EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 3, 2020, 459-471
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Fuzzy duplex UP-algebras†

Aiyared Iampan1, Metawee Songsaeng1, G. Muhiuddin2,∗

1 Department of Mathematics, School of Science, University of Phayao, Phayao 56000,
Thailand
2 Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia

Abstract. Using the concept of a neutrosophic quadruple number to a fuzzy duplex number,
we introduce the concept of a fuzzy duplex UP-algebra, and investigate some related properties.
Also, we find the necessary condition for a fuzzy duplex UP-set to be a fuzzy duplex UP-algebra.
Furthermore, we study the relationship between special subsets of a UP-algebra and special subsets
of a fuzzy duplex UP-set.

2020 Mathematics Subject Classifications: 03G25, 08A72
Key Words and Phrases: UP-algebra, UP-subalgebra, near UP-filter, UP-filter, UP-ideal,
strong UP-ideal, fuzzy duplex UP-set, fuzzy duplex UP-algebra

1. Introduction

The type of the logical algebra, a UP-algebra was introduced by Iampan [9], and it is
known that the class of KU-algebras is a proper subclass of the class of UP-algebras. Later
Somjanta et al. [29] studied fuzzy UP-subalgebras, fuzzy UP-ideals and fuzzy UP-filters of
UP-algebras. Guntasow et al. [7] studied fuzzy translations of a fuzzy set in UP-algebras.
Kesorn et al. [16] studied intuitionistic fuzzy sets in UP-algebras. Kaijae et al. [15]
studied anti-fuzzy UP-ideals and anti-fuzzy UP-subalgebras. Tanamoon et al. [35] and
Sripaeng et al. [34] introduced the concept of Q-fuzzy sets in UP-algebras, and studied
anti Q-fuzzy UP-ideals and anti Q-fuzzy UP-subalgebras of UP-algebras. Dokkhamdang et
al. [6] introduced the concept of fuzzy UP-subalgebras (fuzzy UP-filters, fuzzy UP-ideals,
fuzzy strong UP-ideals) with thresholds of UP-algebras.

Ansari et al. [3] introduced the concept of graphs associated with commutative
UP-algebras and defined a graph of equivalence classes of commutative UP-algebras.
Songsaeng and Iampan [31–33] studied N -fuzzy sets, fuzzy proper UP-filters, and neu-
trosophic sets in UP-algebras. Senapati et al. [26, 27] studies applied cubic set and
interval-valued intuitionistic fuzzy structure in UP-algebras.

†This work was supported by the Unit of Excellence, University of Phayao.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i3.3752

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

https://www.ejpam.com 459 c© 2020 EJPAM All rights reserved.



A. Iampan, M. Songsaeng, G. Muhiuddin / Eur. J. Pure Appl. Math, 13 (3) (2020), 459-471 460

More concepts on UP-algebras are discussed in [4, 5, 17].
A fuzzy set f in a nonempty set S is a function from S to the closed interval [0, 1].

The concept of a fuzzy set in a nonempty set was first considered by Zadeh [36] in 1965.
The fuzzy set theories developed by Zadeh and others have found many applications in
the domain of mathematics and elsewhere.

The concept of a neutrosophic set was introduced by Smarandache [28] in 1999. Neu-
trosophic algebraic structures in BCK/BCI-algebras are discussed in [11, 12, 19, 21, 30].
Neutrosophic quadruple algebraic structures and hyperstructure are discussed in [1, 2].
Neutrosophic quadruple algebraic structures in BCK/BCI-algebras are discussed in [13,
14, 18, 20, 22].

In this paper, we apply the concept of a neutrosophic quadruple number to a fuzzy
duplex number, introduce the concept of a fuzzy duplex set base on a UP-algebra, which
is called a fuzzy duplex UP-set, and investigate some related properties. We find the
necessary conditions that a fuzzy duplex UP-set form a UP-algebra, which is called a
fuzzy duplex UP-algebra. Furthermore, we study the relationship between special subsets
of a UP-algebra and the same special subsets of a fuzzy duplex UP-set.

