EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 2, 2020, 200-215
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Applications of Neutrosophic N -Structures in n-Ary
Groupoids

Amornrat Rattana1,∗, Ronnason Chinram1,2

1 Department of Mathematics and Statistics, Faculty of Science, Prince of Songkla
University, Hat Yai, Songkhla 90110, Thailand
2 Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand

Abstract. This paper includes the notions of neutrosophic n-aryN -subgroupoids of n-ary groupoids
and some properties.

2020 Mathematics Subject Classifications: 20N15, 03B80

Key Words and Phrases: neutrosophic N -structures, n-ary groupoids, neutrosophic n-ary N -
subgroupoids, ε-neutrosophic n-ary N -subgroupoids.

1. Introduction

In 1965, the degree of membership/truth (t) and the fuzzy set were introduced by
Zadeh [12]. Atanassov [1] introduced the degree of nonmembership/falsehood (f) and
defined the intuitionistic fuzzy set in 1986. Neutrosophy, means knowledge of neutral,
is a branch of philosophy introduced as a theory of generalization of dialectic in 1995
by Smarandache. He proposed the term neutrosophic because neutrosophic originally
comes from neutrosophy. In 1999, he introduced the concept of neutrosophic logics [9] and
introduced the degree of indeterminancy/neuterality (i) and proposed the neutrosophic
set on three components

(t, i, f) =(truth, indeterminacy, falsehood).

Jun et al. [11] introduced a negative-valued function and defined N -structures in 2009.
Khan et al. [4] investigated the notion of neutrosophic N -structures and their applications
in semigroups in 2017. Jun et al. [10, 11] considered neutrosophic N -structures applied
to BCK/BCI-algebras. Song et al. [8] proposed neutrosophic commutative N -ideals in
BCK-algebras in 2017. Rangsuk et al. [6] discussed neutrosophic N -structures and their
applications in UP-algebras.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i2.3634

Email addresses: amornrat.r@psu.ac.th (A. Rattana), ronnason.c@psu.ac.th (R. Chinram)

http://www.ejpam.com 200 c© 2020 EJPAM All rights reserved.



A. Rattana, R. Chinram / Eur. J. Pure Appl. Math, 13 (2) (2020), 200-215 201

Algebraic systems with one n-ary operation, for n > 2, have been widely investigated
(see, e.g., [2, 3, 5, 7]). Algebraic n-ary systems have been applied in several fields of
mathematics.

The purpose of this paper is to investigate the extension of neutrosophic N -structures
in semigroups [4] to n-ary groupoids. Some basic notations and definitions will be pre-
sented in section 2. In section 3, we extend the results of neutrosophic N -structures and
their applications in semigroups to n-ary groupoids. Section 4 contains a brief summary
of this paper.

2. Preliminaries

The aim of this section is to review some notations and definitions of n-ary groupoids
and neutrosophic N -structures which can be also found in [2–4].

2.1. n-ary groupoids

Definition 1. Let S be a nonempty set. The set S together with an n-ary operation
f : Sn → S, where n ≥ 2, is called an n-ary groupoid and is denoted by (S, f).

According to the general convention used in the theory of n-ary groupoids, the sequence
of elements xi, xi+1, . . . , xj is denoted by xji . In the case j < i, it is the empty symbol.
If xi+1 = xi+2 = . . . = xi+t = x, then we write x(t) instead of xi+ti+1. In this convention,
f(x1, x2, . . . , xn) = f(xn1 ), and

f(x1, . . . , xi, x, . . . , x︸ ︷︷ ︸
t

, xi+t+1, . . . , xn) = f(xi1, x
(t), xni+t+1).

Definition 2. A nonempty subset T of an n-ary groupoids (S, f) is an n-ary subgroupoid
of S if (T, f) is an n-ary groupoid, i.e., if it is closed under the operation f .

2.2. Neutrosophic N -structures

Definition 3. A neutrosophic N -structure over X is defined to be the structure

XN :=
X

(TN , IN , FN )
=

{
x

(TN (x), IN (x), FN (x))
| x ∈ X

}
where TN , IN and FN are N -functions on X which are called the truth membership func-
tion, the indeterminacy membership function and the falsity membership function on X,
respectively.

Definition 4. Let XN =
X

(TN , IN , FN )
and XM =

X

(TM , IM , FM )
be neutrosophic N -

structures over X.



A. Rattana, R. Chinram / Eur. J. Pure Appl. Math, 13 (2) (2020), 200-215 202

(1) XN is a neutrosophic N -substructure of XM over X, denoted by XN ⊆ XM , if it
satisfies the conditions

TN (x) ≥ TM (x), IN (x) ≤ IM (x), FN (x) ≥ FM (x)

for all x ∈ X.
We have that XN ⊆ XM and XM ⊆ XN if and only if XN = XM .

