of] U([A.sub.T],[[epsilon].sub.T]). Similarly, we show that if x [??] y [much less than] U([A.sub.I],[[epsilon].sub.I]) and y [member of] U([A.sub.I],[[epsilon].sub.I]), then x [member of] U([A.sub.I],[[epsilon].sub.I]). Hence U([A.sub.T],[[epsilon].sub.T]) and U([A.sub.I],[[epsilon].sub.I]) are hyper BCK-ideals of H. Let x,y [member of] H be such that x [??] y [much less than] L([A.sub.F],[[epsilon].sub.F]) and y [member of] L([A.sub.F],[[epsilon].sub.F]). Then [A.sub.F] (y) [less than or equal to] [[epsilon].sub.F]. Let b [member of] x [??] y. Then there exists [b.sub.0] [member of] L([A.sub.F],[[epsilon].sub.F]) such that b [much less than] [b.sub.0], which implies from (3.1) that [A.sub.F](b) [less than or equal to] [A.sub.F]([b.sub.0]) [less than or equal to] [[epsilon].sub.F]. Thus *[A.sub.F](x [??] y) [less than or equal to] [[epsilon].sub.F], and so

[A.sub.F](x) [less than or equal to] max {*[A.sub.F](x [??] y), [A.sub.F](y)} [less than or equal to] [[epsilon].sub.F].

Hence x [member of] L([A.sub.F], [[epsilon].sub.F]), and therefore L([A.sub.F], [[epsilon].sub.F]) is a hyper BCK-ideal of H.

Conversely, suppose that the nonempty sets U([A.sub.T], [[epsilon].sub.T]), U([A.sub.I], [[epsilon].sub.I]) and L([A.sub.F], [[epsilon].sub.F]) are hyper BCK-ideals of H for all [[epsilon].sub.T], [[epsilon].sub.I], [[epsilon].sub.F] [member of] [0,1]. Let x,y [member of] H be such that x [much less than] y. Then

y [member of] U ([A.sub.T],[A.sub.T] (y)) [intersection] U ([A.sub.I], [A.sub.I] (y)) [intersection] L([A.sub.F], [A.sub.F] (y)),

and thus x [much less than] U([A.sub.T],[A.sub.T](y)), x [much less than] U([A.sub.I],[A.sub.I](y)) and x [much less than] L([A.sub.F],[A.sub.F](y)). According to Lemma 3.4 we have x [member of] U([A.sub.T], [A.sub.T](y)), x [member of] U([A.sub.I], [A.sub.I](y)) and x [member of] L([A.sub.F], [A.sub.F](y)) which imply that [A.sub.T](x) [greater than or equal to] [A.sub.T](y), [A.sub.I](x) [greater than or equal to] [A.sub.I](y) and [A.sub.F](x) [less than or equal to] [A.sub.F](y). For any x,y [member of] H, let [[epsilon].sub.T] := min {*[A.sub.T](x [??] y),[A.sub.T](y)} , [[epsilon].sub.I] := min {*[A.sub.I](x [??] y),[A.sub.I](y)} and [[epsilon].sub.F] := max {*[A.sub.F](x [??] y),[A.sub.F](y)}. Then

y [member of] U ([A.sub.T], [[epsilon].sub.T]) [intersection] U ([A.sub.I],[[epsilon].sub.I]) [intersection] L([A.sub.F],[[epsilon].sub.F]),

and for each [A.sub.T], [b.sub.I], [c.sub.F] [member of] x [??] y we have

[A.sub.T] ([a.sub.T]) [greater than or equal to] *[A.sub.T] (x [??] y) [greater than or equal to] min {*[A.sub.T](x [??] y), [A.sub.T] (y)} = [[epsilon].sub.T],

[A.sub.I] ([b.sub.I]) [greater than or equal to] *[A.sub.I](x [??] y) [greater than or equal to] min {*[A.sub.I](x [??] y), [A.sub.I](y)} = [[epsilon].sub.I]

and

[A.sub.F] ([C.sub.F]) [less than or equal to] * [A.sub.F] (x [??] y) [less than or equal to] max {*[A.sub.F] (x [??] y), [A.sub.F] (y)} = [[epsilon].sub.F].

Hence [a.sub.T] [member of] U([A.sub.T],[[epsilon].sub.T]), [b.sub.I] [member of] U([A.sub.I],[[epsilon].sub.I]) and [c.sub.F] [member of] L([A.sub.F], [[epsilon].sub.F]), and so x [??] y [??] U([A.sub.T], [[epsilon].sub.T]), x [??] y [??] U([A.sub.I], [[epsilon].sub.I]) and x [omicron] y [??] L([A.sub.F], [[epsilon].sub.F]). By (2.3), we have x [??] y [much less than] U([A.sub.T],[[epsilon].sub.T]), x [??] y [much less than] U([A.sub.I],[[epsilon].sub.I]) and x [??] y [much less than] L([A.sub.F],[[epsilon].sub.F]). It follows from (2.7) that

x [member of] U([A.sub.T],[[epsilon].sub.T]) [intersection] U([A.sub.I],[[epsilon].sub.I]) [intersection] L([A.sub.F], [[epsilon].sub.F]).

