EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 4, 2019, 1382-1409
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Neutrosophic Set Theory Applied to UP-Algebras†

Metawee Songsaeng1, Aiyared Iampan1,∗

1 Department of Mathematics, School of Science, University of Phayao, Phayao 56000,
Thailand

Abstract. The notions of neutrosophic UP-subalgebras, neutrosophic near UP-filters, neutro-
sophic UP-filters, neutrosophic UP-ideals, and neutrosophic strongly UP-ideals of UP-algebras
are introduced, and several properties are investigated. Conditions for neutrosophic sets to be
neutrosophic UP-subalgebras, neutrosophic near UP-filters, neutrosophic UP-filters, neutrosophic
UP-ideals, and neutrosophic strongly UP-ideals of UP-algebras are provided. Relations between
neutrosophic UP-subalgebras (resp., neutrosophic near UP-filters, neutrosophic UP-filters, neutro-
sophic UP-ideals, neutrosophic strongly UP-ideals) and their level subsets are considered.

2010 Mathematics Subject Classifications: 03G25, 03B52, 03B60

Key Words and Phrases: UP-algebra, neutrosophic UP-subalgebra, neutrosophic near UP-filter,
neutrosophic UP-filter, neutrosophic UP-ideal, neutrosophic strongly UP-ideal

1. Introduction

Among many algebraic structures, algebras of logic form important class of algebras.
Examples of these are BCK-algebras [7], BCI-algebras [8], BCH-algebras [4], KU-algebras
[18], SU-algebras [13] UP-algebras [5] and so on. They are strongly connected with logic.
For example, BCI-algebras were introduced by Iséki [8] in 1966 have connections with BCI-
logic being the BCI-system in combinatory logic which has application in the language of
functional programming. BCK and BCI-algebras are two classes of logical algebras. They
were introduced by Imai and Iséki [7, 8] in 1966 and have been extensively investigated
by many researchers. It is known that the class of BCK-algebras is a proper subclass of
the class of BCI-algebras. The above-mentioned section has been derived from [12].

The branch of the logical algebra, UP-algebras were introduced by Iampan [5]. Later
Somjanta et al. [23] studied fuzzy UP-subalgebras, fuzzy UP-ideals and fuzzy UP-filters of
UP-algebras. Guntasow et al. [3] introduced and studied fuzzy translations of a fuzzy set
in UP-algebras. Kesorn et al. [14] studied intuitionistic fuzzy sets in UP-algebras. Kaijae
et al. [11] introduced and investigated anti-fuzzy UP-ideals and anti-fuzzy UP-subalgebras.

†This work was supported by the Unit of Excellence, University of Phayao.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3543

Email addresses: metawee.faith@gmail.com (M. Songsaeng), aiyared.ia@up.ac.th (A. Iampan)

http://www.ejpam.com 1382 c© 2019 EJPAM All rights reserved.



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1383

Tanamoon et al. [26] introduced and studied Q-fuzzy sets in UP-algebras. Sripaeng et
al. [25] studied anti Q-fuzzy UP-ideals and anti Q-fuzzy UP-subalgebras of UP-algebras.
Dokkhamdang et al. [2] studied Generalized fuzzy sets in UP-algebras. Songsaeng and
Iampan [24] studied N -fuzzy UP-algebras and their level subsets.

The notion of neutrosophic sets was introduced by Smarandache [22] in 1999. Wang et
al. [28] introduced the notion of interval neutrosophic sets in 2005. The notion of neutro-
sophic N -structures and their applications in semigroups was introduced by Khan et al.
[15] in 2017. Jun et al. [9] applied the notion of neutrosophic N -structures to BCK/BCI-
algebras in 2017. Khan et al. [15] discussed neutrosophic N -structures and their appli-
cations in semigroups in 2017. Jun et al. [10] studied neutrosophic positive implicative
N -ideals in BCK-algebras in 2018. Kim et al. [16] studied generalizations of neutrosophic
subalgebras in BCK/BCI-algebras based on neutrosophic points in 2018. Rangsuk et al.
[19] introduced the notions of (special) neutrosophic N -UP-subalgebras, (special) neu-
trosophic N -near UP-filters, (special) neutrosophic N -UP-filters, (special) neutrosophic
N -UP-ideals, and (special) neutrosophic N -strongly UP-ideals of UP-algebras in 2019.

In this paper, the notions of neutrosophic UP-subalgebras, neutrosophic near UP-
filters, neutrosophic UP-filters, neutrosophic UP-ideals, and neutrosophic strongly UP-
ideals of UP-algebras are introduced, and several properties are investigated. Conditions
for neutrosophic sets to be neutrosophic UP-subalgebras, neutrosophic near UP-filters,
neutrosophic UP-filters, neutrosophic UP-ideals, and neutrosophic strongly UP-ideals of
UP-algebras are provided. Relations between neutrosophic UP-subalgebras (resp., neu-
trosophic near UP-filters, neutrosophic UP-filters, neutrosophic UP-ideals, neutrosophic
strongly UP-ideals) and their level subsets are considered.

2. Basic results on UP-algebras

Before we begin our study, we will give the definition and useful properties of UP-
algebras.

Definition 1. [5] An algebra X = (X, ·, 0) of type (2, 0) is called a UP-algebra where X
is a nonempty set, · is a binary operation on X, and 0 is a fixed element of X (i.e., a
nullary operation) if it satisfies the following axioms:

(UP-1) (∀x, y, z ∈ X)((y · z) · ((x · y) · (x · z)) = 0),

(UP-2) (∀x ∈ X)(0 · x = x),

(UP-3) (∀x ∈ X)(x · 0 = 0), and

(UP-4) (∀x, y ∈ X)(x · y = 0, y · x = 0⇒ x = y).

From [5], we know that the notion of UP-algebras is a generalization of KU-algebras
(see [18]).

Example 1. [21] Let X be a universal set and let Ω ∈ P(X) where P(X) means the power
set of X. Let PΩ(X) = {A ∈ P(X) | Ω ⊆ A}. Define a binary operation · on PΩ(X) by



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1384

putting A · B = B ∩ (AC ∪ Ω) for all A,B ∈ PΩ(X) where AC means the complement of
a subset A. Then (PΩ(X), ·,Ω) is a UP-algebra and we shall call it the generalized power
UP-algebra of type 1 with respect to Ω. Let PΩ(X) = {A ∈ P(X) | A ⊆ Ω}. Define a
binary operation ∗ on PΩ(X) by putting A ∗ B = B ∪ (AC ∩ Ω) for all A,B ∈ PΩ(X).
Then (PΩ(X), ∗,Ω) is a UP-algebra and we shall call it the generalized power UP-algebra
of type 2 with respect to Ω. In particular, (P(X), ·, ∅) is a UP-algebra and we shall call it
the power UP-algebra of type 1, and (P(X), ∗, X) is a UP-algebra and we shall call it the
power UP-algebra of type 2.

Example 2. [2] Let N be the set of all natural numbers with two binary operations ◦ and
• defined by

(∀x, y ∈ N)

(
x ◦ y =

{
y if x < y,
0 otherwise

)
and

(∀x, y ∈ N)

(
x • y =

{
y if x > y or x = 0,
0 otherwise

)
.

Then (N, ◦, 0) and (N, •, 0) are UP-algebras.

Example 3. [17] Let X = {0, 1, 2, 3, 4, 5} be a set with a binary operation · defined by the
following Cayley table:

· 0 1 2 3 4 5

0 0 1 2 3 4 5
1 0 0 2 3 2 5
2 0 1 0 3 1 5
3 0 1 2 0 4 5
4 0 0 0 3 0 5
5 0 0 2 0 2 0

Then (X, ·, 0) is a UP-algebra.

For more examples of UP-algebras, see [1, 6, 20, 21].

In a UP-algebra X = (X, ·, 0), the following assertions are valid (see [5, 6]).

(∀x ∈ X)(x · x = 0), (2.1)

(∀x, y, z ∈ X)(x · y = 0, y · z = 0⇒ x · z = 0), (2.2)

(∀x, y, z ∈ X)(x · y = 0⇒ (z · x) · (z · y) = 0), (2.3)

(∀x, y, z ∈ X)(x · y = 0⇒ (y · z) · (x · z) = 0), (2.4)

(∀x, y ∈ X)(x · (y · x) = 0), (2.5)

(∀x, y ∈ X)((y · x) · x = 0⇔ x = y · x), (2.6)

(∀x, y ∈ X)(x · (y · y) = 0), (2.7)



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1385

(∀a, x, y, z ∈ X)((x · (y · z)) · (x · ((a · y) · (a · z))) = 0), (2.8)

(∀a, x, y, z ∈ X)((((a · x) · (a · y)) · z) · ((x · y) · z) = 0), (2.9)

(∀x, y, z ∈ X)(((x · y) · z) · (y · z) = 0), (2.10)

(∀x, y, z ∈ X)(x · y = 0⇒ x · (z · y) = 0), (2.11)

(∀x, y, z ∈ X)(((x · y) · z) · (x · (y · z)) = 0), and (2.12)

(∀a, x, y, z ∈ X)(((x · y) · z) · (y · (a · z)) = 0). (2.13)

On a UP-algebra X = (X, ·, 0), we define a binary relation ≤ on X [5] as follows:

(∀x, y ∈ X)(x ≤ y ⇔ x · y = 0).

Definition 2. [3, 5, 23] A nonempty subset S of a UP-algebra (X, ·, 0) is called

(1) a UP-subalgebra of X if (∀x, y ∈ S)(x · y ∈ S).

(2) a near UP-filter of X if

(i) the constant 0 of X is in S, and

(ii) (∀x, y ∈ X)(y ∈ S ⇒ x · y ∈ S).

(3) a UP-filter of X if

(i) the constant 0 of X is in S, and

(ii) (∀x, y ∈ X)(x · y ∈ S, x ∈ S ⇒ y ∈ S).

(4) a UP-ideal of X if

(i) the constant 0 of X is in S, and

(ii) (∀x, y, z ∈ X)(x · (y · z) ∈ S, y ∈ S ⇒ x · z ∈ S).

(5) a strongly UP-ideal of X if

(i) the constant 0 of X is in S, and

(ii) (∀x, y, z ∈ X)((z · y) · (z · x) ∈ S, y ∈ S ⇒ x ∈ S).