(2) The union of XN and XM , denoted it briefly by XN∪M , is defined to be a neutro-
sophic N -structure

XN∪M =
X

(TN∪M , IN∪M , FN∪M )

where

TN∪M (x) =
∧
{TN (x), TM (x)} ,

IN∪M (x) =
∨
{IN (x), IM (x)} ,

FN∪M (x) =
∧
{FN (x), FM (x)} .

(3) The intersection of XN and XM , written it simply as XN∩M , is defined to be a
neutrosophic N -structure

XN∩M =
X

(TN∩M , IN∩M , FN∩M )

where

TN∩M (x) =
∨
{TN (x), TM (x)} ,

IN∩M (x) =
∧
{IN (x), IM (x)} ,

FN∩M (x) =
∨
{FN (x), FM (x)} .

Definition 5. Let XN :=
X

(TN , IN , FN )
be a neutrosophic N -structure over X. The

complement of XN , denoted by XNc, is defined to be a neutrosophic N -structure

XNc :=
X

(TNc , INc , FNc)

over X, where

TNc(x) = −1− TN (x), INc(x) = −1− IN (x), FNc(x) = −1− FN (x)

for all x ∈ X.



A. Rattana, R. Chinram / Eur. J. Pure Appl. Math, 13 (2) (2020), 200-215 203

Definition 6. Let XN be a neutrosophic N -structure over X and let α, β, γ be real numbers
such that α, β, γ ∈ [−1, 0]. Define the sets

TαN = {x ∈ X | TN (x) ≤ α},

IβN = {x ∈ X | IN (x) ≥ β},
F γN = {x ∈ X | FN (x) ≤ γ}.

We call a set

XN (α, β, γ) = {x ∈ X | TN (x) ≤ α, IN (x) ≥ β, FN (x) ≤ γ}

an (α, β, γ)-level set of XN .

For the convenience, we note that

XN (α, β, γ) = TαN ∩ I
β
N ∩ F

γ
N .

From now on, an n-ary groupoid X denotes the universe of discourse unless otherwise
specified.

3. Main results

In this section, we will look closely at neutrosophic n-aryN -subgroupoids, the (α, β, γ)-
level set, the intersection of neutrosophic n-ary N -subgroupoids, neutrosophic n-ary N -
subgroupoid products, ε-neutrosophic n-ary N -subgroupoids, homomorphic preimage of
the neutrosophic n-ary N -subgroupoids and onto homomorphic image of the neutrosophic
n-ary N -subgroupoids.

Definition 7. Let XN :=
X

(TN , IN , FN )
be a neutrosophic structure over an n-ary groupoid

X. Then XN is called a neutrosophic n-ary N -subgoupoid of X if the following conditions
are valid:

TN (f(xn1 )) ≤
∨
{TN (x1), . . . , TN (xn)},

IN (f(xn1 )) ≥
∧
{IN (x1), . . . , IN (xn)},

FN (f(xn1 )) ≤
∨
{FN (x1), . . . , FN (xn)},

for all x1, x2, . . . , xn ∈ X.

Theorem 1. Let XN be a neutrosophic n-ary N -subgroupoid of an n-ary groupoid X
and let α, β, γ ∈ [−1, 0]. If the (α, β, γ)-level set of XN is nonempty, then it is an n-ary
subgroupoid of X.



A. Rattana, R. Chinram / Eur. J. Pure Appl. Math, 13 (2) (2020), 200-215 204

Proof. Let x1, . . . , xn ∈ XN (α, β, γ). Then

TN (x1) ≤ α, IN (x1) ≥ β, FN (x1) ≤ γ, . . . , TN (xn) ≤ α, IN (xn) ≥ β, FN (xn) ≤ γ.

It follows that

TN (f(xn1 )) ≤
∨
{TN (x1), . . . , TN (xn)} ≤ α,

IN (f(xn1 )) ≥
∧
{IN (x1), . . . , IN (xn)} ≥ β,

FN (f(xn1 )) ≤
∨
{FN (x1), . . . , FN (xn)} ≤ γ.

Therefore f(xn1 ) ∈ XN (α, β, γ). This implies that XN (α, β, γ) is an n-ary subgroupoid of
X.

Theorem 2. Let XN be a neutrosophic N -structure over an n-ary groupoid X. If TαN , I
β
N

and F γN are n-ary subgroupoids of X for all α, β, γ ∈ [−1, 0], then XN is a neutrosophic
n-ary N -subgroupoid of X.