Hence

[A.sub.T] (x) > [[epsilon].sub.T] = min {*[A.sub.T] (x [??] y), [A.sub.T] (y)}, [A.sub.I] (x) [greater than or equal to] [[epsilon].sub.I] = min {*[A.sub.I] (x [omicron] y),[A.sub.I] (y)}

and

[A.sub.F](x) [less than or equal to] [[epsilon].sub.I] = max {*[A.sub.F](x [omicron] y), [A.sub.F](y)}.

Therefore A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic hyper BCK-ideal of H.

Theorem 3.6. If A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic hyper BCK-ideal of H, then the set

J := {x [member of] H | [A.sub.T](x) = [A.sub.T](0), [A.sub.I](x) = [A.sub.I](0), [A.sub.F](x) = [A.sub.F](0)} (3.6)

is a hyper BCK-ideal of H.

Proof. It is easy to check that 0 [member of] J. Let x, y [member of] H be such that x [omicron] y [much less than] J and y [member of] J. Then [A.sub.T](y) = [A.sub.T](0), [A.sub.I](y) = [A.sub.I](0) and [A.sub.F](y) = [A.sub.F](0). Let a [member of] x [omicron] y. Then there exists [a.sub.0] [member of] J such that a [much less than] [a.sub.0], and thus by (3.1), [A.sub.T](a) [greater than or equal to] [A.sub.T]([a.sub.0]) = [A.sub.T](0), [A.sub.I](a) [greater than or equal to] [A.sub.I]([a.sub.0]) = [A.sub.I](0) and [A.sub.F](a) [less than or equal to] [A.sub.F]([a.sub.0]) = [A.sub.F](0). It follows from (3.2) that

[A.sub.T] (x) [greater than or equal to] min {*[A.sub.T] (x [omicron] y), [A.sub.T](y)} [greater than or equal to] [A.sub.T] (0), [A.sub.I] (x) [greater than or equal to] min {*[A.sub.I](x [omicron] y),[A.sub.I](y)} [greater than or equal to] [A.sub.I] (0)

and

[A.sub.F](x) [less than or equal to] max {*[A.sub.F](x [omicron] y),[A.sub.F](y)} [less than or equal to] [A.sub.F](0).

Hence [A.sub.T](x) = [A.sub.T](0), [A.sub.I](x) = [A.sub.I](0) and [A.sub.F](x) = [A.sub.F](0), that is, x [member of] J. Therefore J is a hyper BCK-ideal of H.

We provide conditions for a neutrosophic set A = (, [A.sub.I], [A.sub.F]) to be a neutrosophic hyper BCK-ideal of H.

Theorem 3.7. Let H satisfy |x [omicron] y|< oo for all x,y [member of] H, and let {[J.sub.T]| t [member of] [LAMBDA] [??] [0,0.5)]} be a collection of hyper BCK-ideals of H such that

[mathematical expression not reproducible] (3.7)

([for all]s, t [member of] A)(s>t [??] [J.sub.s] [subset] [J.sub.t]). (3.8)

Then a neutrosophic set A = ([A.sub.T], [A.sub.I], [A.sub.F]) in H defined by

[mathematical expression not reproducible]

is a neutrosophic hyper BCK-ideal of H.

Proof. We first shows that

[mathematical expression not reproducible] (3.9)

It is clear that 0 [mathematical expression not reproducible] [J.sub.p] for all q [member of] [0,1]. Let x,y [member of] H be such that x [omicron] y = {[a.sub.1],[a.sub.2],***,[a.sub.n]}, [mathematical expression not reproducible]and [mathematical expression not reproducible]. Then y [member of] [J.sub.r] for some r v [LAMBDA] with q [less than or equal to] r, and for any [a.sub.i] [member of] x [omicron] y there exists [mathematical expression not reproducible], and so [mathematical expression not reproducible] for some [l.sub.i] [member of] [LAMBDA] with q [less than or equal to] [l.sub.i], such that [a.sub.i] [much less than] [b.sub.i]. If we let t := min{[t.sub.i] | i [member of] {1, 2,***,n}}, then [mathematical expression not reproducible] for all i [member of] {1, 2,***,n} and so x [omicron] y [much less than] [J.sub.t] with q [less than or equal to] t. We may assume that r > t without loss of generality, and so [J.sub.r] [subset] [J.sub.t]. By (2.7), we have [mathematical expression not reproducible].