Guntasow et al. [3] proved that the notion of UP-subalgebras is a generalization of
near UP-filters, the notion of near UP-filters is a generalization of UP-filters, the notion of
UP-filters is a generalization of UP-ideals, and the notion of UP-ideals is a generalization
of strongly UP-ideals. Moreover, they also proved that a UP-algebra X is the only one
strongly UP-ideal of itself.



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1386

3. NSs in UP-algebras

In 1965, Zadeh [29] introduced the notion of fuzzy sets as the following definition.
A fuzzy set (briefly, FS) in a nonempty set X (or a fuzzy subset of X) is an arbitrary

function f : X → [0, 1] where [0, 1] is the unit segment of the real line, and the fuzzy set
f defined by f(x) = 1− f(x) for all x ∈ X is said to be the complement of f in X.

In 1999, Smarandache [22] introduced the notion of neutrosophic sets as the following
definition.

A neutrosophic set (briefly, NS) in a nonempty set X is a structure of the form:

Λ = {(x, λT (x), λI(x), λF (x)) | x ∈ X} (3.1)

where λT : X → [0, 1] is a truth membership function, λI : X → [0, 1] is an indeterminate
membership function, and λF : X → [0, 1] is a false membership function.

For our convenience, we will denote a NS as Λ = (X,λT , λI , λF ) = (X,λT,I,F ) =
{(x, λT (x), λI(x), λF (x)) | x ∈ X}.

Definition 3. [22] Let Λ be a NS in a nonempty set X. The NS Λ = (X,λT,I,F ) in X
defined by

(∀x ∈ X)

λT (x) = 1− λT (x)

λI(x) = 1− λI(x)

λF (x) = 1− λF (x)

 (3.2)

is called the complement of Λ in X.

Remark 1. For all NS Λ in a nonempty set X, we have Λ = Λ.

Lemma 1. [27] Let a, b, c ∈ R. Then the following statements hold:

(1) a−min{b, c} = max{a− b, a− c}, and

(2) a−max{b, c} = min{a− b, a− c}.

The following lemma is easily proved.

Lemma 2. Let f be a fuzzy set in a nonempty set X. Then the following statements hold:

(1) (∀x, y, z ∈ X)(f(x) ≥ min{f(y), f(z)} ⇔ f(x) ≤ max{f(y), f(z)}),

(2) (∀x, y, z ∈ X)(f(x) ≤ min{f(y), f(z)} ⇔ f(x) ≥ max{f(y), f(z)}),

(3) (∀x, y, z ∈ X)(f(x) ≥ max{f(y), f(z)} ⇔ f(x) ≤ min{f(y), f(z)}), and

(4) (∀x, y, z ∈ X)(f(x) ≤ max{f(y), f(z)} ⇔ f(x) ≥ min{f(y), f(z)}).



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1387

In what follows, let X denote a UP-algebra (X, ·, 0) unless otherwise specified.

Now, we introduce the notions of neutrosophic UP-subalgebras, neutrosophic near UP-
filters, neutrosophic UP-filters, neutrosophic UP-ideals, and neutrosophic strongly UP-
ideals of UP-algebras, provide the necessary examples, investigate their properties, and
prove their generalizations.

Definition 4. A NS Λ in X is called a neutrosophic UP-subalgebra of X if it satisfies the
following conditions:

(∀x, y ∈ X)(λT (x · y) ≥ min{λT (x), λT (y)}), (3.3)

(∀x, y ∈ X)(λI(x · y) ≤ max{λI(x), λI(y)}), (3.4)

(∀x, y ∈ X)(λF (x · y) ≥ min{λF (x), λF (y)}). (3.5)

Example 4. Let X = {0, 1, 2, 3, 4} be a UP-algebra with a fixed element 0 and a binary
operation · defined by the following Cayley table:

· 0 1 2 3 4

0 0 1 2 3 4
1 0 0 2 2 4
2 0 0 0 2 4
3 0 0 0 0 4
4 0 1 2 3 0

We define a NS Λ in X as follows:

λT =

(
0

0.9

1

0.7

2

0.5

3

0.3

4

0.3

)
, λI =

(
0

0

1

0.8

2

0.4

3

0.2

4

0.4

)
, λF =

(
0

1

1

0.6

2

0.8

3

0.3

4

0.2

)
.

Hence, Λ is a neutrosophic UP-subalgebra of X.

Definition 5. A NS Λ in X is called a neutrosophic near UP-filter of X if it satisfies the
following conditions:

(∀x ∈ X)(λT (0) ≥ λT (x)), (3.6)

(∀x ∈ X)(λI(0) ≤ λI(x)), (3.7)

(∀x ∈ X)(λF (0) ≥ λF (x)), (3.8)

(∀x, y ∈ X)(λT (x · y) ≥ λT (y)), (3.9)

(∀x, y ∈ X)(λI(x · y) ≤ λI(y)), (3.10)

(∀x, y ∈ X)(λF (x · y) ≥ λF (y)). (3.11)

Example 5. Let X = {0, 1, 2, 3, 4} be a UP-algebra with a fixed element 0 and a binary
operation · defined by the following Cayley table:

· 0 1 2 3 4

0 0 1 2 3 4
1 0 0 1 2 4
2 0 0 0 1 4
3 0 0 0 0 4
4 0 1 2 3 0



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1388

We define a NS Λ in X as follows:

λT =

(
0

1

1

0.7

2

0.5

3

0.4

4

0.8

)
, λI =

(
0

0.1

1

0.2

2

0.3

3

0.7

4

0.6

)
, λF =

(
0

0.9

1

0.8

2

0.4

3

0.3

4

0.5

)
.

Hence, Λ is a neutrosophic near UP-filter of X.

Definition 6. A NS Λ in X is called a neutrosophic UP-filter of X if it satisfies the
following conditions: (3.6), (3.7), (3.8), and

(∀x, y ∈ X)(λT (y) ≥ min{λT (x · y), λT (x)}), (3.12)

(∀x, y ∈ X)(λI(y) ≤ max{λI(x · y), λI(x)}), (3.13)

(∀x, y ∈ X)(λF (y) ≥ min{λF (x · y), λF (x)}). (3.14)

Example 6. Let X = {0, 1, 2, 3, 4} be a UP-algebra with a fixed element 0 and a binary
operation · defined by the following Cayley table:

· 0 1 2 3 4

0 0 1 2 3 4
1 0 0 2 3 4
2 0 0 0 3 3
3 0 1 2 0 3
4 0 1 2 0 0

We define a NS Λ in X as follows:

λT =

(
0

0.9

1

0.4

2

0.3

3

0.1

4

0.1

)
, λI =

(
0

0.2

1

0.3

2

0.7

3

0.8

4

0.8

)
, λF =

(
0

0.8

1

0.7

2

0.4

3

0.3

4

0.3

)
.

Hence, Λ is a neutrosophic UP-filter of X.

Definition 7. A NS Λ in X is called a neutrosophic UP-ideal of X if it satisfies the
following conditions: (3.6), (3.7), (3.8), and

(∀x, y, z ∈ X)(λT (x · z) ≥ min{λT (x · (y · z)), λT (y)}), (3.15)

(∀x, y, z ∈ X)(λI(x · z) ≤ max{λI(x · (y · z)), λI(y)}), (3.16)

(∀x, y, z ∈ X)(λF (x · z) ≥ min{λF (x · (y · z)), λF (y)}). (3.17)

Example 7. Let X = {0, 1, 2, 3, 4} be a UP-algebra with a fixed element 0 and a binary
operation · defined by the following Cayley table:

· 0 1 2 3 4

0 0 1 2 3 4
1 0 0 2 3 4
2 0 0 0 2 4
3 0 0 0 0 4
4 0 1 2 3 0



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1389

We define a NS Λ in X as follows:

λT =

(
0

1

1

0.7

2

0.6

3

0.6

4

0.4

)
, λI =

(
0

0

1

0.3

2

0.5

3

0.5

4

0.7

)
, λF =

(
0

1

1

0.8

2

0.7

3

0.7

4

0.5

)
.

Hence, Λ is a neutrosophic UP-ideal of X.

Definition 8. A NS Λ in X is called a neutrosophic strongly UP-ideal of X if it satisfies
the following conditions: (3.6), (3.7), (3.8), and

(∀x, y, z ∈ X)(λT (x) ≥ min{λT ((z · y) · (z · x)), λT (y)}), (3.18)

(∀x, y, z ∈ X)(λI(x) ≤ max{λI((z · y) · (z · x)), λI(y)}), (3.19)

(∀x, y, z ∈ X)(λF (x) ≥ min{λF ((z · y) · (z · x)), λF (y)}). (3.20)

Example 8. Let X = {0, 1, 2, 3, 4} be a UP-algebra with a fixed element 0 and a binary
operation · defined by the following Cayley table:

· 0 1 2 3 4

0 0 1 2 3 4
1 0 0 2 3 4
2 0 1 0 2 4
3 0 1 0 0 4
4 0 1 0 3 0

We define a NS Λ in X as follows:

(∀x ∈ X)

λT (x) = 1

λI(x) = 0.2

λF (x) = 0.8

 .

Hence, Λ is a neutrosophic strongly UP-ideal of X.

Definition 9. A NS Λ in X is said to be constant if Λ is a constant function from X to
[0, 1]3. That is, λT , λI , and λF are constant functions from X to [0, 1].

Theorem 1. Every neutrosophic UP-subalgebra of X satisfies the conditions (3.6), (3.7),
and (3.8).

Proof. Assume that Λ is a neutrosophic UP-subalgebra of X. Then for all x ∈ X,

λT (0) = λT (x · x) ≥ min{λT (x), λT (x)} = λT (x), (2.1) and (3.3)

λI(0) = λI(x · x) ≤ max{λI(x), λI(x)} = λI(x), (2.1) and (3.4)

λF (0) = λF (x · x) ≥ min{λF (x), λF (x)} = λF (x). (2.1) and (3.5)

Hence, Λ satisfies the conditions (3.6), (3.7), and (3.8).



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1390

Theorem 2. A NS Λ in X is constant if and only if it is a neutrosophic strongly UP-ideal
of X.