Proof. We prove this theorem by contradiction. Assume that there exist x1, . . . , xn ∈ X
such that TN (f(xn1 )) >

∨
{TN (x1), . . . , TN (xn)}. Then

TN (f(xn1 )) > tα ≥
∨
{TN (x1), . . . , TN (xn)}

for some tα ∈ [−1, 0). Thus x1, . . . , xn ∈ T tαN but f(xn1 ) /∈ T tαN , which is a contradiction.
Thus

TN (f(xn1 )) ≤
∨
{TN (x1), . . . , TN (xn)}

for all x1, . . . , xn ∈ X.
We now assume that IN (f(xn1 )) <

∨
{IN (x1), . . . , IN (xn)} for some x1, . . . , xn ∈ X.

Then
IN (f(xn1 )) < tβ ≤

∨
{IN (x1), . . . , IN (xn)}

for some tβ ∈ [−1, 0). Thus x1, . . . , xn ∈ I
tβ
N but f(xn1 ) /∈ I

tβ
N . This is a contradiction.

Hence
IN (f(xn1 )) ≥

∧
{IN (x1), . . . , IN (xn)}

for all x1, . . . , xn ∈ X.
It remains to prove that

FN (f(xn1 )) ≤
∨
{FN (x1), . . . , FN (xn)}

for all x1, . . . , xn ∈ X. Suppose contrary to our claim that there are x1, . . . , xn ∈ X such
that FN (f(xn1 )) >

∨
{FN (x1), . . . , FN (xn)}. Then

FN (f(xn1 )) > tγ ≥
∨
{FN (x1), . . . , FN (xn)}

for some tγ ∈ [−1, 0). Thus x1, . . . , xn ∈ F
tγ
N but f(xn1 ) /∈ F tγN , which is a contradiction.

Therefore XN is a neutrosophic n-ary N -subgroupoid of X.



A. Rattana, R. Chinram / Eur. J. Pure Appl. Math, 13 (2) (2020), 200-215 205

Theorem 3. Let XN :=
X

(TN , IN , FN )
and XM :=

X

(TM , IM , FM )
be two neutrosophic

n-ary N -subgroupoids over X. Then XN∩M is also a neutrosophic n-ary N -subgroupoid
of X.

Proof. Let x1, . . . , xn ∈ X. We obtain

TN∩M (f(xn1 )) =
∨
{TN (f(xn1 )), TM (f(xn1 ))}

≤
∨{∨

{TN (x1), . . . , TN (xn)} ,
∨
{TM (x1), . . . , TM (xn)}

}
=
∨{∨

{TN (x1), TM (x1)} , . . . ,
∨
{TN (xn), TM (xn)}

}
=
∨
{TN∩M (x1), . . . , TN∩M (xn)} ,

IN∩M (f(xn1 )) =
∧
{IN (f(xn1 )), IM (f(xn1 ))}

≥
∧ {∧

{IN (x1), . . . , IN (xn)} ,
∧
{IM (x1), . . . , IM (xn)}

}
=
∧ {∧

{IN (x1), IM (x1)} , . . . ,
∧
{IN (xn), IM (xn)}

}
=
∧
{IN∩M (x1), . . . , IN∩M (xn)} ,

FN∩M (f(xn1 )) =
∨
{FN (f(xn1 )), FM (f(xn1 ))}

≤
∨{∨

{FN (x1), . . . , FN (xn)} ,
∨
{FM (x1), . . . , FM (xn)}

}
=
∨{∨

{FN (x1), FM (x1)} , . . . ,
∨
{FN (xn), FM (xn)}

}
=
∨
{FN∩M (x1), . . . , FN∩M (xn)}

for all x1, . . . , xn ∈ X. Therefore XN∩M is a neutrosophic n-ary N -subgroupoid of X.

Corollary 1. Let {XNi | i ∈ N} be a family of neutrosophic n-ary N -subgroupoids of an
n-ary groupoid X.Then

⋂
i∈NXNi is also a neutrosophic n-ary N -subgroupoid of X.

For each i ∈ {1, 2, . . . , n}, let XNi :=
X

(TNi , INi , FNi)
be a neutrosophic N -structure

over an n-ary groupoid (X, f). Then a neutrosophic N -structure over X

XN1 � . . .�XNn =
X

(TN1 � . . .� TNn , IN1 � . . .� INn , FN1 � . . .� FNn)

=

{
x

TN1 � . . .� TNn(x), IN1 � . . .� INn(x), FN1 � . . .� FNn(x)

∣∣∣x ∈ X}
is defined to be a neutrosophic N -product of XN1 , XN2 , . . . , XNn where

TN1�. . .�TNn(x) =


∧

x=f(xn1 )

{TN1(x1) ∨ ... ∨ TNn(xn)} , if x = f(xn1 ) ∃x1, ..., xn ∈ X,

0, otherwise,



A. Rattana, R. Chinram / Eur. J. Pure Appl. Math, 13 (2) (2020), 200-215 206

IN1 � . . .� INn(x) =


∨

x=f(xn1 )

{IN1(x1) ∧ ... ∧ INn(xn)} , if x = f(xn1 ) ∃x1, ..., xn ∈ X,

−1, otherwise,

and

FN1�. . .�FNn(x) =


∧

x=f(xn1 )

{FN1(x1) ∨ ... ∨ FNn(xn)} , if x = f(xn1 ) ∃x1, ..., xn ∈ X,

0, otherwise.