Hence [mathematical expression not reproducible] is a hyper BCK-ideal of H. Next, we consider the following two cases:

(i) t = sup{q [member of] [LAMBDA] | q <t}, (ii) t [not equal to] sup{q [member of] [LAMBDA] | q <t}. (3.10)

If the first case is valid, then

[mathematical expression not reproducible]

and so [mathematical expression not reproducible] which is a hyper BCK-ideal of H. Similarly, we know that U([A.sub.I],t) is a hyper BCK-ideal of H. For the second case, we will show that [mathematical expression not reproducible]. If [mathematical expression not reproducible], then x [member of] [J.sub.p] for some q [greater than or equal to] t. Thus A[tau](x) [greater than or equal to] q [greater than or equal to] t, and so x [member of] U([A.sub.T],t) which shows that [mathematical expression not reproducible]. Assume that [mathematical expression not reproducible]. Then x [??] [J.sub.p] for all q [greater than or equal to] t, and so there exist [delta] > 0 such that (t - [delta],t) [intersection] [LAMBDA] = [phi]. Thus x [??] [J.sub.q] for all q > t - [delta], that is, if x [member of] [J.sub.q] then q [less than or equal to] t - [delta] < t. Hence x [??] U([A.sub.T],t). This shows that [mathematical expression not reproducible] which is a hyper BCK-ideal of H by (3.9). Similarly we can prove that U([A.sub.I],t) is a hyper BCK-ideal of H. Now we consider the following two cases:

s = inf{r [member of] [LAMBDA] I s < r} and s [not equal to] inf{r [member of] [LAMBDA] | s < r}. (3.11)

The first case implies that

[mathematical expression not reproducible]

and so [mathematical expression not reproducible] which is a hyper BCK-ideal of H. For the second case, there exists [delta] > 0 such that (s, s + [delta]) [intersection] [LAMBDA] = [phi]. If [mathematical expression not reproducible], then x [member of] [J.sub.r] for some s [greater than or equal to] r. Thus [A.sub.F](x) [less than or equal to] r [less than or equal to] s, that is, x [member of] L([A.sub.F], s). Hence [mathematical expression not reproducible]. If [mathematical expression not reproducible], then x [??] [J.sub.r] for all r [less than or equal to] s and thus x [??] [J.sub.r] for all r < s + [delta]. This shows that if x [member of] [J.sub.r] then r [greater than or equal to] s + [delta]. Hence [A.sub.F](x) [greater than or equal to] s + [delta] > s, i.e., x [??] L([A.sub.F],s). Therefore [mathematical expression not reproducible]. Consequently, [mathematical expression not reproducible] which is a hyper BCK-ideal of H by (3.9). It follows from Theorem 3.5 that A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic hyper BCK-ideal of H.

Definition 3.8. A neutrosophic set A = ([A.sub.T], [A.sub.I], [A.sub.F]) in H is called a neutrosophic strong hyper BCK-ideal of H if it satisfies the following assertions.

[mathematical expression not reproducible] (3.12)

for all x, y [member of] H.

Example 3.9. Consider a hyper BCK-algebra H = {0, a, b} with the hyper operation "[omicron]" which is given by Table 3.

Let A = ([A.sub.T], [A.sub.I], [A.sub.F]) be a neutrosophic set in H which is described in Table 4.

It is routine to verify that A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic strong hyper BCK-ideal of H.

Theorem 3.10. For any neutrosophic strong hyper BCK-ideal A = ([A.sub.T], [A.sub.I], [A.sub.F]) of H, the following assertions are valid.

(1) A = ([A.sub.T], [A.sub.I], [A.sub.F]) satisfies the conditions (3.1) and (3.3).

(2) A = ([A.sub.T], [A.sub.I], [A.sub.F]) satisfies

[mathematical expression not reproducible] (3.13)

Proof. (1) Since x [much less than] x, i.e., 0 [member of] x [??] x for all x [member of] H, we get

[mathematical expression not reproducible]

which shows that (3.3) is valid. Let x, y [member of] H be such that x < y. Then 0 [member of] x [??] y, and so

*[A.sub.T](x [??] y) [greater than or equal to] [A.sub.T](0), *[A.sub.I](x [??] y) [greater than or equal to] [A.sub.I](0) and *[A.sub.F](x [??] y) [less than or equal to] [A.sub.F](0).

It follows from (3.3) that

[mathematical expression not reproducible]

Hence A = ([A.sub.T], [A.sub.I], [A.sub.F]) satisfies the condition (3.1).

(2) Let x,y,a,b,c [member of] H be such that a,b,c [member of] x [??] y. Then

[mathematical expression not reproducible]

This completes the proof.

Theorem 3.11. If a neutrosophic set A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic strong hyper BCK-ideal of H, then the nonempty sets U([A.sub.T],[[epsilon].sub.T]), U([A.sub.I,[[epsilon].sub.I) and L([A.sub.F], [[epsilon].sub.F]) are strong hyper BCK-ideals of H for all [[epsilon].sub.T] (,) [[epsilon].sub.I (,) [[epsilon].sub.F [member of] [0, 1].