Proof. Assume that Λ is constant. Then for all x ∈ X, λT (x) = λT (0), λI(x) = λI(0),
and λF (x) = λF (0) and so λT (0) ≥ λT (x), λI(0) ≤ λI(x), and λF (0) ≥ λF (x). Next, for
all x, y, z ∈ X,

λT (x) = λT (0) = min{λT (0), λT (0)} = min{λT ((z · y) · (z · x)), λT (y)},
λI(x) = λI(0) = max{λI(0), λI(0)} = max{λI((z · y) · (z · x)), λI(y)},

λF (x) = λF (0) = min{λF (0), λF (0)} = min{λF ((z · y) · (z · x)), λF (y)}.

Hence, Λ is a neutrosophic strongly UP-ideal of X.

Conversely, assume that Λ is a neutrosophic strongly UP-ideal of X. For any x ∈ X,
we have

λT (x) ≥ min{λT ((x · 0) · (x · x)), λT (0)} (3.18)

= min{λT (0 · (x · x)), λT (0)} (UP-3)

= min{λT (x · x), λT (0)} (UP-2)

= min{λT (0), λT (0)} (2.1)

= λT (0),

λI(x) ≤ max{λI((x · 0) · (x · x)), λI(0)} (3.19)

= max{λI(0 · (x · x)), λI(0)} (UP-3)

= max{λI(x · x), λI(0)} (UP-2)

= max{λI(0), λI(0)} (2.1)

= λI(0),

λF (x) ≥ min{λF ((x · 0) · (x · x)), λF (0)} (3.20)

= min{λF (0 · (x · x)), λF (0)} (UP-3)

= min{λF (x · x), λF (0)} (UP-2)

= min{λF (0), λF (0)} (2.1)

= λF (0).

Thus λT (x) = λT (0), λI(x) = λI(0), and λF (x) = λF (0) for all x ∈ X. Hence, Λ is
constant.

Theorem 3. Every neutrosophic strongly UP-ideal of X is a neutrosophic UP-ideal.

Proof. Assume that Λ is a neutrosophic strong UP-ideal of X. Then Λ satisfies the
conditions (3.6), (3.7), and (3.8). By Theorem 2, we have Λ is constant. Then for all
x ∈ X, λT (x) = λT (0), λI(x) = λI(0), and λF (x) = λF (0). Thus

λT (x · z) = min{λT ((z · y) · (z · (x · z))), λT (y)} (3.18)



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1391

= min{λT ((z · y) · 0), λT (y)} (2.5)

= min{λT (0), λT (y)} (UP-3)

= λT (y) (3.6)

≥ min{λT (x · (y · z)), λT (y)},
λI(x · z) = max{λI((z · y) · (z · (x · z))), λI(y)} (3.19)

= max{λI((z · y) · 0), λI(y)} (2.5)

= max{λI(0), λI(y)} (UP-3)

= λI(y) (3.7)

≤ max{λI(x · (y · z)), λI(y)},
λF (x · z) = min{λF ((z · y) · (z · (x · z))), λF (y)} (3.20)

= min{λF ((z · y) · 0), λF (y)} (2.5)

= min{λF (0), λF (y)} (UP-3)

= λF (y) (3.8)

≥ min{λF (x · (y · z)), λF (y)}.

Hence, Λ is a neutrosophic UP-ideal of X.

The following example show that the converse of Theorem 3 is not true.

Example 9. From Example 7, we have Λ is a neutrosophic UP-ideal of X. Since Λ is
not constant, it follows from Theorem 2 that it is not a neutrosophic strongly UP-ideal of
X.

Theorem 4. Every neutrosophic UP-ideal of X is a neutrosophic UP-filter.

Proof. Assume that Λ is a neutrosophic UP-ideal of X. Then Λ satisfies the conditions
(3.6), (3.7), and (3.8). Next, let x, y ∈ X. Then

λT (y) = λT (0 · y) (UP-2)

≥ min{λT (0 · (x · y)), λT (x)} (3.15)

= min{λT (x · y), λT (x)}, (UP-2)

λI(y) = λI(0 · y) (UP-2)

≤ max{λI(0 · (x · y)), λI(x)} (3.16)

= max{λI(x · y), λI(x)}, (UP-2)

λF (y) = λF (0 · y) (UP-2)

≥ min{λF (0 · (x · y)), λF (x)} (3.17)

= min{λF (x · y), λF (x)}. (UP-2)

Hence, Λ is a neutrosophic UP-filter of X.

The following example show that the converse of Theorem 4 is not true.



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1392

Example 10. From Example 6, we have Λ is a neutrosophic UP-filter of X. Since λF (3 ·
4) = 0.3 < 0.4 = min{λF (3 · (2 · 4)), λF (2)}, we have Λ is not a neutrosophic UP-ideal of
X.

Theorem 5. Every neutrosophic UP-filter of X is a neutrosophic near UP-filter.

Proof. Assume that Λ is a neutrosophic UP-filter. Then Λ satisfies the conditions
(3.6), (3.7), and (3.8). Next, let x, y ∈ X. Then

λT (x · y) ≥ min{λT (y · (x · y)), λT (y)} (3.12)

= min{λT (0), λT (y)} (2.5)

= λT (y), (3.6)

λI(x · y) ≤ max{λI(y · (x · y)), λI(y)} (3.13)

= max{λI(0), λI(y)} (2.5)

= λI(y), (3.7)

λF (x · y) ≥ min{λF (y · (x · y)), λF (y)} (3.14)

= min{λF (0), λF (y)} (2.5)

= λF (y). (3.8)

Hence, Λ is a neutrosophic near UP-filter of X.

The following example show that the converse of Theorem 5 is not true.

Example 11. From Example 5, we have Λ is a neutrosophic near UP-filter of X. Since
λI(3) = 0.7 > 0.3 = max{λI(2 · 3), λI(2)}, we have Λ is not a neutrosophic UP-filter of
X.

Theorem 6. Every neutrosophic near UP-filter of X is a neutrosophic UP-subalgebra.

Proof. Assume that Λ is a neutrosophic near UP-filter of X. Then for all x, y ∈ X

λT (x · y) ≥ λT (y) ≥ min{λT (x), λT (y)}, (3.9)

λI(x · y) ≤ λI(y) ≤ max{λI(x), λI(y)}, (3.10)

λF (x · y) ≥ λF (y) ≥ min{λF (x), λF (y)}. (3.11)

Hence, Λ is a neutrosophic UP-subalgebra of X.

The following example show that the converse of Theorem 6 is not true.

Example 12. From Example 4, we have Λ is a neutrosophic UP-subalgebra of X. Since
λI(2 · 3) = 0.4 > 0.2 = λI(3), we have Λ is not a neutrosophic near UP-filter of X.

By Theorems 3, 4, 5, and 6 and Examples 9, 10, 11, and 12, we have that the notion
of neutrosophic UP-subalgebras is a generalization of neutrosophic near UP-filters, the
notion of neutrosophic near UP-filters is a generalization of neutrosophic UP-filters, the
notion of neutrosophic UP-filters is a generalization of neutrosophic UP-ideals, and the
notion of neutrosophic UP-ideals is a generalization of neutrosophic strongly UP-ideals.
Moreover, by Theorem 2, we obtain that neutrosophic strongly UP-ideals and constant
neutrosophic set coincide.



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1393

Theorem 7. If Λ is a neutrosophic UP-subalgebra of X satisfying the following condition:

(∀x, y ∈ X)

x · y 6= 0⇒


λT (x) ≥ λT (y)

λI(x) ≤ λI(y)

λF (x) ≥ λF (y)

 , (3.21)

then Λ is a neutrosophic near UP-filter of X.

Proof. Assume that Λ is a neutrosophic UP-subalgebra of X satisfying the condition
(3.21). By Theorem 1, we have Λ satisfies the conditions (3.6), (3.7), and (3.8). Next, let
x, y ∈ X.

Case 1: x · y = 0. Then

λT (x · y) = λT (0) ≥ λT (y), (3.6)

λI(x · y) = λI(0) ≤ λI(y), (3.7)

λF (x · y) = λF (0) ≥ λF (y). (3.8)

Case 2: x · y 6= 0. Then

λT (x · y) ≥ min{λT (x), λT (y)} = λT (y), (3.3) and (3.21) for λT

λI(x · y) ≤ max{λI(x), λI(y)} = λI(y), (3.4) and (3.21) for λI

λF (x · y) ≥ min{λF (x), λF (y)} = λF (y). (3.5) and (3.21) for λF

Hence, Λ is a neutrosophic near UP-filter of X.

Theorem 8. If Λ is a neutrosophic near UP-filter of X satisfying the following condition:

λT = λI = λF , (3.22)

then Λ is a neutrosophic UP-filter of X.

Proof. Assume that Λ is a neutrosophic near UP-filter of X satisfying the condition
(3.22). Then Λ satisfies the conditions (3.6), (3.7), and (3.8). Next, let x, y ∈ X. Then

min{λT (x · y), λT (x)} = min{λI(x · y), λT (x)} (3.22)

≤ min{λI(y), λT (x)} (3.10)

= min{λT (y), λT (x)} (3.22)

≤ λT (y),

max{λI(x · y), λI(x)} = max{λT (x · y), λI(x)} (3.22)

≥ max{λT (y), λI(x)} (3.9)

= max{λI(y), λI(x)} (3.22)

≥ λI(y),

min{λF (x · y), λF (x)} = min{λI(x · y), λF (x)} (3.22)



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1394

≤ min{λI(y), λF (x)} (3.10)

= min{λF (y), λF (x)} (3.22)

≤ λF (y).

Hence, Λ is a neutrosophic UP-filter of X.

Theorem 9. If Λ is a neutrosophic UP-filter of X satisfying the following condition:

(∀x, y, z ∈ X)

λT (y · (x · z)) = λT (x · (y · z))
λI(y · (x · z)) = λI(x · (y · z))
λF (y · (x · z)) = λF (x · (y · z))

 , (3.23)

then Λ is a neutrosophic UP-ideal of X.