2. Basic concepts and preliminary notes on UP-algebras

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

Definition 1. [9] 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 [9], we know that the concept of UP-algebras is a generalization of KU-algebras
(see [23]).

For more examples of UP-algebras, see [6, 10, 24–27].

In a UP-algebra X = (X, ·, 0), the following assertions are valid (see [9, 10]).

(∀x ∈ X)(x · x = 0), (1)

(∀x, y, z ∈ X)(x · y = 0, y · z = 0⇒ x · z = 0), (2)

(∀x, y, z ∈ X)(x · y = 0⇒ (z · x) · (z · y) = 0), (3)

(∀x, y, z ∈ X)(x · y = 0⇒ (y · z) · (x · z) = 0), (4)

(∀x, y ∈ X)(x · (y · x) = 0), (5)



A. Iampan, M. Songsaeng, G. Muhiuddin / Eur. J. Pure Appl. Math, 13 (3) (2020), 459-471 461

(∀x, y ∈ X)((y · x) · x = 0⇔ x = y · x), (6)

(∀x, y ∈ X)(x · (y · y) = 0), (7)

(∀a, x, y, z ∈ X)((x · (y · z)) · (x · ((a · y) · (a · z))) = 0), (8)

(∀a, x, y, z ∈ X)((((a · x) · (a · y)) · z) · ((x · y) · z) = 0), (9)

(∀x, y, z ∈ X)(((x · y) · z) · (y · z) = 0), (10)

(∀x, y, z ∈ X)(x · y = 0⇒ x · (z · y) = 0), (11)

(∀x, y, z ∈ X)(((x · y) · z) · (x · (y · z)) = 0), and (12)

(∀a, x, y, z ∈ X)(((x · y) · z) · (y · (a · z)) = 0). (13)

From [9], the binary relation ≤ on a UP-algebra X = (X, ·, 0) defined as follows:

(∀x, y ∈ X)(x ≤ y ⇔ x · y = 0). (14)

Definition 2. [7–9, 29] A nonempty subset S of a UP-algebra X = (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 strong UP-ideal (renamed from 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. [7] and Iampan [8] proved that the concept of UP-subalgebras is a
generalization of near UP-filters, near UP-filters is a generalization of UP-filters, UP-filters
is a generalization of UP-ideals, and UP-ideals is a generalization of strong UP-ideals.
Furthermore, they proved that the only strong UP-ideal of a UP-algebra X is X.



A. Iampan, M. Songsaeng, G. Muhiuddin / Eur. J. Pure Appl. Math, 13 (3) (2020), 459-471 462

3. Fuzzy duplex UP-algebras

In this section, we introduce the concepts of fuzzy duplex UP-numbers and fuzzy
duplex UP-sets, and investigate some properties. We find the necessary conditions that a
fuzzy duplex UP-set form a UP-algebra. Furthermore, we study the relationship between
special subsets of a UP-algebra and the same special subsets of a fuzzy duplex UP-set.

Definition 3. Let X and Y be nonempty sets and T : X → Y be a function. A fuzzy
duplex X-number is an ordered pair (x, yT ), where x, y ∈ X, and T (y) denoted by yT .
The Cartesian product X × Im(T ) is called the fuzzy duplex set based on X. If X is a
UP-algebra, a fuzzy duplex X-number is called a fuzzy duplex UP-number and we say that
X × Im(T ) is the fuzzy duplex UP-set. For any two nonempty subsets A and B of X,
we see that A × T (B) is a nonempty subset of X × Im(T ). If (a, yT ) ∈ A × T (B), then
(x, yT ) ∈ A× T (B) for all x ∈ A.

In what follows, X will denote a UP-algebra (X, ·, 0), Y will denote a nonempty set,
and T : X → Y will be a function.