Proof. Let A = ([A.sub.T], [A.sub.I], [A.sub.F]) be a neutrosophic strong hyper BCK-ideal of H. Then A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic hyper BCK-ideal of H. Assume that U([A.sub.T], [A.sub.I], [A.sub.F]), U([A.sub.I,[[epsilon].sub.I) and L([A.sub.F], [[epsilon].sub.F]) are nonempty for all [[epsilon].sub.T] [[epsilon].sub.I], [[epsilon].sub.F] [member of] [0, 1]. Then there exist a [member of] U([A.sub.T],[[epsilon].sub.T]), b [member of] U([A.sub.I],[[epsilon].sub.I]) and c [member of] L([A.sub.F], [[epsilon].sub.F]), that is, [A.sub.T](a) > [[epsilon].sub.T], [A.sub.I](b) [greater than or equal to] [[epsilon].sub.I] and [A.sub.F](c) [less than or equal to] [[epsilon].sub.F]. It follows from (3.3) that [A.sub.T](0) > [A.sub.T](a) > [[epsilon].sub.T], [A.sub.I](0) [greater than or equal to] [A.sub.I](b) [greater than or equal to] [[epsilon].sub.I] and [A.sub.F](0) [less than or equal to] [A.sub.F](c) [less than or equal to] [[epsilon].sub.F]. Hence

0 [member of] U ([A.sub.T], [[epsilon].sub.T]) [intersection] U ([A.sub.I],[[epsilon].sub.I]) [intersection] L([A.sub.F], [[epsilon].sub.F]).

Let x,y,a,b,u,v [member of] H be such that (x [omicron] y) [intersection] U([A.sub.T],[[epsilon].sub.T]) [not equal to] [phi], y [member of] U([A.sub.T],[[epsilon].sub.T]), (a [omicron] b) [intersection] U([A.sub.I],[[epsilon].sub.I]) [not equal to] [phi], b [member of] U([A.sub.I],[[epsilon].sub.I]), (u [omicron] v) [intersection] L([A.sub.F], [[epsilon].sub.F]) = 0 and v [member of] L([A.sub.F], [[epsilon].sub.F]). Then there exist [x.sub.0] [member of] (x [omicron] y) [intersection] U([A.sub.T],[[epsilon].sub.T]), [a.sub.0] [member of] (a [omicron] b) [intersection] U([A.sub.I],[[epsilon].sub.I]) and [u.sub.0] [member of] (u [omicron] v) [intersection] L([A.sub.F], [[epsilon].sub.F]). It follows that

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

Hence x [member of] U([A.sub.T],[[epsilon].sub.T]), a [member of] U([A.sub.I],[[epsilon].sub.I]) and u [member of] L([A.sub.F], [[epsilon].sub.F]). Therefore U([A.sub.T],[[epsilon].sub.T]), U([A.sub.I],[[epsilon].sub.I]) and L([A.sub.F], [[epsilon].sub.F]) are strong hyper BCK-ideals of H.

Theorem 3.12. For any neutrosophic set A = ([A.sub.T], [A.sub.I], [A.sub.F]) in H satisfying the condition

[mathematical expression not reproducible] (3.14)

if the nonempty sets U([A.sub.T],[[epsilon].sub.T]), U([A.sub.I],[[epsilon].sub.I]) and L([A.sub.F],[[epsilon].sub.F]) are strong hyper BCK-ideals of H for all [[epsilon].sub.T], [[epsilon].sub.I], [[epsilon].sub.F] [member of] [0, 1], then A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic strong hyper BCK-ideal of H.

Proof. Assume that U([A.sub.T],[[epsilon].sub.T]), U([A.sub.I],[[epsilon].sub.I]) and L([A.sub.F], [[epsilon].sub.F]) are nonempty and strong hyper BCK-ideals of H for all [[epsilon].sub.T], [[epsilon].sub.I], [[epsilon].sub.F] [member of] [0, 1]. For any x, y, z [member of] H, such that x [member of] U([A.sub.T],[A.sub.T](x)), y [member of] U([A.sub.I], [A.sub.I](y)) and z [member of] L([A.sub.F], [A.sub.F](z)), since xo x [much less than] x, yo y [much less than] y and zo z [much less than] z by (2.1), we have xo x [much less than] U([A.sub.T], [A.sub.T](x)), yo y [much less than] U ([A.sub.I], [A.sub.I] (y)) and z [omicron] z [much less than] L([A.sub.F], [A.sub.F] (z)). By Lemma 3.4, x [omicron] x [??] U ([A.sub.T], [A.sub.T] (x)), y [omicron] y [??] U ([A.sub.I], [A.sub.I] (y)) and z [omicron] z [??] L([A.sub.F], [A.sub.F] (z)). Hence a [member of] U ([A.sub.T], [A.sub.T] (x)), b [member of] U ([A.sub.I], [A.sub.I] (y)) and c [member of] L([A.sub.F], [A.sub.F] (z)) for all a [member of] x [omicron] x, b [member of] y [omicron] y and c [member of] z [omicron] z. Therefore *[A.sub.T](x [omicron] x) [greater than or equal to] [A.sub.T](x), *[A.sub.I](y [omicron] y) [greater than or equal to] [A.sub.I](y) and *[A.sub.F](z [omicron] z) [less than or equal to] [A.sub.F](z). Now, let [[epsilon].sub.T] := min {*[A.sub.T] (x [omicron] y), [A.sub.T] (y)}, [[epsilon].sub.I] := min {*[A.sub.I] (x [omicron] y), [A.sub.I] (y)} and [[epsilon].sub.F] := max {* [A.sub.F] (x [omicron] y), [A.sub.F] (y)}. By (3.14), we have