Proof. Assume that Λ is a neutrosophic UP-filter of X satisfying the condition (3.23).
Then Λ satisfies the conditions (3.6), (3.7), and (3.8). Next, let x, y, z ∈ X. Then

λT (x · z) ≥ min{λT (y · (x · z)), λT (y)} (3.12)

= min{λT (x · (y · z)), λT (y)}, (3.23) for λT

λI(x · z) ≤ max{λI(y · (x · z)), λI(y)} (3.13)

= max{λI(x · (y · z)), λI(y)}, (3.23) for λI

λF (x · z) ≥ min{λF (y · (x · z)), λF (y)} (3.14)

= min{λF (x · (y · z)), λF (y)}. (3.23) for λF

Hence, Λ is a neutrosophic UP-ideal of X.

Theorem 10. If Λ is a NS in X satisfying the following condition:

(∀x, y, z ∈ X)

z ≤ x · y ⇒

λT (z) ≥ min{λT (x), λT (y)}
λI(z) ≤ max{λI(x), λI(y)}
λF (z) ≥ min{λF (x), λF (y)}

 , (3.24)

then Λ is a neutrosophic UP-subalgebra of X.

Proof. Assume that Λ is a NS in X satisfying the condition (3.24). Let x, y ∈ X. By
(2.1), we have (x · y) · (x · y) = 0, that is, x · y ≤ x · y. It follows from (3.24) that

λT (x · y) ≥ min{λT (x), λT (y)},
λI(x · y) ≤ max{λI(x), λI(y)},
λF (x · y) ≥ min{λF (x), λF (y)}.

Hence, Λ is a neutrosophic UP-subalgebra of X.



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1395

Theorem 11. If Λ is a NS in X satisfying the following condition:

(∀x, y, z ∈ X)

z ≤ x · y ⇒

λT (z) ≥ λT (y)

λI(z) ≤ λI(y)

λF (z) ≥ λF (y)

 , (3.25)

then Λ is a neutrosophic near UP-filter of X.

Proof. Assume that Λ is a NS in X satisfying the condition (3.25). Let x ∈ X. By
(UP-2) and (2.1), we have 0 · (x · x) = 0, that is, 0 ≤ x · x. It follows from (3.25) that
λT (0) ≥ λT (x), λI(0) ≤ λI(x), and λF (0) ≥ λF (x). Next, let x, y ∈ X. By (2.1), we
have (x · y) · (x · y) = 0, that is, x · y ≤ x · y. It follows from (3.25) that λT (x · y) ≥
λT (y), λI(x · y) ≤ λI(y), and λF (x · y) ≥ λF (y). Hence, Λ is a neutrosophic near UP-filter
of X.

Theorem 12. If Λ is a NS in X satisfying the following condition:

(∀x, y, z ∈ X)

z ≤ x · y ⇒

λT (y) ≥ min{λT (z), λT (x)}
λI(y) ≤ max{λI(z), λI(x)}
λF (y) ≥ min{λF (z), λF (x)}

 , (3.26)

then Λ is a neutrosophic UP-filter of X.

Proof. Assume that Λ is a NS in X satisfying the condition (3.26). Let x ∈ X. By
(UP-3), we have x · (x · 0) = 0, that is, x ≤ x · 0. It follows from (3.26) that

λT (0) ≥ min{λT (x), λT (x)} = λT (x),

λI(0) ≤ max{λI(x), λI(x)} = λI(x),

λF (0) ≥ min{λF (x), λF (x)} = λF (x).

Next, let x, y ∈ X. By (2.1), we have (x · y) · (x · y) = 0, that is, x · y ≤ x · y. It follows
from (3.26) that

λT (y) ≥ min{λT (x · y), λT (x)},
λI(y) ≤ max{λI(x · y), λI(x)},
λF (y) ≥ min{λF (x · y), λF (x)}.

Hence, Λ is a neutrosophic UP-filter of X.

Theorem 13. If Λ is a NS in X satisfying the following condition:

(∀a, x, y, z ∈ X)

a ≤ x · (y · z)⇒

λT (x · z) ≥ min{λT (a), λT (y)}
λI(x · z) ≤ max{λI(a), λI(y)}
λF (x · z) ≥ min{λF (a), λF (y)}

 , (3.27)

then Λ is a neutrosophic UP-ideal of X.



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1396

Proof. Assume that Λ is a NS in X satisfying the condition (3.27). Let x ∈ X. By
(UP-3), we have x · (0 · (x · 0) = 0, that is, x ≤ 0 · (x · 0). It follows from (3.27) that

λT (0) = λT (0 · 0) ≥ min{λT (x), λT (x)} = λT (x), (UP-2)

λI(0) = λI(0 · 0) ≤ max{λI(x), λI(x)} = λI(x), (UP-2)

λF (0) = λF (0 · 0) ≥ min{λF (x), λF (x)} = λF (x). (UP-2)

Next, let x, y, z ∈ X. By (2.1), we have (x·(y ·z))·(x·(y ·z)) = 0, that is, x·(y ·z) ≤ x·(y ·z).
It follows from (3.27) that

λT (x · z) ≥ min{λT (x · (y · z)), λT (y)},
λI(x · z) ≤ max{λI(x · (y · z)), λI(y)},
λF (x · z) ≥ min{λF (x · (y · z)), λF (y)}.

Hence, Λ is a neutrosophic UP-ideal of X.

For any fixed numbers α+, α−, β+, β−, γ+, γ− ∈ [0, 1] such that α+ > α−, β+ >

β−, γ+ > γ− and a nonempty subsetG ofX, a NS ΛG[α
+,β−,γ+

α−,β+,γ− ] = (X,λGT [α
+

α− ], λGI [β
−

β+ ], λGF [γ
+

γ− ])

in X where λGT [α
+

α− ], λGI [β
−

β+ ], and λGF [γ
+

γ− ] are functions on X which are given as follows:

λGT [α
+

α− ](x) =

{
α+ if x ∈ G,
α− otherwise,

λGI [β
−

β+ ](x) =

{
β− if x ∈ G,
β+ otherwise,

λGF [γ
+

γ− ](x) =

{
γ+ if x ∈ G,
γ− otherwise.

Lemma 3. If the constant 0 of X is in a nonempty subset G of X, then a NS ΛG[α
+,β−,γ+

α−,β+,γ− ]

in X satisfies the conditions (3.6), (3.7), and (3.8).

Proof. If 0 ∈ G, then λGT [α
+

α− ](0) = α+, λGI [β
−

β+ ](0) = β−, λGF [γ
+

γ− ](0) = γ+. Thus

(∀x ∈ X)


λGT [α

+

α− ](0) = α+ ≥ λGT [α
+

α− ](x)

λGI [β
−

β+ ](0) = β− ≤ λGI [β
−

β+ ](x)

λGF [γ
+

γ− ](0) = γ+ ≥ λGF [γ
+

γ− ](x)

 .

Hence, ΛG[α
+,β−,γ+

α−,β+,γ− ] satisfies the conditions (3.6), (3.7), and (3.8).

Lemma 4. If a NS ΛG[α
+,β−,γ+

α−,β+,γ− ] in X satisfies the condition (3.6) (resp., (3.7), (3.8)),
then the constant 0 of X is in a nonempty subset G of X.



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1397

Proof. Assume that the NS ΛG[α
+,β−,γ+

α−,β+,γ− ] in X satisfies the condition (3.6). Then

λGT [α
+

α− ](0) ≥ λGT [α
+

α− ](x) for all x ∈ X. Since G is nonempty, there exists g ∈ G. Thus

λGT [α
+

α− ](g) = α+ and so λGT [α
+

α− ](0) ≥ λGT [α
+

α− ](g) = α+ ≥ λGT [α
+

α− ](0), that is, λGT [α
+

α− ](0) = α+.
Hence, 0 ∈ G.

Theorem 14. A NS ΛG[α
+,β−,γ+

α−,β+,γ− ] in X is a neutrosophic UP-subalgebra of X if and only
if a nonempty subset G of X is a UP-subalgebra of X.

Proof. Assume that ΛG[α
+,β−,γ+

α−,β+,γ− ] is a neutrosophic UP-subalgebra of X. Let x, y ∈ G.

Then λGT [α
+

α− ](x) = α+ = λGT [α
+

α− ](y). Thus

λGT [α
+

α− ](x · y) ≥ min{λGT [α
+

α− ](x), λGT [α
+

α− ](y)} = α+ ≥ λGT [α
+

α− ](x · y) (3.3)

and so λGT [α
+

α− ](x · y) = α+. Thus x · y ∈ G. Hence, G is a UP-subalgebra of X.

Conversely, assume that G is a UP-subalgebra of X. Let x, y ∈ X.
Case 1: x, y ∈ G. Then

λGT [α
+

α− ](x) = α+ = λGT [α
+

α− ](y),

λGI [β
−

β+ ](x) = β− = λGI [β
−

β+ ](y),

λGF [γ
+

γ− ](x) = γ+ = λGF [γ
+

γ− ](y).

Thus

min{λGT [α
+

α− ](x), λGT [α
+

α− ](y)} = α+,

max{λGI [β
−

β+ ](x), λGI [β
−

β+ ](y)} = β−,

min{λGF [γ
+

γ− ](x), λGF [γ
+

γ− ](y)} = γ+.

Since G is a UP-subalgebra of X, we have x·y ∈ G and so λGT [α
+

α− ](x·y) = α+, λGI [β
−

β+ ](x·y) =

β−, and λGF [γ
+

γ− ](x · y) = γ+. Hence,

λGT [α
+

α− ](x · y) = α+ ≥ α+ = min{λGT [α
+

α− ](x), λGT [α
+

α− ](y)},

λGI [β
−

β+ ](x · y) = β− ≤ β− = max{λGI [β
−

β+ ](x), λGI [β
−

β+ ](y)},

λGF [γ
+

γ− ](x · y) = γ+ ≥ γ+ = min{λGF [γ
+

γ− ](x), λGF [γ
+

γ− ](y)}.

Case 2: x 6∈ G or y 6∈ G. Then

λGT [α
−

α− ](x) = α− or λGT [α
+

α− ](y) = α−,

λGI [β
−

β+ ](x) = β+ or λGI [β
−

β+ ](y) = β+,

λGF [γ
+

γ− ](x) = γ− or λGF [γ
+

γ− ](y) = γ−.