We define the binary operation � on the fuzzy duplex UP-set X × Im(T ) by

(∀(a, xT ), (b, yT ) ∈ X × Im(T ))((a, xT )� (b, yT ) = (a · b, (x · y)T )). (15)

If the algebra (X × Im(T ),�, 0̃) is a UP-algebra, then it is called the fuzzy duplex UP-
algebra. We denote by ã the fuzzy duplex UP-number, that is, ã = (a1, a2T ) for some
a1, a2 ∈ X, and the zero fuzzy duplex UP-number (0, 0T ) is denoted by 0̃. We define the
binary relation � and the equality

.
= on X × Im(T ) as follows:

(∀(a, xT ), (b, yT ) ∈ X × Im(T ))

(
(a, xT )� (b, yT )⇔ a ≤ b, x ≤ y

(a, xT )
.
= (b, yT )⇔ (a, xT )� (b, yT ), (b, yT )� (a, xT )

)
.

Then we can easily prove that the binary relation � is an order relation on X × Im(T )
and

(∀(a, xT ), (b, yT ) ∈ X × Im(T ))

(
(a, xT )� (b, yT )⇔ (a, xT )� (b, yT ) = 0̃

(a, xT )
.
= (b, yT )⇔ a = b, x = y

)
.

Hence,
.
=⊆= on X × Im(T ).

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

· 0 a b c

0 0 a b c
a 0 0 b b
b 0 a 0 b
c 0 a 0 0



A. Iampan, M. Songsaeng, G. Muhiuddin / Eur. J. Pure Appl. Math, 13 (3) (2020), 459-471 463

Let T : X → {0.5, 1} be a function defined by

0T = aT = bT = 0.5, cT = 1.

Then the axiom (UP-4) is not satisfied. Indeed, there are (0, aT ), (0, cT ) ∈ X × {0.5, 1}
such that (0, aT ) = (0, 0.5) 6= (0, 1) = (0, cT ) but

(0, aT )� (0, cT ) = (0 · 0, (a · c)T ) = (0, bT ) = (0, 0T ) = 0̃

and

(0, cT )� (0, aT ) = (0 · 0, (c · a)T ) = (0, aT ) = (0, 0T ) = 0̃.

Hence, the algebra (X × {0.5, 1},�, 0̃) is not a UP-algebra.

Theorem 1. The algebra (X × Im(T ),�, 0̃) satisfies the axioms (UP-1), (UP-2), and
(UP-3).

Proof. (UP-1) Let x̃, ỹ, z̃ ∈ X × Im(T ) where x̃ = (x1, x2T ), ỹ = (y1, y2T ), and
z̃ = (z1, z2T ). Then

(ỹ � z̃)� ((x̃� ỹ)� (x̃� z̃))

= ((y1, y2T )� (z1, z2T ))� (((x1, x2T )� (y1, y2T ))� ((x1, x2T )�(z1, z2T )))

= (y1 · z1, (y2 · z2)T )� ((x1 · y1, (x2 · y2)T )� (x1 · z1, (x2 · z2)T ))

= (y1 · z1, (y2 · z2)T )� ((x1 · y1) · (x1 · z1), ((x2 · y2) · (x2 · z2))T )

= ((y1 · z1) · ((x1 · y1) · (x1 · z1)), ((y2 · z2) · ((x2 · y2) · (x2 · z2)))T )

= (0, 0T ) ((UP-1))

= 0̃.

(UP-2) Let x̃ ∈ X × Im(T ) where x̃ = (x1, x2T ). Then

0̃� x̃ = (0, 0T )� (x1, x2T )

= (0 · x1, (0 · x2)T )

= (x1, x2T ) ((UP-2))

= x̃.

(UP-3) Let x̃ ∈ X × Im(T ) where x̃ = (x1, x2T ). Then

x̃� 0̃ = (x1, x2T )� (0, 0T )

= (x1 · 0, (x2 · 0)T )

= (0, 0T ) ((UP-3))

= 0̃.