[A.sub.T] ([a.sub.0]) = *[A.sub.T] (x [omicron] y) [greater than or equal to] min {*[A.sub.T] (x [omicron] y), [A.sub.T] (y)} = [[epsilon].sub.T], [A.sub.I] ([b.sub.0]) = *[A.sub.I] (x [omicron] y) [greater than or equal to] min {* [A.sub.I] (x [omicron] y), [A.sub.I] (y)} = [[epsilon].sub.I]

and

[A.sub.F] ([c.sub.0]) = *[A.sub.F] (x [omicron] y) [less than or equal to] max {* [A.sub.F] (x [omicron] y), [A.sub.F] (y)} = [[epsilon].sub.F]

for some [a.sub.0], [b.sub.0], [c.sub.0] [member of] x [omicron] y. Hence [a.sub.0] [member of] U([A.sub.T],[[epsilon].sub.T]), [b.sub.0] [member of] U([A.sub.I],[[epsilon].sub.I]) and [c.sub.0] [member of] L([A.sub.F], [[epsilon].sub.F]) which imply that (x [omicron] y) [intersection] U([A.sub.T],[[epsilon].sub.T]), (x [omicron] y) [intersection] U([A.sub.I],[[epsilon].sub.I]) and (x [omicron] y) [intersection] L([A.sub.F], [[epsilon].sub.F])

are nonempty. Since y [member of] U([A.sub.T], [[epsilon].sub.T]) [intersection] U([A.sub.I],[[epsilon].sub.I]) [intersection] L([A.sub.F], [[epsilon].sub.F]), it follows from (2.9) that x [member of] U([A.sub.T], [[epsilon].sub.T]) [intersection] U([A.sub.I], [[epsilon].sub.I]) [eta] L([A.sub.F], [[epsilon].sub.F]). Thus

[A.sub.T] (x) [greater than or equal to] [[epsilon].sub.T] = min {*[A.sub.T] (x [omicron] y), [A.sub.T] (y)}, [A.sub.I] (x) [greater than or equal to] [[epsilon].sub.I] = min {*[A.sub.I] (x [omicron] y),[A.sub.I] (y)}

and

[A.sub.F](x) [less than or equal to] [[epsilon].sub.F] = max {*[A.sub.F](x [omicron] y), [A.sub.F](y)}.

Consequently, A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic strong hyper BCK-ideal of H.

Since any neutrosophic set A = ([A.sub.T], [A.sub.I], [A.sub.F]) satisfies the condition (3.14) in a finite hyper BCKalgebra, we have the following corollary.

Corollary 3.13. Let A = ([A.sub.T], [A.sub.I], [A.sub.F]) be a neutrosophic set in a finite hyper BCK-algebra H. Then A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic strong hyper BCK-ideal of H if and only if the nonempty sets U([A.sub.T],[[epsilon].sub.T]), U([A.sub.I],[[epsilon].sub.I]) and L([A.sub.F],[[epsilon].sub.F]) are strong hyper BCK-ideals of H for all [[epsilon].sub.T], [[epsilon].sub.I], [[epsilon].sub.F] [member of] [0,1].

Definition 3.14. A neutrosophic set A = ([A.sub.T], [A.sub.I], [A.sub.F]) in H is called a neutrosophic weak hyper BCK-ideal of H if it satisfies the following assertions.

[mathematical expression not reproducible] (3.15)

for all x, y [member of] H.

Definition 3.15. A neutrosophic set A = ([A.sub.T], [A.sub.I], [A.sub.F]) in H is called a neutrosophic s-weak hyper BCK-ideal of H if it satisfies the conditions (3.3) and (3.5).

Example 3.16. Consider a hyper BCK-algebra H = {0, a, b, c} with the hyper operation "[omicron]" which is given by Table 5.

Let A = ([A.sub.T], [A.sub.I], [A.sub.F]) be a neutrosophic set in H which is described in Table 6.

It is routine to verify that A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic weak hyper BCK-ideal of H.