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1398

Thus

min{λGT [α
+

α− ](x), λGT [α
+

α− ](y)} = α−,

max{λGI [β
−

β+ ](x), λGI [β
−

β+ ](y)} = β+,

min{λGF [γ
+

γ− ](x), λGF [γ
+

γ− ](y)} = γ−.

Therefore,

λGT [α
+

α− ](x · y) ≥ α− = min{λGT [α
+

α− ](x), λGT [α
+

α− ](y)},

λGI [β
−

β+ ](x · y) ≤ β+ = max{λGI [β
−

β+ ](x), λGI [β
−

β+ ](y)},

λGF [γ
+

γ− ](x · y) ≥ γ− = min{λGF [γ
+

γ− ](x), λGF [γ
+

γ− ](y)}.

Hence, ΛG[α
+,β−,γ+

α−,β+,γ− ] is a neutrosophic UP-subalgebra of X.

Theorem 15. A NS ΛG[α
+,β−,γ+

α−,β+,γ− ] in X is a neutrosophic near UP-filter of X if and only
if a nonempty subset G of X is a near UP-filter of X.

Proof. Assume that ΛG[α
+,β−,γ+

α−,β+,γ− ] is neutrosophic near UP-filter ofX. Since ΛG[α
+,β−,γ+

α−,β+,γ− ]

satisfies the condition (3.6), it follows from Lemma 4 that 0 ∈ G. Next, let x ∈ X and
y ∈ G. Then λGT [α

+

α− ](y) = α+. Thus

λGT [α
+

α− ](x · y) ≥ λGT [α
+

α− ](y) = α+ ≥ λGT [α
+

α− ](x · y) (3.9)

and so λGT [α
+

α− ](x · y) = α+. Thus x · y ∈ G. Hence, G is a near UP-filter of X.

Conversely, assume that G is a near UP-filter of X. Since 0 ∈ G, it follows from Lemma

3 that ΛG[α
+,β−,γ+

α−,β+,γ− ] satisfies the conditions (3.6), (3.7), and (3.8). Next, let x, y ∈ X.

Case 1: y ∈ G. Then λGT [α
+

α− ](y) = α+, λGI [β
−

β+ ](y) = β−, and λGF [γ
+

γ− ](y) = γ+. Since G

is a near UP-filter of X, we have x · y ∈ G and so λGT [α
+

α− ](x · y) = α+, λGI [β
−

β+ ](x · y) = β−,

and λGF [γ
+

γ− ](x · y) = γ+. Thus

λGT [α
+

α− ](x · y) = α+ ≥ α+ = λGT [α
+

α− ](y),

λGI [β
−

β+ ](x · y) = β− ≤ β− = λGI [β
−

β+ ](y),

λGF [γ
+

γ− ](x · y) = γ+ ≥ γ+ = λGF [γ
+

γ− ](y).

Case 2: y 6∈ G. Then λGT [α
+

α− ](y) = α−, λGI [β
−

β+ ](y) = β+, and λGF [γ
+

γ− ](y) = γ−. Thus

λGT [α
+

α− ](x · y) ≥ α− = λGT [α
+

α− ](y),

λGI [β
−

β+ ](x · y) ≤ β+ = λGI [β
−

β+ ](y),



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1399

λGF [γ
+

γ− ](x · y) ≥ γ− = λGF [γ
+

γ− ](y).

Hence, ΛG[α
+,β−,γ+

α−,β+,γ− ] is a neutrosophic near UP-filter of X.

Theorem 16. A NS ΛG[α
+,β−,γ+

α−,β+,γ− ] in X is a neutrosophic UP-filter of X if and only if a
nonempty subset G of X is a UP-filter of X.

Proof. Assume that ΛG[α
+,β−,γ+

α−,β+,γ− ] is a neutrosophic UP-filter of X. Since ΛG[α
+,β−,γ+

α−,β+,γ− ]

satisfies the condition (3.6), it follows from Lemma 4 that 0 ∈ G. Next, let x, y ∈ X be
such that x · y ∈ G and x ∈ G. Then λGT [α

+

α− ](x · y) = α+ = λGT [α
+

α− ](x). Thus

λGT [α
+

α− ](y) ≥ min{λGT [α
+

α− ](x · y), λGT [α
+

α− ](x)} = α+ ≥ λGT [α
+

α− ](y) (3.12)

and so λGT [α
+

α− ](y) = α+. Thus y ∈ G. Hence, G is a UP-filter of X.

Conversely, assume that G is a UP-filter of X. Since 0 ∈ G, it follows from Lemma 3

that ΛG[α
+,β−,γ+

α−,β+,γ− ] satisfies the conditions (3.6), (3.7), and (3.8). Next, let x, y ∈ X.
Case 1: x · y ∈ G and x ∈ G. Then

λGT [α
+

α− ](x · y) = α+ = λGT [α
+

α− ](x),

λGI [β
−

β+ ](x · y) = β− = λGI [β
−

β+ ](x),

λGF [γ
+

γ− ](x · y) = γ+ = λGF [γ
+

γ− ](x).

Since G is a UP-filter of X, we have y ∈ G and so λGT [α
+

α− ](y) = α+, λGI [β
−

β+ ](y) = β−, and

λGF [γ
+

γ− ](y) = γ+. Thus

λGT [α
+

α− ](y) = α+ ≥ α+ = min{λGT [α
+

α− ](x · y), λGT [α
+

α− ](x)},

λGI [β
−

β+ ](y) = β− ≤ β− = max{λGI [β
−

β+ ](x · y), λGI [β
−

β+ ](x)},

λGF [γ
+

γ− ](y) = γ+ ≥ γ+ = min{λGF [γ
+

γ− ](x · y), λGF [γ
+

γ− ](x)}.

Case 2: x · y 6∈ G or x 6∈ G. Then

λGT [α
+

α− ](x · y) = α− or λGT [α
+

α− ](x) = α−,

λGI [β
−

β+ ](x · y) = β+ or λGI [β
−

β+ ](x) = β+,

λGF [γ
+

γ− ](x · y) = γ− or λGF [γ
+

γ− ](x) = γ−.

Thus

min{λGT [α
+

α− ](x · y), λGT [α
+

α− ](x)} = α−,

max{λGI [β
−

β+ ](x · y), λGI [β
−

β+ ](x)} = β+,



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1400

min{λGF [γ
+

γ− ](x · y), λGF [γ
+

γ− ](x)} = γ−.

Therefore,

λGT [α
+

α− ](y) ≥ α− = min{λGT [α
+

α− ](x · y), λGT [α
+

α− ](x)},

λGI [β
−

β+ ](y) ≤ β+ = max{λGI [β
−

β+ ](x · y), λGI [β
−

β+ ](x)},

λGF [γ
+

γ− ](y) ≥ γ− = min{λGF [γ
+

γ− ](x · y), λGF [γ
+

γ− ](x)}.

Hence, ΛG[α
+,β−,γ+

α−,β+,γ− ] is a neutrosophic UP-filter of X.

Theorem 17. A NS ΛG[α
+,β−,γ+

α−,β+,γ− ] in X is a neutrosophic UP-ideal of X if and only if a
nonempty subset G of X is a UP-ideal of X.

Proof. Assume that ΛG[α
+,β−,γ+

α−,β+,γ− ] is a neutrosophic UP-ideal of X. Since ΛG[α
+,β−,γ+

α−,β+,γ− ]

satisfies the condition (3.6), it follows from Lemma 4 that 0 ∈ G. Next, let x, y, z ∈ X be
such that x · (y · z) ∈ G and y ∈ G. Then λGT [α

+

α− ](x · (y · z)) = α+ = λGT [α
+

α− ](y). Thus

λGT [α
+

α− ](x · z) ≥ min{λGT [α
+

α− ](x · (y · z)), λGT [α
+

α− ](y)} = α+ ≥ λGT [α
+

α− ](x · z) (3.18)

and so λGT [α
+

α− ](x · z) = α+. Thus x · z ∈ G. Hence, G is a UP-ideal of X.

Conversely, assume that G is a UP-ideal of X. Since 0 ∈ G, it follows from Lemma 3

that ΛG[α
+,β−,γ+

α−,β+,γ− ] satisfies the conditions (3.6), (3.7), and (3.8). Next, let x, y, z ∈ X.

Case 1: x · (y · z) ∈ G and y ∈ G. Then

λGT [α
+

α− ](x · (y · z)) = α+ = λGT [α
+

α− ](y),

λGI [β
−

β+ ](x · (y · z)) = β− = λGI [β
−

β+ ](y),

λGF [γ
+

γ− ](x · (y · z)) = γ+ = λGF [γ
+

γ− ](y).

Thus

min{λGT [α
+

α− ](x · (y · z)), λGT [α
+

α− ](y)} = α+,

max{λGI [β
−

β+ ](x · (y · z)), λGI [β
−

β+ ](y)} = β−,

min{λGF [γ
+

γ− ](x · (y · z)), λGF [γ
+

γ− ](y)} = γ+.

Since G is a UP-ideal of X, we have x ·z ∈ G and so λGT [α
+

α− ](x ·z) = α+, λGI [β
−

β+ ](x ·z) = β−,

and λGF [γ
+

γ− ](x · z) = γ+. Thus

λGT [α
+

α− ](x · z) = α+ ≥ α+ = min{λGT [α
+

α− ](x · (y · z)), λGT [α
+

α− ](y)},

λGI [β
−

β+ ](x · z) = β− ≤ β− = max{λGI [β
−

β+ ](x · (y · z)), λGI [β
−

β+ ](y)},



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1401

λGF [γ
+

γ− ](x · z) = γ+ ≥ γ+ = min{λGF [γ
+

γ− ](x · (y · z)), λGF [γ
+

γ− ](y)}.

Case 2: x · (y · z) 6∈ G or y 6∈ G. Then

λGT [α
+

α− ](x · (y · z)) = α− or λGT [α
+

α− ](y) = α−,

λGI [β
−

β+ ](x · (y · z)) = β+ or λGI [β
−

β+ ](y) = β+,

λGF [γ
+

γ− ](x · (y · z)) = γ− or λGF [γ
+

γ− ](y) = γ−.

Thus

min{λGT [α
+

α− ](x · (y · z)), λGT [α
+

α− ](y)} = α−,

max{λGI [β
−

β+ ](x · (y · z)), λGI [β
−

β+ ](y)} = β+,

min{λGF [γ
+

γ− ](x · (y · z)), λGF [γ
+

γ− ](y)} = γ−.