Hence, (UP-1), (UP-2), and (UP-3) are valid.



A. Iampan, M. Songsaeng, G. Muhiuddin / Eur. J. Pure Appl. Math, 13 (3) (2020), 459-471 464

Proposition 1. The algebra (X × Im(T ),�, 0̃) satisfies the following properties:

(1) (∀ã ∈ X × Im(T ))(ã� ã),

(2) (∀ã, b̃, c̃ ∈ X × Im(T ))(ã� b̃, b̃� c̃⇒ ã� c̃),

(3) (∀ã, b̃, c̃ ∈ X × Im(T ))(ã� b̃⇒ c̃� ã� c̃� b̃),

(4) (∀ã, b̃, c̃ ∈ X × Im(T ))(ã� b̃⇒ b̃� c̃� ã� c̃),

(5) (∀ã, b̃ ∈ X × Im(T ))(ã� b̃� ã),

(6) (∀ã, b̃ ∈ X × Im(T ))(ã� b̃� b̃),

(7) (∀x̃, ã, b̃, c̃ ∈ X × Im(T ))(ã� (b̃� c̃)� ã� ((x̃� b̃)� (x̃� c̃))),

(8) (∀x̃, ã, b̃, c̃ ∈ X × Im(T ))(((x̃� ã)� (x̃� b̃))� c̃� (ã� b̃)� c̃),

(9) (∀ã, b̃, c̃ ∈ X × Im(T ))((ã� b̃)� c̃� b̃� c̃),

(10) (∀ã, b̃, c̃ ∈ X × Im(T ))(ã� b̃⇒ ã� c̃� b̃),

(11) (∀ã, b̃, c̃ ∈ X × Im(T ))((ã� b̃)� c̃� ã� (b̃� c̃)), and

(12) (∀x̃, ã, b̃, c̃ ∈ X × Im(T ))((ã� b̃)� c̃� b̃� (x̃� c̃)).

Proof. By Theorem 1, the algebra (X × Im(T ),�, 0̃) satisfies the axioms (UP-1),
(UP-2), and (UP-3).

(1) Let ã ∈ X × Im(T ). Then

0̃ = (0̃� ã)� ((0̃� 0̃)� (0̃� ã)) ((UP-1))

= (0̃� ã)� (0̃� ã) ((UP-2))

= ã� ã. ((UP-2))

Hence, ã� ã.

(2) Let ã, b̃, c̃ ∈ X×Im(T ) be such that ã� b̃ and b̃� c̃. Then ã� b̃ = 0̃ and b̃� c̃ = 0̃.
Thus

ã� c̃ = 0̃� (0̃� (ã� c̃)) ((UP-2))

= (b̃� c̃)� ((ã� b̃)� (ã� c̃))

= 0̃. ((UP-1))

Hence, ã� c̃.

(3) Let ã, b̃ ∈ X × Im(T ) be such that ã� b̃. Then ã� b̃ = 0̃.

(c̃� ã)� (c̃� b̃) = 0̃� ((c̃� ã)� (c̃� b̃)) ((UP-2))



A. Iampan, M. Songsaeng, G. Muhiuddin / Eur. J. Pure Appl. Math, 13 (3) (2020), 459-471 465

= (ã� b̃)� ((c̃� ã)� (c̃� b̃))

= 0̃. ((UP-1))

Hence, c̃� ã� c̃� b̃.

(4) Let ã, b̃ ∈ X × Im(T ) be such that ã� b̃. Then ã� b̃ = 0̃.

(b̃� c̃)� (ã� c̃) = (b̃� c̃)� (0̃� (ã� c̃)) ((UP-2))

= (b̃� c̃)� ((ã� b̃)� (ã� c̃))

= 0̃. ((UP-1))

Hence, b̃� c̃� ã� c̃.