Theorem 3.17. Every neutrosophic s-weak hyper BCK-ideal is a neutrosophic weak hyper BCK-ideal.

Proof. Let A = ([A.sub.T], [A.sub.I], [A.sub.F]) be a neutrosophic s-weak hyper BCK-ideal of H and let x,y [member of] H. Then there exist a, b, c [member of] x [omicron] y such that

[mathematical expression not reproducible]

Hence A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic weak hyper BCK-ideal of H.

We can conjecture that the converse of Theorem 3.17 is not true. But it is not easy to find an example of a neutrosophic weak hyper BCK-ideal which is not a neutrosophic s-weak hyper BCK-ideal.

Now we provide a condition for a neutrosophic weak hyper BCK-ideal to be a neutrosophic s-weak hyper BCK-ideal.

Theorem 3.18. If A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic weak hyper BCK-ideal of H which satisfies the condition (3.4), then A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic s-weak hyper BCK-ideal of H.

Proof. Let A = ([A.sub.T], [A.sub.I], [A.sub.F]) be a neutrosophic weak hyper BCK-ideal of H in which the condition (3.4) is true. Then there exist [a.sub.0], [b.sub.0], [c.sub.0] [member of] x [omicron] y such that [A.sub.T] ([a.sub.0]) = *[A.sub.T] (x [omicron] y), [A.sub.T]([b.sub.0],) = *[A.sub.I](x [omicron] y) and [A.sub.F]([c.sub.0],) = *[A.sub.F](x [omicron] y). Hence

[mathematical expression not reproducible]

Therefore A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic s-weak hyper BCK-ideal of H.

Remark 3.19. In a finite hyper BCK-algebra, every neutrosophic set satisfies the condition (3.4). Hence the concept of neutrosophic s-weak hyper BCK-ideal and neutrosophic weak hyper BCK-ideal coincide in a finite hyper BCK-algebra.

Theorem 3.20. A neutrosophic set A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic weak hyper BCK-ideal of H if and only if the nonempty sets U([A.sub.T], [[epsilon].sub.T]), U([A.sub.I], [[epsilon].sub.I]) and L([A.sub.F], [[epsilon].sub.F]) are weak hyper BCK-ideals of H for all [[epsilon].sub.T], [[epsilon].sub.I], [[epsilon].sub.F] [member of] [0, 1].

Proof. The proof is similar to the proof of Theorem 3.5.

Definition 3.21. A neutrosophic set A = ([A.sub.T], [A.sub.I], [A.sub.F]) in H is called a reflexive neutrosophic hyper BCK-ideal of H if it satisfies

[mathematical expression not reproducible] (3.16)

and

[mathematical expression not reproducible] (3.17)

Theorem 3.22. Every reflexive neutrosophic hyper BCK-ideal is a neutrosophic strong hyper BCK-ideal.

Proof. Straightforward.

Theorem 3.23. If A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a reflexive neutrosophic hyper BCK-ideal of H, then the nonempty sets U([A.sub.T], [[epsilon].sub.T]), U([A.sub.I], [[epsilon].sub.I]) and L([A.sub.F], [[epsilon].sub.F]) are reflexive hyper BCK-ideals of H for all [[epsilon].sub.T], [[epsilon].sub.I], [[epsilon].sub.F] [member of] [0,1].

Proof. Assume that U([A.sub.T], [[epsilon].sub.T]), U([A.sub.I], [[epsilon].sub.I]) and L([A.sub.F], [[epsilon].sub.F]) are nonempty for all [[epsilon].sub.T], [[epsilon].sub.I], [[epsilon].sub.F] [member of] [0, 1]. Let a [member of] U([A.sub.T],[[epsilon].sub.T]), b [member of] U([A.sub.I],[[epsilon].sub.I]) and c [member of] L([A.sub.F],[[epsilon].sub.F]). If A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a reflexive neutrosophic hyper BCK-ideal of H, then by Theorem 3.22, A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic strong hyper BCK-ideal of H, and so it is a neutrosophic hyper BCK-ideal of H. It follows from Theorem 3.5 that U([A.sub.T], [[epsilon].sub.T]), U([A.sub.I], [[epsilon].sub.I]) and L([A.sub.F], [[epsilon].sub.F]) are hyper BCK-ideals of H. For each x [member of] H, let [a.sub.0],[b.sub.0],[c.sub.0] [member of] x [??] x. Then

[mathematical expression not reproducible]

and so [a.sub.0] [member of] U([A.sub.T], [[epsilon].sub.T]), [b.sub.0] [member of] U([A.sub.I],[[epsilon].sub.I]) and [c.sub.0] [member of] L([A.sub.F],[[epsilon].sub.F]). Hence x [??] x [??] U([A.sub.T],[[epsilon].sub.T]), x [??] x [??] U([A.sub.I],[[epsilon].sub.I]) and x [??] x [??] L([A.sub.F],[[epsilon].sub.F]). Therefore U([A.sub.T],[[epsilon].sub.T]), U([A.sub.I],[[epsilon].sub.I]) and L([A.sub.F],[[epsilon].sub.F]) are reflexive hyper BCK-ideals of H.