If XN = XN1 = XN2 = . . . = XNn , then XN1 � . . .�XNn is denoted by �(XN )(n). For

any x ∈ X, the element
x

�(TN )(n)(x),�(IN )(n)(x),�(FN )(n)(x)
is denoted by

�(XN )(n)(x) :=
(
�(TN )(n)(x),�(IN )(n)(x),�(FN )(n)(x)

)
.

Theorem 4. A neutrosophic N -structure XN over X is a neutrosophic N -subgroupoid of
X if and only if �(XN )(n) ⊆ XN .

Proof. We first prove that if a neutrosophic N -structure XN over X is a neutrosophic
n-ary N -subgroupoid of X, then �(XN )(n) ⊆ XN . We assume that XN is a neutrosophic
n-ary N -subgroupoid of X and let x ∈ X. If x 6= f(xn1 ) for all x1, . . . , xn ∈ X, then
this clearly forces �(XN )(n) ⊆ XN . Suppose that there are x1, . . . , xn ∈ X such that
x = f(xn1 ), we obtain

�(TN )(n)(x) =
∧

x=f(xn1 )

{TN (x1) ∨ . . . ∨ TN (xn)} ≥
∧

x=f(xn1 )

TN (f(xn1 )) = TN (x),

�(IN )(n)(x) =
∨

x=f(xn1 )

{IN (x1) ∧ . . . ∧ IN (xn)} ≤
∨

x=f(xn1 )

IN (f(xn1 )) = IN (x),

�(FN )(n)(x) =
∧

x=f(xn1 )

{FN (x1) ∨ . . . ∨ FN (xn)} ≥
∧

x=f(xn1 )

FN (f(xn1 )) = FN (x).

Therefore �(XN )(n) ⊆ XN .
Conversely, letXN be any neutrosophic n-aryN -subgroupoid ofX such that�(XN )(n) ⊆

XN . We only need to show that XN is a neutrosophic n-ary N -subgroupoid of X. Let
x1, . . . , xn be elements of X and let x = f(xn1 ). Then

TN (f(xn1 )) = TN (x) ≤ �(TN )(n)(x) =
∧

x=f(xn1 )

{TN (x1) ∨ . . . ∨ TN (xn)}

≤ TN (x1) ∨ . . . ∨ TN (xn),

IN (f(xn1 )) = IN (x) ≥ �(IN )(n)(x) =
∨

x=f(xn1 )

{IN (x1) ∧ . . . ∧ IN (xn)}

≥ IN (x1) ∧ . . . ∧ IN (xn),



A. Rattana, R. Chinram / Eur. J. Pure Appl. Math, 13 (2) (2020), 200-215 207

FN (f(xn1 )) = FN (x) ≤ �(FN )(n)(x) =
∧

x=f(xn1 )

{FN (x1) ∨ . . . ∨ FN (xn)}

≤ FN (x1) ∨ . . . ∨ FN (xn).

Therefore XN is a neutrosophic n-ary N -subgroupoid of X.

Theorem 5. Let X be an n-ary groupoid with identity e and let XN :=
X

(TN , IN , FN )
be

a neutrosophic n-ary N -subgroupoid over X such that

XN (e) ≥ XN (x)

for all x ∈ X, that is, TN (e) ≤ TN (x), IN (e) ≥ IN (x) and FN (e) ≤ FN (x) for all x ∈ X.
If XN is a neutrosophic n-ary N -subgroupoid of X, then �(XN )(n) = XN .

Proof. For any x ∈ X, we have

�(TN )(n)(x) =
∧

x=f(xn1 )

{TN (x1) ∨ . . . ∨ TN (xn)} ≤ TN (x) ∨ TN (e) = TN (x),

�(IN )(n)(x) =
∨

x=f(xn1 )

{IN (x1) ∧ . . . ∧ IN (xn)} ≥ IN (x) ∧ IN (e) = IN (x),

�(FN )(n)(x) =
∧

x=f(xn1 )

{FN (x1) ∨ . . . ∨ FN (xn)} ≤ FN (x) ∨ FN (e) = FN (x).

This shows that XN ⊆ �(XN )(n). From Theorem 4, we already have �(XN )(n) ⊆ XN .
Then �(XN )(n) = XN .