(5) Let ã, b̃ ∈ X × Im(T ). Then

ã� (b̃� ã) = (0̃� ã)� (0̃� (b̃� ã)) ((UP-2))

= (0̃� ã)� ((b̃� 0̃)� (b̃� ã)) ((UP-3))

= 0̃. ((UP-1))

Hence, ã� b̃� ã.

(6) Let ã, b̃ ∈ X × Im(T ). By (UP-3) and (1), we have ã� (b̃� b̃) = ã� 0̃ = 0̃. Hence,
ã� b̃� b̃.

(7) Let x̃, ã, b̃, c̃ ∈ X × Im(T ). By (UP-1), we have (b̃ � c̃) � ((x̃ � b̃) � (x̃ � c̃)) = 0̃.
Thus b̃� c̃� (x̃� b̃)� (x̃� c̃). By (3), we have ã� (b̃� c̃)� ã� ((x̃� b̃)� (x̃� c̃)).

(8) Let x̃, ã, b̃, c̃ ∈ X × Im(T ). By (UP-1), we have (ã � b̃) � ((x̃ � ã) � (x̃ � b̃)) = 0̃.
Thus ã� b̃� (x̃� ã)� (x̃� b̃). By (4), we have ((x̃� ã)� (x̃� b̃))� c̃� (ã� b̃)� c̃.

(9) Let ã, b̃, c̃ ∈ X × Im(T ). Then

0̃ = (((ã� 0̃)� (ã� b̃))� c̃)� ((0̃� b̃)� c̃) ((8))

= ((0̃� (ã� b̃))� c̃)� (b̃� c̃) ((UP-2), (UP-3))

= ((ã� b̃)� c̃)� (b̃� c̃). ((UP-2))

Hence, (ã� b̃)� c̃� b̃� c̃.

(10) Let ã, b̃, c̃ ∈ X × Im(T ) be such that ã� b̃. By (3), we have (c̃� ã)� (c̃� b̃) = 0̃.
Thus

ã� (c̃� b̃) = 0̃� (ã� (c̃� b̃)) ((UP-2))

= ((c̃� ã)� (c̃� b̃))� (ã� (c̃� b̃))

= 0̃. ((9))

Hence, ã� c̃� b̃.



A. Iampan, M. Songsaeng, G. Muhiuddin / Eur. J. Pure Appl. Math, 13 (3) (2020), 459-471 466

(11) Let ã, b̃, c̃ ∈ X × Im(T ). By (9), we have (ã � b̃) � c̃ � b̃ � c̃. By (5), we have
b̃� c̃� ã� (b̃� c̃). It follows from (2) that (ã� b̃)� c̃� ã� (b̃� c̃).

(12) Let x̃, ã, b̃, c̃ ∈ X × Im(T ). By (5), we have b̃� ã� b̃ and ã� b̃� x̃� (ã� b̃). By
(2), we have b̃� x̃� (ã� b̃). By (4), we have

(x̃� (ã� b̃))� (x̃� c̃)� b̃� (x̃� c̃).

By (UP-1), we have ((ã� b̃)� c̃)� ((x̃� (ã� b̃))� (x̃� c̃)) = 0̃. Then

(ã� b̃)� c̃� (x̃� (ã� b̃))� (x̃� c̃).

It follows from (2) that (ã� b̃)� c̃� b̃� (x̃� c̃).

Theorem 2. If T : X → Y is a constant function, that is, the inverse image T−1({0T}) =
X, then the algebra (X × Im(T ),�, 0̃) is a UP-algebra which is UP-isomorphic to X.

Proof. (UP-4) Let x̃, ỹ ∈ X × Im(T ) be such that x̃ � ỹ = 0̃ and ỹ � x̃ = 0̃ where
x̃ = (x1, x2T ), ỹ = (y1, y2T ). Then

(x1 · y1, (x2 · y2)T ) = (x1, x2T )� (y1, y2T ) = (0, 0T )

and

(y1 · x1, (y2 · x2)T ) = (y1, y2T )� (x1, x2T ) = (0, 0T ).