Lemma 3.24 ([15]). Every reflexive hyper BCK-ideal is a strong hyper BCK-ideal.

We consider the converse of Theorem 3.23 by adding a condition.

Theorem 3.25. Let A = ([A.sub.T], [A.sub.I], [A.sub.F]) be a neutrosophic set in H satisfying the condition (3.14). If the nonempty sets U([A.sub.T],[[epsilon].sub.T]), U([A.sub.I],[[epsilon].sub.I]) and L([A.sub.F], [[epsilon].sub.F]) are reflexive hyper BCK-ideals of H for all [[epsilon].sub.T], [[epsilon].sub.I], [[epsilon].sub.F] [member of] [0,1], then A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a reflexive neutrosophic hyper BCK-ideal of H.

Proof. If the nonempty sets U([A.sub.T],[[epsilon].sub.T]), U([A.sub.I],[[epsilon].sub.I]) and L([A.sub.F], [[epsilon].sub.F]) are reflexive hyper BCK-ideals of H, then by Lemma 3.24 they are strong hyper BCK-ideals of H. By Theorem 3.12 that A = ([A.sub.T], [A.sub.I], [A.sub.F] is a neutrosophic strong hyper BCK-ideal of H. Hence the condition (3.17) is valid. Let x, y [member of] H. Then the sets U([A.sub.T], [A.sub.T](y)), U([A.sub.I],[A.sub.I](y)) and L([A.sub.F],[A.sub.F](y)) are reflexive hyper BCK-ideals of H, and so x [omicron] x [??] U([A.sub.T],[A.sub.T](y)), x [omicron] x [??] U([A.sub.I], [A.sub.I](y)) and x [omicron] x [??] L([A.sub.F],[A.sub.F](y)). Hence [A.sub.T](a) [greater than or equal to] [A.sub.T](y), [A.sub.I](b) [greater than or equal to] [A.sub.I](y) and [A.sub.F](c) [less than or equal to] [A.sub.F](y) for all a, b, c [member of] x [omicron] x and so [A.sub.T](x [omicron] x) [greater than or equal to] [A.sub.T](y), [A.sub.I](x [omicron] x) [greater than or equal to] [A.sub.I](y) and * [A.sub.F](x [omicron] x) [less than or equal to] [A.sub.F](y). Therefore A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a reflexive neutrosophic hyper BCK-ideal of H.

We provide conditions for a neutrosophic strong hyper BCK-ideal to be a reflexive neutrosophic hyper BCK-ideal.

Theorem 3.26. Let A = ([A.sub.T], [A.sub.I], [A.sub.F]) be a neutrosophic strong hyper BCK-ideal of H which satisfies the condition (3.14). Then A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a reflexive neutrosophic hyper BCK-ideal of H if and only if the following assertion is valid.

[mathematical expression not reproducible] (3.18)

Proof. It is clear that if A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a reflexive neutrosophic hyper BCK-ideal of H, then the condition (3.18) is valid.

Conversely, assume that A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a neutrosophic strong hyper BCK-ideal of H which satisfies the conditions (3.14) and (3.18). Then [A.sub.T] (0) [greater than or equal to] [A.sub.T] (y), [A.sub.I] (0) [greater than or equal to] [A.sub.I] (y) and [A.sub.F] (0) [less than or equal to] [A.sub.F] (y) for all y [member of] H. Hence

*[A.sub.T](x [omicron] x) [greater than or equal to] [A.sub.T](y), *[A.sub.I](x [omicron] x) [greater than or equal to] [A.sub.I](y) and *[A.sub.F](x [omicron] x) [less than or equal to] [A.sub.F](y). For any x, y [member of] H, let

[mathematical expression not reproducible]

Then U([A.sub.T],[[epsilon].sub.T]), U([A.sub.I],[[epsilon].sub.I]) and L([A.sub.F], [[epsilon].sub.F]) are strong hyper BCK-ideals of H by Theorem 3.11. Since A = ([A.sub.T], [A.sub.I], [A.sub.F]) satisfies the condition (3.14), there exist [a.sub.0], [b.sub.0],[c.sub.0] [member of] x [omicron] y such that

[A.sub.T] ([a.sub.0]) = *[A.sub.T] (x [omicron] y), [A.sub.I] ([b.sub.0]) = *[A.sub.I] (x [omicron] y), [A.sub.F] ([c.sub.0]) = *[A.sub.F] (x [omicron] y).