Definition 8. A neutrosophic N -structure XN over X is said to be an ε-neutrosophic
n-ary N -subgroupoid of X if the conditions

TN (f(xn1 )) ≤
∨
{TN (x1), . . . , TN (xn), εT } ,

IN (f(xn1 )) ≥
∧
{IN (x1), . . . , IN (xn), εI} ,

FN (f(xn1 )) ≤
∨
{FN (x1), . . . , FN (xn), εF } ,

hold for all x1, . . . , xn ∈ X where εT , εI , εF ∈ [−1, 0].

Proposition 1. Let XN be an ε-neutrosophic n-ary N -subgroupoid of X. If XN (x) ≤
(εT , εI , εF ), that is, TN (x) ≥ εT , IN (x) ≤ εI , FN (x) ≥ εF for all x ∈ X, then XN is a
neutrosophic n-ary N -subgroupoid of X.

Theorem 6. Let XN be a neutrosophic N -structure over X and let α, β, γ be real numbers
on the interval [−1, 0]. If XN is an ε-neutrosophic n-ary N -subgroupoid of X, then the
(α, β, γ)-level set of XN is an n-ary subgroupoid of X whenever (α, β, γ) ≤ (εT , εI , εF ),
that is α ≥ εT , β ≤ εI , and γ ≥ εF .



A. Rattana, R. Chinram / Eur. J. Pure Appl. Math, 13 (2) (2020), 200-215 208

Proof. Assume that XN (α, β, γ) 6= ∅ for α, β, γ ∈ [−1, 0]. Let x1, . . . , xn ∈ XN (α, β, γ).
Then TN (x1) ≤ α, IN (x1) ≥ β, FN (x1) ≤ γ, . . . , TN (xn) ≤ α, IN (xn) ≥ β, FN (xn) ≤ γ. It
follows that

TN (f(xn1 )) ≤
∨
{TN (x1), . . . , TN (xn), εT } ≤

∨
{α, εT } = α,

IN (f(xn1 )) ≥
∧
{IN (x1), . . . , IN (xn), εI} ≥

∧
{β, εI} = β,

FN (f(xn1 )) ≤
∨
{FN (x1), . . . , FN (xn), εF } ≤

∨
{γ, εF } = γ.

Hence f(xn1 ) ∈ XN (α, β, γ). It follows that XN (α, β, γ) is an n-ary subgroupoid of X.

Theorem 7. Let XN be a neutrosophic N -structure over X and let α, β, γ be real numbers
on the interval [−1, 0]. If TαN , I

β
N and F γN are n-ary subgroupoids of X for all εT , εI , εF ∈

[−1, 0] and (α, β, γ) ≤ (εT , εI , εF ), then XN is an ε-neutrosophic n-ary N -subgroupoid of
X.

Proof. We prove this theorem by contradiction. We begin the proof by assuming that

TN (f(xn1 )) >
∨
{TN (x1), . . . , TN (xn), εT } .

for some x1, . . . , xn ∈ X. Then

TN (f(xn1 )) > tα ≥
∨
{TN (x1), . . . , TN (xn), εT }

for some tα ∈ [−1, 0). It follows that x1, . . . , xn ∈ T tαN , f(xn1 ) /∈ T tαN and tα ≥ εT . This is
a contradiction since T tαN is an n-ary subgroupoid of X by hypothesis. Thus

TN (f(xn1 )) ≤
∨
{TN (x1), . . . , TN (xn), εT }

for all x1, . . . , xn ∈ X.
Suppose now that there are x1, . . . , xn ∈ X such that

IN (f(xn1 )) <
∧
{IN (x1), . . . , IN (xn), εI} .

Then
IN (f(xn1 )) < tβ ≤

∧
{IN (x1), . . . , IN (xn), εI}

for some tβ ∈ [−1, 0). It follows that x1, . . . , xn ∈ I
tβ
N , f(xn1 ) /∈ I

tβ
N and tβ ≤ εI . This

contradicts to the fact that I
tβ
N is an n-ary subgroupoid of X. Thus

IN (f(xn1 )) ≥
∧
{IN (x1), . . . , IN (xn), εI}

for all x1, . . . , xn ∈ X.
Similarly, assume that

FN (f(xn1 )) >
∨
{FN (x1), . . . , FN (xn), εF }



A. Rattana, R. Chinram / Eur. J. Pure Appl. Math, 13 (2) (2020), 200-215 209

for some x1, . . . , xn ∈ X. Then

FN (xn1 ) > tγ ≥
∨
{FN (x1), . . . , FN (xn), εF }

for some tγ ∈ [−1, 0). It implies that x1, . . . , xn ∈ F
tγ
N , f(xn1 ) /∈ F tγN and tγ ≥ εF . This is

a contradiction since F
tγ
N is an n-ary subgroupoid of X. Thus

FN (f(xn1 )) ≤
∨
{FN (x1), . . . , FN (xn), εF }

for all x1, . . . , xn ∈ X. Therefore XN is an ε-neutrosophic n-ary N -subgroupoid of X.