It follows that x1 ·y1 = 0 and y1 ·x1 = 0. By (UP-4), we have x1 = y1. Since T is constant,
we have x2T = y2T . Thus x̃ = (x1, x2T ) = (y1, y2T ) = ỹ, (UP-4) holding. By Theorem 1,
we have (X × Im(T ),�, 0̃) is a UP-algebra. Finally, X and X × Im(T ) are UP-isomorphic
under the UP-isomorphism sending x 7→ (x, 0T ).

Corollary 1. If Y is a singleton set, then the algebra (X × Im(T ),�, 0̃) is a UP-algebra.

Proof. If Y is a singleton set, then T : X → Y is a constant function. By Theorem 2,
we have the algebra (X × Im(T ),�, 0̃) is a UP-algebra.

Theorem 3. If T : X → Y is a function with the inverse image T−1({0T}) = {0}, then
the algebra (X × Im(T ),�, 0̃) is a UP-algebra.

Proof. (UP-4) Let x̃, ỹ ∈ X × Im(T ) be such that x̃ � ỹ = 0̃ and ỹ � x̃ = 0̃ where
x̃ = (x1, x2T ), ỹ = (y1, y2T ). Then

(x1 · y1, (x2 · y2)T ) = (x1, x2T )� (y1, y2T ) = (0, 0T )

and

(y1 · x1, (y2 · x2)T ) = (y1, y2T )� (x1, x2T ) = (0, 0T ).

It follows that x1 · y1 = 0 and y1 · x1 = 0, and x2 · y2, y2 · x2 ∈ T−1({0T}) = {0}, that is,
x2·y2 = 0 and y2·x2 = 0. By (UP-4), we have x1 = y1 and x2 = y2. Thus x2T = y2T and so
x̃ = (x1, x2T ) = (y1, y2T ) = ỹ, (UP-4) holding. By Theorem 1, we have (X × Im(T ),�, 0̃)
is a UP-algebra.



A. Iampan, M. Songsaeng, G. Muhiuddin / Eur. J. Pure Appl. Math, 13 (3) (2020), 459-471 467

Corollary 2. If T : X → Y is an injective function, then the algebra (X × Im(T ),�, 0̃)
is a UP-algebra.

Proof. If T : X → Y is an injective function, then the inverse image T−1({0T}) = {0}.
By Theorem 3, we have the algebra (X × Im(T ),�, 0̃) is a UP-algebra.

Theorem 4. Let A and B be nonempty subsets of a UP-algebra X and (X× Im(T ),�, 0̃)
be a fuzzy duplex UP-algebra.

(1) If A and B are UP-subalgebras of X, then A×T (B) is a UP-subalgebra of X×Im(T ).

(2) If A× T (B) is a UP-subalgebra of X × Im(T ), then A is a UP-subalgebra of X.

Proof. (1) Assume that A and B are UP-subalgebras of X and let x̃, ỹ ∈ A × T (B)
where x̃ = (a1, b1T ) and ỹ = (a2, b2T ). Then a1 · a2 ∈ A and b1 · b2 ∈ B. Thus x̃ � ỹ =
(a1, b1T )�(a2, b2T ) = (a1 ·a2, (b1 ·b2)T ) ∈ A×T (B). Hence, A×T (B) is a UP-subalgebra
of X × Im(T ).

(2) Assume that A × T (B) is a UP-subalgebra of X × Im(T ). Let x, y ∈ A. Since
(0, 0T ) ∈ A × T (B), we have Then (x, 0T ), (y, 0T ) ∈ A × T (B). Thus (x · y, 0T ) =
(x · y, (0 · 0)T ) = (x, 0T )� (y, 0T ) ∈ A× T (B), so x · y ∈ A. Hence, A is a UP-subalgebra
of X.