Therefore,

λGT [α
+

α− ](x · z) ≥ α− = min{λGT [α
+

α− ](x · (y · z)), λGT [α
+

α− ](y)},

λGI [β
−

β+ ](x · z) ≤ β+ = max{λGI [β
−

β+ ](x · (y · z)), λGI [β
−

β+ ](y)},

λGF [γ
+

γ− ](x · z) ≥ γ− = min{λGF [γ
+

γ− ](x · (y · z)), λGF [γ
+

γ− ](y)}.

Hence, ΛG[α
+,β−,γ+

α−,β+,γ− ] is a neutrosophic UP-ideal of X.

Theorem 18. A NS ΛG[α
+,β−,γ+

α−,β+,γ− ] in X is a neutrosophic strongly UP-ideal of X if and
only if a nonempty subset G of X is a strongly UP-ideal of X.

Proof. Assume that ΛG[α
+,β−,γ+

α−,β+,γ− ] is a neutrosophic strongly UP-ideal of X. By Theo-

rem 2, we have ΛG[α
+,β−,γ+

α−,β+,γ− ] is constant, that is, λGT [α
+

α− ] is constant. Since G is nonempty,

we have λGT [α
+

α− ](x) = α+ for all x ∈ X. Thus G = X. Hence, G is a strongly UP-ideal of
X.

Conversely, assume that G is a strongly UP-ideal of X. Then G = X, so

(∀x ∈ X)


λGT [α

+

α− ](x) = α+

λGI [β
−

β+ ](x) = β−

λGF [γ
+

γ− ](x) = γ+

 .

Thus λGT [α
+

α− ], λGI [β
−

β+ ], and λGF [γ
+

γ− ] are constant, that is, ΛG[α
+,β−,γ+

α−,β+,γ− ] is constant. By The-

orem 2, we have ΛG[α
+,β−,γ+

α−,β+,γ− ] is a neutrosophic strongly UP-ideal of X.



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1402

4. Level subsets of a NS

In this section, we discuss the relationships between neutrosophic UP-subalgebras
(resp., neutrosophic near UP-filters, neutrosophic UP-filters, neutrosophic UP-ideals, neu-
trosophic strongly UP-ideals) of UP-algebras and their level subsets.

Definition 10. [23] Let f be a fuzzy set in A. For any t ∈ [0, 1], the sets

U(f ; t) = {x ∈ X | f(x) ≥ t},
L(f ; t) = {x ∈ X | f(x) ≤ t},
E(f ; t) = {x ∈ X | f(x) = t}

are called an upper t-level subset, a lower t-level subset, and an equal t-level subset of f ,
respectively.

Theorem 19. A NS Λ in X is a neutrosophic UP-subalgebra of X if and only if for
all α, β, γ ∈ [0, 1], the sets U(λT ;α), L(λI ;β), and U(λF ; γ) are UP-subalgebras of X if
U(λT ;α), L(λI ;β), and U(λF ; γ) are nonempty.

Proof. Assume that Λ is a neutrosophic UP-subalgebra of X. Let α, β, γ ∈ [0, 1] be
such that U(λT ;α), L(λI ;β), and U(λF ; γ) are nonempty.

Let x, y ∈ U(λT ;α). Then λT (x) ≥ α and λT (y) ≥ α, so α is an lower bound of
{λT (x), λT (y)}. By (3.3), we have λT (x · y) ≥ min{λT (x), λT (y)} ≥ α. Thus x · y ∈
U(λT ;α).

Let x, y ∈ L(λI ;β). Then λI(x) ≤ β and λI(y) ≤ β, so β is a upper bound of
{λI(x), λI(y)}. By (3.4), we have λI(x·y) ≤ max{λI(x), λI(y)} ≤ β. Thus x·y ∈ L(λI ;β).

Let x, y ∈ U(λF ; γ). Then λF (x) ≥ γ and λF (y) ≥ γ, so γ is an lower bound of
{λF (x), λF (y)}. By (3.5), we have λF (x · y) ≥ min{λF (x), λF (y)} ≥ γ. Thus x · y ∈
U(λF ; γ).

Hence, U(λT ;α), L(λI ;β), and U(λF ; γ) are UP-subalgebras of X.

Conversely, assume that for all α, β, γ ∈ [0, 1], the sets U(λT ;α), L(λI ;β), and U(λF ; γ)
are UP-subalgebras of X if U(λT ;α), L(λI ;β), and U(λF ; γ) are nonempty.

Let x, y ∈ X. Then λT (x), λT (y) ∈ [0, 1]. Choose α = min{λT (x), λT (y)}. Thus
λT (x) ≥ α and λT (y) ≥ α, so x, y ∈ U(λT ;α) 6= ∅. By assumption, we have U(λT ;α) is a
UP-subalgebra of X and so x · y ∈ U(λT ;α). Thus λT (x · y) ≥ α = min{λT (x), λT (y)}.

Let x, y ∈ X. Then λI(x), λI(y) ∈ [0, 1]. Choose β = max{λI(x), λI(y)}. Thus
λI(x) ≤ β and λI(y) ≤ β, so x, y ∈ L(λI ;β) 6= ∅. By assumption, we have L(λI ;β) is a
UP-subalgebra of X and so x · y ∈ L(λI ;β). Thus λI(x · y) ≤ β = max{λI(x), λI(y)}.

Let x, y ∈ X. Then λF (x), λF (y) ∈ [0, 1]. Choose γ = min{λF (x), λF (y)}. Thus
λF (x) ≥ γ and λF (y) ≥ γ, so x, y ∈ U(λF ; γ) 6= ∅. By assumption, we have U(λF ; γ) is a
UP-subalgebra of X and so x · y ∈ U(λF ; γ). Thus λF (x · y) ≥ γ = min{λF (x), λF (y)}.

Therefore, Λ is a neutrosophic UP-subalgebra of X.



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1403

Theorem 20. A NS Λ in X is a neutrosophic near UP-filter of X if and only if for
all α, β, γ ∈ [0, 1], the sets U(λT ;α), L(λI ;β), and U(λF ; γ) are near UP-filters of X if
U(λT ;α), L(λI ;β), and U(λF ; γ) are nonempty.

Proof. Assume that Λ is a neutrosophic near UP-filter of X. Let α, β, γ ∈ [0, 1] be
such that U(λT ;α), L(λI ;β), and U(λF ; γ) are nonempty.

Let x ∈ U(λT ;α). Then λT (x) ≥ α. By (3.6), we have λT (0) ≥ λT (x) ≥ α. Thus
0 ∈ U(λT ;α). Next, let x ∈ X and y ∈ U(λT ;α). Then λT (y) ≥ α. By (3.9), we have
λT (x · y) ≥ λT (y) ≥ α. Thus x · y ∈ U(λT ;α).

Let x ∈ L(λI ;β). Then λI(x) ≤ β. By (3.7), we have λI(0) ≤ λI(x) ≤ β. Thus
0 ∈ L(λI ;β). Next, let x ∈ X and y ∈ L(λI ;β). Then λI(y) ≤ β. By (3.10), we have
λI(x · y) ≤ λI(y) ≤ β. Thus x · y ∈ L(λI ;β).

Let x ∈ U(λF ; γ). Then λF (x) ≥ γ. By (3.8), we have λF (0) ≥ λF (x) ≥ γ. Thus
0 ∈ U(λF ; γ). Next, let x ∈ X and y ∈ U(λF ; γ). Then λF (y) ≥ γ. By (3.11), we have
λF (x · y) ≥ λF (y) ≥ γ. Thus x · y ∈ U(λF ; γ).

Hence, U(λT ;α), L(λI ;β), and U(λF ; γ) are near UP-filters of X.

Conversely, assume that for all α, β, γ ∈ [0, 1], the sets U(λT ;α), L(λI ;β), and U(λF ; γ)
are near UP-filters of X if U(λT ;α), L(λI ;β), and U(λF ; γ) are nonempty.

Let x ∈ X. Then λT (x) ∈ [0, 1]. Choose α = λT (x). Thus λT (x) ≥ α, so x ∈
U(λT ;α) 6= ∅. By assumption, we have U(λT ;α) is a near UP-filter of X and so 0 ∈
U(λT ;α). Thus λT (0) ≥ α = λT (x). Next, let x, y ∈ X. Then λT (y) ∈ [0, 1]. Choose
α = λT (y). Thus λT (y) ≥ α, so y ∈ U(λT ;α) 6= ∅. By assumption, we have U(λT ;α) is a
near UP-filter of X and so x · y ∈ U(λT ;α). Thus λT (x · y) ≥ α = λT (y).

Let x ∈ X. Then λI(x) ∈ [0, 1]. Choose β = λI(x). Thus λI(x) ≤ β, so x ∈ L(λI ;β) 6=
∅. By assumption, we have L(λI ;β) is a near UP-filter of X and so 0 ∈ L(λI ;β). Thus
λI(0) ≤ β = λI(x). Next, let x, y ∈ X. Then λI(y) ∈ [0, 1]. Choose β = λI(y). Thus
λI(y) ≤ β, so y ∈ L(λI ;β) 6= ∅. By assumption, we have L(λI ;β) is a near UP-filter of X
and so x · y ∈ L(λI ;β). Thus λI(x · y) ≤ β = λI(y).

Let x ∈ X. Then λF (x) ∈ [0, 1]. Choose γ = λF (x). Thus λF (x) ≥ γ, so x ∈
U(λF ; γ) 6= ∅. By assumption, we have U(λF ; γ) is a near UP-filter of X and so 0 ∈
U(λF ; γ). Thus λF (0) ≥ γ = λF (x). Next, let x, y ∈ X. Then λF (y) ∈ [0, 1]. Choose
γ = λF (y). Thus λF (y) ≥ γ, so y ∈ U(λF ; γ) 6= ∅. By assumption, we have L(λF ; γ) is a
near UP-filter of X and so x · y ∈ U(λF ; γ). Thus λF (x · y) ≥ γ = λF (y).

Therefore, Λ is a neutrosophic near UP-filter of X.