Theorem 8. Let εT , εI , εF , δT , δI , δF ∈ [−1, 0]. Let XN and XM be an ε-neutrosophic
n-ary N -subgroupoid and a δ-neutrosophic n-ary N -subgroupoid of X, respectively. The
intersection of XN and XM is a ξ-neutrosophic n-ary N -subgroupoid of X for ξ := ε ∧ δ
where (ξT , ξI , ξF ) = (εT ∨ δT , εI ∧ δI , εF ∨ δF ).

Proof. For any x1, . . . , xn ∈ X, we have

TN∩M (f(xn1 )) =
∨
{TN (f(xn1 )), TM (f(xn1 ))}

≤
∨{∨

{TN (x1), . . . , TN (xn), εT } ,
∨
{TM (x1), . . . , TM (xn), δT }

}
≤
∨{∨

{TN (x1), . . . , TN (xn), ξT } ,
∨
{TM (x1), . . . , TM (xn), ξT }

}
=
∨{∨

{TN (x1), TM (x1), ξT } , . . . ,
∨
{TN (xn), TM (xn), ξT }

}
=
∨{∨

{TN (x1), TM (x1)} , . . . ,
∨
{TN (xn), TM (xn)} , ξT

}
=
∨
{TN∩M (x1), . . . , TN∩M (xn), ξT } ,

IN∩M (f(xn1 )) =
∧
{IN (f(xn1 )), IM (f(xn1 ))}

≥
∧{∧

{IN (x1), . . . , IN (xn), εI} ,
∧
{IM (x1), . . . , IM (xn), δI}

}
≥
∧{∧

{IN (x1), . . . , IN (xn), ξI} ,
∧
{IM (x1), . . . , IM (xn), ξI}

}
=
∧{∧

{IN (x1), IM (x1), ξI} , . . . ,
∧
{IN (xn), IM (xn), ξI}

}
=
∧{∧

{IN (x1), IM (x1)} , . . . ,
∧
{IN (xn), IM (xn)} , ξI

}
=
∧
{IN∩M (x1), . . . , IN∩M (xn), ξI} ,

FN∩M (f(xn1 )) =
∨
{FN (f(xn1 )), FM (f(xn1 ))}

≤
∨{∨

{FN (x1), . . . , FN (xn), εF } ,
∨
{FM (x1), . . . , FM (xn), δF }

}
≤
∨{∨

{FN (x1), . . . , FN (xn), ξF } ,
∨
{FM (x1), . . . , FM (xn), ξF }

}
=
∨{∨

{FN (x1), FM (x1), ξF } , . . . ,
∨
{FN (xn), FM (xn), ξF }

}



A. Rattana, R. Chinram / Eur. J. Pure Appl. Math, 13 (2) (2020), 200-215 210

=
∨{∨

{FN (x1), FM (x1)} , . . . ,
∨
{FN (xn), FM (xn)} , ξF

}
=
∨
{FN∩M (x1), . . . , FN∩M (xn), ξF } .

Therefore XN∩M is a ξ-neutrosophic n-ary N -subgroupoid of X.

Theorem 9. Let XN be an ε-neutrosophic n-ary N -subgroupoid of X. If

κ := (κT , κI , κF ) =

(∨
x∈X
{TN (x)},

∧
x∈X
{IN (x)},

∨
x∈X
{FN (x)}

)

then the set

Ω := {x ∈ X | TN (x) ≤ κT ∨ εT , IN (x) ≥ κI ∧ εI , FN (x) ≤ κF ∨ εF }

is an n-ary subgroupoid of X.

Proof. Let x1, . . . , xn ∈ Ω for any x1, . . . , xn ∈ X. Then

TN (x1) ≤ κT ∨ εT =
∨
x1∈X

{TN (x1)} ∨ εT ,

IN (x1) ≥ κI ∧ εI =
∧
x1∈X

{IN (x1)} ∧ εI ,

FN (x1) ≤ κF ∨ εF =
∨
x1∈X

{FN (x1)} ∨ εF ,

...

TN (xn) ≤ κT ∨ εT =
∨
xn∈X

{TN (xn)} ∨ εT ,

IN (xn) ≥ κI ∧ εI =
∧
xn∈X

{IN (xn)} ∧ εI ,

FN (xn) ≤ κF ∨ εF =
∨
xn∈X

{FN (xn)} ∨ εF .

It follows that

TN (f(xn1 )) ≤
∨
{TN (x1), . . . , TN (xn), εT }

≤
∨
{κT ∨ εT , . . . , κT ∨ εT , εT }

= κT ∨ εT ,

IN (f(xn1 )) ≥
∧
{IN (x1), . . . , IN (xn), εI}

≥
∧
{κI ∧ εI , . . . , κI ∧ εI , εI}

= κI ∧ εI ,



A. Rattana, R. Chinram / Eur. J. Pure Appl. Math, 13 (2) (2020), 200-215 211

FN (f(xn1 )) ≤
∨
{FN (x1), . . . , FN (xn), εF }

≤
∨
{κF ∨ εF , . . . , κF ∨ εF , εF }

= κF ∨ εF ,

Then f(xn1 ) ∈ Ω. Hence Ω is an n-ary subgroupoid of X.