Hence [A.sub.T]([a.sub.0]) [greater than or equal to] [[epsilon].sub.T], [A.sub.I]([b.sub.0]) [greater than or equal to] [[epsilon].sub.I] and [A.sub.F]([c.sub.0]) [less than or equal to] [[epsilon].sub.F], that is, [a.sub.0] [member of] U([A.sub.T],[[epsilon].sub.T]), [b.sub.0] [member of] U([A.sub.I],[[epsilon].sub.I]) and [c.sub.0] [member of] L([A.sub.F], [[epsilon].sub.F]). Hence (x [omicron] y) [intersection] U ([A.sub.T],[[epsilon].sub.T]) [not equal to] [phi], (x [omicron] y) [intersection] U ([A.sub.I],[[epsilon].sub.I]) [not equal to] [phi] and (x [omicron] y) [intersection] L([A.sub.F], [[epsilon].sub.F]) [not equal to] [phi]. Since y [member of] U ([A.sub.T],[[epsilon].sub.T]) [intersection] U ([A.sub.I],[[epsilon].sub.I]) [intersection] L([A.sub.F], [[epsilon].sub.F]), by (2.9), x [member of] U ([A.sub.T],[[epsilon].sub.T]) [intersection] U ([A.sub.I],[[epsilon].sub.I]) [intersection] L([A.sub.F], [[epsilon].sub.F]). Thus

[mathematical expression not reproducible]

Therefore A = ([A.sub.T], [A.sub.I], [A.sub.F]) is a reflexive neutrosophic hyper BCK-ideal of H.

4 Conclusions

We have introduced the notions of neutrosophic (strong, weak, s-weak) hyper BCK-ideal and reflexive neutrosophic hyper BCK-ideal. We have considered their relations and related properties. We have discussed characterizations of neutrosophic (weak) hyper BCK-ideal, and have given conditions for a neutrosophic set to be a (reflexive) neutrosophic hyper BCK-ideal and a neutrosophic strong hyper BCK-ideal. We have provided conditions for a neutrosophic weak hyper BCK-ideal to be a neutrosophic s-weak hyper BCK-ideal, and have provided conditions for a neutrosophic strong hyper BCK-ideal to be a reflexive neutrosophic hyper BCK-ideal.

Acknowledgement

The authors wish to thank the anonymous reviewers for their valuable suggestions.

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Received: February 16, 2019. Accepted: April 330, 2019.

S. Khademan (1), M. M. Zahedi (2,*), R. A. Borzooei (3), Y. B. Jun (3,4)

(1) Department of Mathematics, Tarbiat Modares University, Tehran, Iran.

E-mail: somayeh.khademan@modares.ac.ir, Khademans@gmail.com

(2) Department of Mathematics, Graduate University of Advanced Technology, Kerman, Iran.

E-mail: zahed_mm@kgut.ac.ir, zahedi_mm@yahoo.com

(3) Department of Mathematics, Shahid Beheshti University, Tehran 1983963113, Iran.

E-mail: borzooei@sbu.ac.ir

(4) Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea.

E-mail: skywine@gmail.com

(*) Correspondence: M. M. Zahedi (zahedi_mm@kgut.ac.ir, zahedi_mm@yahoo.com)
Table 1: Cayley table for the binary operation "[omicron]"

[omicron]   0     a       b

0          {0}   {0}     {0}
a          {a}   {0,a}   {0,a}
b          {b}   {a,b}   {0,a,b}

Table 2: Tabular representation of A = ([A.sub.T], [A.sub.I], [A.sub.F])

H   [A.sub.T] (x)   [A.sub.I] (x)   [A.sub.F] (x)

0       0.77            0.65            0.08
a       0.55            0.47            0.57
b       0.11            0.27            0.69

Table 3: Cayley table for the binary operation "[omicron]"

[omicron]    0     a     b

0           {0}   {0}   {0}
a           {a}   {0}   {a}
b           {b}   {b}   {0,b}

Table 4: Tabular representation of A = ([A.sub.T], [A.sub.I], [A.sub.F])

H   [A.sub.T] (x)   [A.sub.I] (x)   [A.sub.F] (x)

0       0.86            0.75            0.09
a       0.65            0.57            0.17
b       0.31            0.37            0.29

Table 5: Cayley table for the binary operation "[omicron]"

[omicron]    0     a     b        c

0           {0}   {0}   {0}      {0}
a           {a}   {0}   {0}      {0}
b           {b}   {b}   {0}      {0}
c           {c}   {c}   {b, c}   {0, b, c}

Table 6: Tabular representation of A = ([A.sub.T], [A.sub.I], [A.sub.F])

H   [A.sub.T] (x)   [A.sub.I] (x)   [A.sub.F] (x)

0      0.98            0.85            0.02
a      0.81            0.69            0.19
b      0.56            0.43            0.32
c      0.34            0.21            0.44
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Author:Khademan, S.; Zahedi, M.M.; Borzooei, R.A.; Jun, Y.B.
Publication:Neutrosophic Sets and Systems
Date:Aug 27, 2019
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