Theorem 21. A NS Λ in X is a neutrosophic UP-filter of X if and only if for all α, β, γ ∈
[0, 1], the sets U(λT ;α), L(λI ;β), and U(λF ; γ) are UP-filters of X if U(λT ;α), L(λI ;β),
and U(λF ; γ) are nonempty.

Proof. Assume that Λ is a neutrosophic UP-filter of X. Let α, β, γ ∈ [0, 1] be such
that U(λT ;α), L(λI ;β), and U(λF ; γ) are nonempty.

Let x ∈ U(λT ;α). Then λT (x) ≥ α. By (3.6), we have λT (0) ≥ λT (x) ≥ α. Thus
0 ∈ U(λT ;α). Next, let x, y ∈ X be such that x · y ∈ U(λT ;α) and x ∈ U(λT ;α). Then



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1404

λT (x · y) ≥ α and λT (x) ≥ α, so α is an lower bound of {λT (x · y), λT (x)}. By (3.12), we
have λT (y) ≥ min{λT (x · y), λT (x)} ≥ α. Thus y ∈ U(λT ;α).

Let x ∈ L(λI ;β). Then λI(x) ≤ β. By (3.7), we have λI(0) ≤ λI(x) ≤ β. Thus
0 ∈ L(λI ;β). Next, let x, y ∈ X be such that x · y ∈ L(λI ;β) and x ∈ L(λI ;β). Then
λI(x · y) ≤ β and λI(x) ≤ β, so β is a upper bound of {λI(x · y), λI(x)}. By (3.13), we
have λI(y) ≤ max{λI(x · y), λI(x)} ≤ β Thus y ∈ L(λI ;β).

Let x ∈ U(λF ; γ). Then λF (x) ≥ γ. By (3.8), we have λF (0) ≥ λF (x) ≥ γ. Thus
0 ∈ U(λF ; γ). Next, let x, y ∈ X be such that x · y ∈ U(λF ; γ) and x ∈ U(λF ; γ). Then
λF (x · y) ≥ γ and λF (x) ≥ γ, so γ is an lower bound of {λF (x · y), λF (x)}. By (3.14), we
have λF (y) ≥ min{λF (x · y), λF (x)} ≥ γ. Thus y ∈ U(λF ; γ).

Hence, U(λT ;α), L(λI ;β), and U(λF ; γ) are UP-filters of X.

Conversely, assume that for all α, β, γ ∈ [0, 1], the sets U(λT ;α), L(λI ;β), and U(λF ; γ)
are UP-filters of X if U(λT ;α), L(λI ;β), and U(λF ; γ) are nonempty.

Let x ∈ X. Then λT (x) ∈ [0, 1]. Choose α = λT (x). Thus λT (x) ≥ α, so x ∈
U(λT ;α) 6= ∅. By assumption, we have U(λT ;α) is a UP-filter of X and so 0 ∈ U(λT ;α).
Thus λT (0) ≥ α = λT (x). Next, let x, y ∈ X. Then λT (x · y), λT (x) ∈ [0, 1]. Choose
α = min{λT (x · y), λT (x)}. Thus λT (x · y) ≥ α and λT (x) ≥ α, so x · y, x ∈ U(λT ;α) 6= ∅.
By assumption, we have U(λT ;α) is a UP-filter of X and so y ∈ U(λT ;α). Thus λT (y) ≥
α = min{λT (x · y), λT (x)}.

Let x ∈ X. Then λI(x) ∈ [0, 1]. Choose β = λI(x). Thus λI(x) ≤ β, so x ∈
L(λI ;β) 6= ∅. By assumption, we have L(λI ;β) is a UP-filter of X and so 0 ∈ L(λI ;β).
Thus λI(0) ≤ β = λI(x). Next, let x, y ∈ X. Then λI(x · y), λI(x) ∈ [0, 1]. Choose
β = max{λI(x · y), λI(x)}. Thus λI(x · y) ≤ β and λI(x) ≤ β, so x · y, x ∈ L(λI ;β) 6= ∅.
By assumption, we have L(λI ;β) is a UP-filter of X and so y ∈ L(λI ;β). Thus λI(y) ≤
β = max{λI(x · y), λI(x)}.

Let x ∈ X. Then λF (x) ∈ [0, 1]. Choose γ = λF (x). Thus λF (x) ≥ γ, so x ∈
U(λF ; γ) 6= ∅. By assumption, we have U(λF ; γ) is a UP-filter of X and so 0 ∈ U(λF ; γ).
Thus λF (0) ≥ γ = λF (x). Next, let x, y ∈ X. Then λF (x · y), λF (x) ∈ [0, 1]. Choose
γ = min{λF (x · y), λF (x)}. Thus λF (x · y) ≥ γ and λF (x) ≥ γ, so x · y, x ∈ U(λF ; γ) 6= ∅.
By assumption, we have U(λF ; γ) is a UP-filter of X and so y ∈ U(λF ; γ). Thus λF (y) ≥
γ = min{λF (x · y), λF (x)}.

Therefore, Λ is a neutrosophic UP-filter of X.

Theorem 22. A NS Λ in X is a neutrosophic UP-ideal of X if and only if for all α, β, γ ∈
[0, 1], the sets U(λT ;α), L(λI ;β), and U(λF ; γ) are UP-ideals of X if U(λT ;α), L(λI ;β),
and U(λF ; γ) are nonempty.

Proof. Assume that Λ is a neutrosophic UP-ideal of X. Let α, β, γ ∈ [0, 1] be such
that U(λT ;α), L(λI ;β), and U(λF ; γ) are nonempty.

Let x ∈ U(λT ;α). Then λT (x) ≥ α. By (3.6), we have λT (0) ≥ λT (x) ≥ α. Thus
0 ∈ U(λT ;α). Next, let x, y, z ∈ X be such that x · (y · z) ∈ U(λT ;α) and y ∈ U(λT ;α).
Then λT (x · (y · z)) ≥ α and λT (y) ≥ α, so α is an lower bound of {λT (x · (y · z)), λT (y)}.
By (3.15), we have λT (x · z) ≥ min{λT (x · (y · z)), λT (y)} ≥ α. Thus x · z ∈ U(λT ;α).



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1405

Let x ∈ L(λI ;α). Then λI(x) ≤ β. By (3.7), we have λI(0) ≤ λI(x) ≤ β. Thus
0 ∈ L(λI ;β). Next, let x, y, z ∈ X be such that x · (y · z) ∈ L(λI ;β) and y ∈ L(λI ;β).
Then λI(x · (y · z)) ≤ β and λI(y) ≤ β, so β is a upper bound of {λI(x · (y · z)), λI(y)}.
By (3.16), we have λI(x · z) ≤ max{λI(x · (y · z)), λI(y)} ≤ β. Thus x · z ∈ L(λI ;β).

Let x ∈ U(λF ; γ). Then λF (x) ≥ γ. By (3.8), we have λF (0) ≥ λF (x) ≥ γ. Thus
0 ∈ U(λF ; γ). Next, let x, y, z ∈ X be such that x · (y · z) ∈ U(λF ; γ) and y ∈ U(λF ; γ).
Then λF (x · (y · z)) ≥ γ and λF (y) ≥ γ, so γ is an lower bound of {λF (x · (y · z)), λF (y)}.
By (3.17), we have λF (x · z) ≥ min{λF (x · (y · z)), λF (y)} ≥ γ. Thus x · z ∈ U(λF ; γ).

Hence, U(λT ;α), L(λI ;β), and U(λF ; γ) are UP-ideals of X.

Conversely, assume that for all α, β, γ ∈ [0, 1], the sets U(λT ;α), L(λI ;β), and U(λF ; γ)
are UP-ideals of X if U(λT ;α), L(λI ;β), and U(λF ; γ) are nonempty.

Let x ∈ X. Then λT (x) ∈ [0, 1]. Choose α = λT (x). Thus λT (x) ≥ α, so x ∈
U(λT ;α) 6= ∅. By assumption, we have U(λT ;α) is a UP-ideal of X and so 0 ∈ U(λT ;α).
Thus λT (0) ≥ α = λT (x). Next, let x, y, z ∈ X. Then λT (x · (y · z)), λT (y) ∈ [0, 1].
Choose α = min{λT (x · (y · z)), λT (y)}. Thus λT (x · (y · z)) ≥ α and λT (y) ≥ α, so
x · (y · z), y ∈ U(λT ;α) 6= ∅. By assumption, we have U(λT ;α) is a UP-ideal of X and so
x · z ∈ U(λT ;α). Thus λT (x · z) ≥ α = min{λT (x · (y · z)), λT (y)}.

Let x ∈ X. Then λI(x) ∈ [0, 1]. Choose β = λI(x). Thus λI(x) ≤ β, so x ∈
L(λI ;β) 6= ∅. By assumption, we have L(λI ;β) is a UP-ideal of X and so 0 ∈ L(λI ;β).
Thus λI(0) ≤ β = λI(x). Next, let x, y, z ∈ X. Then λI(x · (y · z)), λI(y) ∈ [0, 1].
Choose β = max{λI(x · (y · z)), λI(y)}. Thus λI(x · (y · z)) ≤ β and λI(y) ≤ β, so
x · (y · z), y ∈ L(λI ;β) 6= ∅. By assumption, we have L(λI ;β) is a UP-ideal of X and so
x · z ∈ L(λI ;β). Thus λI(x · z) ≤ β = max{λI(x · (y · z)), λI(y)}.

Let x ∈ X. Then λF (x) ∈ [0, 1]. Choose γ = λF (x). Thus λF (x) ≥ γ, so x ∈
U(λF ; γ) 6= ∅. By assumption, we have U(λF ; γ) is a UP-ideal of X and so 0 ∈ U(λF ; γ).
Thus λF (0) ≥ γ = λF (x). Next, let x, y, z ∈ X. Then λF (x · (y · z)), λF (y) ∈ [0, 1].
Choose γ = min{λF (x · (y · z)), λF (y)}. Thus λF (x · (y · z)) ≥ γ and λF (y) ≥ γ, so
x · (y · z), y ∈ U(λF ; γ) 6= ∅. By assumption, we have U(λF ; γ) is a UP-ideal of X and so
x · z ∈ U(λF ; γ). Thus λF (x · z) ≥ γ = min{λF (x · (y · z)), λF (y)}.