Let X and Y be sets, g : X → Y be a function, YN :=
Y

(TN , IN , FN )
be a neutrosophic

N -structure over Y with ε = (εT , εI , εF ). An ε-neutrosophic N -structure over X is defined

by Xε
N :=

X

(T εN , I
ε
N , F

ε
N )

where

T εN : X → [−1, 0], x 7→ ∨{TN (g(x)), εT },
IεN : X → [−1, 0], x 7→ ∧{IN (g(x)), εI},
F εN : X → [−1, 0], x 7→ ∨{FN (g(x)), εF }.

Theorem 10. Let X,Y be two n-ary groupoids and g : X → Y be a homomorphism. If

a neutrosophic N -structure YN :=
Y

(TN , IN , FN )
over Y is an ε-neutrosophic n-ary N -

subgroupoid of Y , then Xε
N :=

X

(T εN , I
ε
N , F

ε
N )

is an ε-neutrosophic n-ary N -subgroupoid of

X.

Proof. For any x1, . . . , xn ∈ X, we have

T εN (f(xn1 )) =
∨
{TN (g(f(xn1 ))), εT }

=
∨
{TN (g(x1) . . . g(xn)), εT }

≤
∨
{
∨
{TN (g(x1)), . . . , TN (g(xn)), εT }, εT }

=
∨
{
∨
{TN (g(x1)), εT }, . . . ,

∨
{TN (g(xn)), εT }, εT }

=
∨
{T εN (x1), . . . , T

ε
N (xn), εT },

IεN (f(xn1 )) =
∧
{IN (g(f(xn1 ))), εI}

=
∧
{IN (g(x1) . . . g(xn)), εI}

≥
∧
{
∧
{IN (g(x1)), . . . , IN (g(xn)), εI}, εI}

=
∧
{
∧
{IN (g(x1)), εI}, . . . ,

∧
{IN (g(xn)), εI}, εI}

=
∧
{IεN (x1), . . . , I

ε
N (xn), εI},

F εN (f(xn1 )) =
∨
{FN (g(f(xn1 ))), εF }

=
∨
{FN (g(x1) . . . g(xn)), εF }



A. Rattana, R. Chinram / Eur. J. Pure Appl. Math, 13 (2) (2020), 200-215 212

≤
∨
{
∨
{FN (g(x1)), . . . , FN (g(xn)), εF }, εF }

=
∨
{
∨
{FN (g(x1)), εF }, . . . ,

∨
{FN (g(xn)), εF }, εF }

=
∨
{F εN (x1), . . . , F

ε
N (xn), εF }.

Therefore Xε
N :=

X

(T εN , I
ε
N , F

ε
N )

is an ε-neutrosophic n-ary N -subgroupoid of X.

Let X,Y be two sets and g : X → Y be a function. If YM :=
Y

(TM , IM , FM )
is a

neutrosophic N -structure over Y , then the preimage of YM under g is a neutrosophic
N -structure over X defined by

g−1(YM ) :=
X

(g−1(TM ), g−1(IM ), g−1(FM ))

where g−1(TM )(x) = TM (g(x)), g−1(IM )(x) = IM (g(x)), and g−1(FM )(x) = FM (g(x)) for
all x ∈ X.

Theorem 11. Let X,Y be two n-ary groupoids and g : X → Y be a homomorphism. If

YM :=
Y

(TM , IM , FM )
is a neutrosophic n-ary N -subgroupoid of Y , then the preimage of

YM under g,

g−1(YM ) =
X

(g−1(TM ), g−1(IM ), g−1(FM ))
,

is a neutrosophic n-ary N -subgroupoid of X.

Proof. For any x1, . . . , xn ∈ X, we have

g−1(TM )(f(xn1 )) = TM (g(f(xn1 ))) = TM (g(x1) . . . g(xn))

≤
∨
{TM (g(x1)), . . . , TM (g(xn))}

=
∨{

g−1(TM )(x1), . . . , g
−1(TM )(xn)

}
,

g−1(IM )(f(xn1 )) = IM (g(f(xn1 ))) = IM (g(x1) . . . g(xn))

≥
∧
{IM (g(x1)), . . . , IM (g(xn))}

=
∧{

g−1(IM )(x1), . . . , g
−1(IM )(xn)

}
,

g−1(FM )(f(xn1 )) = FM (g(f(xn1 ))) = FM (g(x1) . . . g(xn))

≤
∨
{FM (g(x1)), . . . , FM (g(xn))}

=
∨{

g−1(FM )(x1), . . . , g
−1(FM )(xn)

}
.