Therefore, Λ is a neutrosophic UP-ideal of X.

Theorem 23. A NS Λ in X is a neutrosophic strongly UP-ideal of X if and only if the
sets E(λT ;λT (0)), E(λI ;λI(0)), and E(λF ;λF (0)) are strongly UP-ideals of X.

Proof. Assume that Λ is a neutrosophic strongly UP-ideal of X. By Theorem 2, we
have Λ is constant, that is, λT , λI , and λF are constant. Thus

(∀x ∈ X)

λT (x) = λT (0)

λI(x) = λI(0)

λF (x) = λF (0)

 .

Hence, E(λT ;λT (0)) = X,E(λI ;λI(0)) = X, and E(λF ;λF (0)) = X and so E(λT ;λT (0)),
E(λI ;λI(0)), and E(λF ;λF (0)) are strongly UP-ideals of X.



M. Songsaeng, A. Iampan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1382-1409 1406

Conversely, assume that E(λT ;λT (0)), E(λI ;λI(0)), and E(λF ;λF (0)) are strongly
UP-ideals of X. Then E(λT ;λT (0)) = X,E(λI ;λI(0)) = X, E(λF ;λF (0)) = X and so

(∀x ∈ X)

λT (x) = λT (0)

λI(x) = λI(0)

λF (x) = λF (0)

 .

Thus λT , λI , and λF are constant, that is, Λ is constant. By Theorem 2, we have Λ is a
neutrosophic strongly UP-ideal of X.

Definition 11. Let Λ be a NS in X. For α, β, γ ∈ [0, 1], the sets

ULUΛ(α, β, γ) = {x ∈ X | λT ≥ α, λI ≤ β, λF ≥ γ},
LULΛ(α, β, γ) = {x ∈ X | λT ≤ α, λI ≥ β, λF ≤ γ},

EΛ(α, β, γ) = {x ∈ X | λT = α, λI = β, λF = γ}

are called a ULU -(α, β, γ)-level subset, a LUL-(α, β, γ)-level subset, and an E-(α, β, γ)-
level subset of Λ, respectively. Then we see that

ULUΛ(α, β, γ) = U(λT ;α) ∩ L(λI ;β) ∩ U(λF ; γ),

LULΛ(α, β, γ) = L(λT ;α) ∩ U(λI ;β) ∩ L(λF ; γ),

EΛ(α, β, γ) = E(λT ;α) ∩ E(λI ;β) ∩ E(λF ; γ).

Corollary 1. A NS Λ in X is a neutrosophic UP-subalgebra of X if and only if for all
α, β, γ ∈ [0, 1], ULUΛ(α, β, γ) is a UP-subalgebra of X where ULUΛ(α, β, γ) is nonempty.

Proof. It is straightforward by Theorem 19.

Corollary 2. A NS Λ in X is a neutrosophic near UP-filter of X if and only if for all
α, β, γ ∈ [0, 1], ULUΛ(α, β, γ) is a near UP-filter of X where ULUΛ(α, β, γ) is nonempty.

Proof. It is straightforward by Theorem 20.

Corollary 3. A NS Λ in X is a neutrosophic UP-filter of X if and only if for all α, β, γ ∈
[0, 1], ULUΛ(α, β, γ) is a UP-filter of X where ULUΛ(α, β, γ) is nonempty.

Proof. It is straightforward by Theorem 21.

Corollary 4. A NS Λ in X is a neutrosophic UP-ideal of X if and only if for all α, β, γ ∈
[0, 1], ULUΛ(α, β, γ) is a UP-ideal of X where ULUΛ(α, β, γ) is nonempty.

Proof. It is straightforward by Theorem 22.

Corollary 5. A NS Λ in X is a neutrosophic strongly UP-ideal of X if and only if
E(λT , λT (0)), E(λI , λI(0)), and E(λF , λF (0)) are strongly UP-ideals of X, that is, E(λT , λT (0)) =
X,E(λI , λI(0)) = X, and E(λF , λF (0)) = X.

Proof. It is straightforward by Theorem 23.



REFERENCES 1407

5. Conclusions

In this paper, we have introduced the notions of neutrosophic UP-subalgebras, neutro-
sophic near UP-filters, neutrosophic UP-filters, neutrosophic UP-ideals, and neutrosophic
strongly UP-ideals of UP-algebras and investigated some of their important properties.
Then, we get the diagram of generalization of NSs in UP-algebras as shown in Figure 1.

Figure 1: NSs in UP-algebras

In our future study, we will apply this notion/results to other type of NSs in UP-
algebras. Also, we will study the soft set theory/cubic set theory of neutrosophic UP-
subalgebras, neutrosophic near UP-filters, neutrosophic UP-filters, neutrosophic UP-ideals,
and neutrosophic strongly UP-ideals.

Acknowledgements

The authors would also like to thank the anonymous referee for giving many helpful
suggestion on the revision of present paper.

References

[1] M. A. Ansari, A. Haidar, and A. N. A. Koam. On a graph associated to UP-algebras.
Math. Comput. Appl., 23(4):61, 2018.

[2] N. Dokkhamdang, A. Kesorn, and A. Iampan. Generalized fuzzy sets in UP-algebras.
Ann. Fuzzy Math. Inform., 16(2):171–190, 2018.



REFERENCES 1408

[3] T. Guntasow, S. Sajak, A. Jomkham, and A. Iampan. Fuzzy translations of a fuzzy
set in UP-algebras. J. Indones. Math. Soc., 23(2):1–19, 2017.

[4] Q. P. Hu and X. Li. On BCH-algebras. Math. Sem. Notes Kobe Univ., 11(2):313–320,
1983.

[5] A. Iampan. A new branch of the logical algebra: UP-algebras. J. Algebra Relat. Top.,
5(1):35–54, 2017.

[6] A. Iampan. Introducing fully UP-semigroups. Discuss. Math., Gen. Algebra Appl.,
38(2):297–306, 2018.

[7] Y. Imai and K. Iséki. On axiom system of propositional calculi, XIV. Proc. Japan
Acad., 42(1):19–22, 1966.

[8] K. Iséki. An algebra related with a propositional calculus. Proc. Japan Acad.,
42(1):26–29, 1966.

[9] Y. B. Jun, F. Smarandache, and H. Bordbar. Neutrosophic N -structures applied to
BCK/BCI-algebras. Inform., 8(4):128, 2017.

[10] Y. B. Jun, F. Smarandache, S.-Z. Song, and M. Khan. Neutrosophic positive implica-
tive N -ideals in BCK-algebras. Axioms, 7(1):3, 2018.

[11] W. Kaijae, P. Poungsumpao, S. Arayarangsi, and A. Iampan. UP-algebras character-
ized by their anti-fuzzy UP-ideals and anti-fuzzy UP-subalgebras. Ital. J. Pure Appl.
Math., 36:667–692, 2016.

[12] K. Kawila, C. Udomsetchai, and A. Iampan. Bipolar fuzzy UP-algebras. Math.
Comput. Appl., 23(4):69, 2018.

[13] S. Keawrahun and U. Leerawat. On isomorphisms of SU-algebras. Sci. Magna.,
7(2):39–44, 2011.

[14] B. Kesorn, K. Maimun, W. Ratbandan, and A. Iampan. Intuitionistic fuzzy sets in
UP-algebras. Ital. J. Pure Appl. Math., 34:339–364, 2015.

[15] M. Khan, S. Anis, F. Smarandache, and Y. B. Jun. Neutrosophic N -structures and
their applications in semigroups. Ann. Fuzzy Math. Inform., 14:583–598, 2017.

[16] S. J. Ki, S.-Z. Song, and Y. B. Jun. Generalizations of neutrosophic subalgebras in
BCK/BCI-algebras based on neutrosophic points. Neutrosophic Sets Syst., 20:26–35,
2018.

[17] T. Klinseesook, S. Bukok, and A. Iampan. Rough set theory applied to UP-algebras.
Manuscript accepted for publication in J. Inf. Optim. Sci., November 2018.

[18] C. Prabpayak and U. Leerawat. On ideals and congruences in KU-algebras. Sci.
Magna, 5(1):54–57, 2009.



REFERENCES 1409

[19] P. Rangsuk, P. Huana, and A. Iampan. Neutrosophic N -structures over UP-algebras.
Neutrosophic Sets Syst., 28:87–127, 2019.

[20] A. Satirad, P. Mosrijai, and A. Iampan. Formulas for finding UP-algebras. Int. J.
Math. Comput. Sci., 14(2):403–409, 2019.

[21] A. Satirad, P. Mosrijai, and A. Iampan. Generalized power UP-algebras. Int. J.
Math. Comput. Sci., 14(1):17–25, 2019.

[22] F. Smarandache. A Unifying Field in Logics: Neutrosophic Logic, Neutrosophy, Neu-
trosophic Set, Neutrosophic Probability. American Research Press, 1999.

[23] J. Somjanta, N. Thuekaew, P. Kumpeangkeaw, and A. Iampan. Fuzzy sets in UP-
algebras. Ann. Fuzzy Math. Inform., 12(6):739–756, 2016.

[24] M. Songsaeng and A. Iampan. N -fuzzy UP-algebras and its level subsets. J. Algebra
Relat. Top., 6(1):1–24, 2018.

[25] S. Sripaeng, K. Tanamoon, and A. Iampan. On anti Q-fuzzy UP-ideals and anti
Q-fuzzy UP-subalgebras of UP-algebras. J. Inf. Optim. Sci., 39(5):1095–1127, 2018.

[26] K. Tanamoon, S. Sripaeng, and A. Iampan. Q-fuzzy sets in UP-algebras. Songk-
lanakarin J. Sci. Technol., 40(1):9–29, 2018.

[27] N. Udten, N. Songseang, and A. Iampan. Translation and density of a bipolar-valued
fuzzy set in UP-algebras. Ital. J. Pure Appl. Math., 41:469–496, 2019.

[28] H. Wang, F. Smarandache, Y. Q. Zhang, and R. Sunderraman. Interval Neutrosophic
Sets and logic: Theory and Applications in Computing. Hexis, Phoenix, Ariz, USA,
2005.

[29] L. A. Zadeh. Fuzzy sets. Inf. Cont., 8:338–353, 1965.