Therefore g−1(YM ) is a neutrosophic n-ary N -subgroupoid of X.



A. Rattana, R. Chinram / Eur. J. Pure Appl. Math, 13 (2) (2020), 200-215 213

Let X,Y be two sets and g : X → Y be an onto function. If XN :=
X

(TN , IN , FN )
is a neutrosophic N -structure over X, then the image of XN under g is a neutrosophic
N -structure over Y defined by

g(XN ) :=
Y

(g(TN ), g(IN ), g(FN ))

where

g(TN )(y) =
∧

x∈g−1(y)

TN (x),

g(IN )(y) =
∨

x∈g−1(y)

IN (x),

g(FN )(y) =
∧

x∈g−1(y)

FN (x).

Theorem 12. Let X,Y be two n-ary groupoids and let g : X → Y be an onto homo-

morphism. Let XN :=
X

(TN , IN , FN )
be a neutrosophic N -structure of X such that for all

A ⊆ X, there is x0 ∈ A such that

TN (x0) =
∧
z∈A

TN (z), IN (x0) =
∨
z∈A

IN (z), FN (x0) =
∧
z∈A

FN (z).

If XN is a neutrosophic n-ary N -subgroupoid of X, then the image of XN under g,

g(XN ) =
Y

(g(TN ), g(IN ), g(FN ))
,

is a neutrosophic n-ary N -subgroupoid of Y .

Proof. Let

g(XN ) =
Y

(g(TN ), g(IN ), g(FN ))

be the image of XN under g. Let y1, . . . , yn ∈ Y . Then g−1(y1) 6= ∅, . . . , , g−1(yn) 6= ∅ in
X which implies that there are xy1 ∈ g−1(y1), . . . , xyn ∈ g−1(yn) such that

TN (xy1) =
∧

z1∈g−1(y1)

TN (z1), IN (xy1) =
∨

z1∈g−1(y1)

IN (z1), FN (xy1) =
∧

z1∈g−1(y1)

FN (z1),

...

TN (xyn) =
∧

zn∈g−1(yn)

TN (zn), IN (xyn) =
∨

zn∈g−1(yn)

IN (zn), FN (xyn) =
∧

zn∈g−1(yn)

FN (zn).



A. Rattana, R. Chinram / Eur. J. Pure Appl. Math, 13 (2) (2020), 200-215 214

Hence

g(TN )(yn1 ) =
∧

x∈g−1(yn1 )

TN (x) ≤ TN (xy1 . . . xyn)

≤
∨
{TN (xy1), . . . , TN (xyn)}

=
∨ ∧

z1∈g−1(y1)

TN (z1), . . . ,
∧

zn∈g−1(yn)

TN (zn)


=
∨
{g(TN )(y1), . . . , g(TN )(yn)} ,

g(IN )(yn1 ) =
∨

x∈g−1(yn1 )

IN (x) ≥ IN (xy1 . . . xyn)

≥
∧
{IN (xy1), . . . , IN (xyn)}

=
∧ ∨

z1∈g−1(y1)

IN (z1), . . . ,
∨

zn∈g−1(yn)

IN (zn)


=
∧
{g(IN )(y1), . . . , g(IN )(yn)} ,

g(FN )(yn1 ) =
∧

x∈g−1(yn1 )

FN (x) ≤ FN (xy1 . . . xyn)

≤
∨
{FN (xy1), . . . , FN (xyn)}

=
∨ ∧

z1∈g−1(y1)

FN (z1), . . . ,
∧

zn∈g−1(yn)

FN (zn)


=
∨
{g(FN )(y1), . . . , g(FN )(yn)} .

Hence g(XN ) is a neutrosophic n-ary N -subgroupoid of Y .

4. Conclusions

We have studied the neutrosophic N -structure and applied it to n-ary groupoids. We
also investigated the notion of neutrosophic N -structures in n-ary groupoids and showed
some properties. We have investigated the conditions for neutrosophic N -structures to be
neutrosophic n-ary N -subgroupiods. A neutrosophic N -product has been introduced. In
addition, we have introduced neutrosophic n-ary N -subgroupoids, ε-neutrosophic n-ary
N -subgroupoids and shown the relation between n-ary subgroupoids and neutrosophic
n-ary N -subgroupoids. Finally, we showed that the homomorphic preimage of the neu-
trosophic n-ary N -subgroupoids is a neutrosophic n-ary N -subgroupoids and the onto



REFERENCES 215

homomorphic image of the neutrosophic n-ary N -subgroupoids is also a neutrosophic n-
ary N -subgroupoids.

Acknowledgments

This paper was supported by Algebra and Applications Research Unit, Prince of
Songkla University.

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