Theorem 5. Let A and B be nonempty subsets of a UP-algebra X and (X× Im(T ),�, 0̃)
be a fuzzy duplex UP-algebra.

(1) If A and B are near UP-filters of X, then A×T (B) is a near UP-filter of X×Im(T ).

(2) If A× T (B) is a near UP-filter of X × Im(T ), then A is a near UP-filter of X.

Proof. (1) Assume that A and B are near UP-filters of X. Since 0 ∈ A and 0 ∈ B, we
have 0̃ = (0, 0T ) ∈ A× T (B). Let x̃ ∈ X × Im(T ) and ỹ ∈ A× T (B) where x̃ = (x1, x2T )
and ỹ = (a, bT ). Thus x1 · a ∈ A and x2 · b ∈ B, so x̃ � ỹ = (x1, x2T ) � (a, bT ) =
(x1 · a, (x2 · b)T ) ∈ A× T (B). Hence, A× T (B) is a near UP-filter of X × Im(T ).

(2) Assume that A × T (B) is a near UP-filter of X × Im(T ). Since 0̃ = (0, 0T ) ∈
A × T (B), we have 0 ∈ A. Let x ∈ X and a ∈ A. Then (x, 0T ) ∈ X × Im(T ) and
(a, 0T ) ∈ A× T (B). Thus (x · a, 0T ) = (x · a, (0 · 0)T ) = (x, 0T )� (a, 0T ) ∈ A× T (B), so
x · a ∈ A. Hence, A is a near UP-filter of X.

Theorem 6. Let A and B be nonempty subsets of a UP-algebra X and (X× Im(T ),�, 0̃)
be a fuzzy duplex UP-algebra. If A × T (B) is a UP-filter of X × Im(T ), then A is a
UP-filter of X.

Proof. Assume that A×T (B) is a UP-filter of X×Im(T ). Since 0̃ = (0, 0T ) ∈ A×T (B),
we have 0 ∈ A. Let x, a ∈ X be such that a · x ∈ A and a ∈ A. Then (a, 0T )� (x, 0T ) =
(a ·x, (0 ·0)T ) = (a ·x, 0T ) ∈ A×T (B) and (a, 0T ) ∈ A×T (B). Thus (x, 0T ) ∈ A×T (B),
so x ∈ A. Hence, A is a UP-filter of X.



A. Iampan, M. Songsaeng, G. Muhiuddin / Eur. J. Pure Appl. Math, 13 (3) (2020), 459-471 468

Theorem 7. Let A and B be nonempty subsets of a UP-algebra X and (X× Im(T ),�, 0̃)
be a fuzzy duplex UP-algebra. If A×T (B) is a UP-ideal of X×Im(T ), then A is a UP-ideal
of X.

Proof. Assume that A × T (B) is a UP-ideal of X × Im(T ). Since 0̃ = (0, 0T ) ∈
A × T (B), we have 0 ∈ A. Let x, y, z ∈ X be such that x · (y · z) ∈ A and y ∈ A. Then
(x, 0T )� ((y, 0T )� (z, 0T )) = (x · (y · z), (0 · (0 · 0))T ) = (x · (y · z), 0T ) ∈ A× T (B) and
(y, 0T ) ∈ A× T (B). Thus (x · z, 0T ) = (x · z, (0 · 0)T ) = (x, 0T )� (z, 0T ) ∈ A× T (B), so
x · z ∈ A. Hence, A is a UP-ideal of X.

The following example shows that the sentence “if A and B are UP-filters (resp., UP-
ideals) of X, then A× T (B) is a UP-filter (resp., UP-ideal) of X × Im(T )” does not hold
in general.

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

· 0 a b c

0 0 a b c
a 0 0 b b
b 0 a 0 a
c 0 0 0 0

Let T : X → {0.5, 0.7, 1} be a function defined by

0T = 0.5, aT = bT = 0.7, cT = 1.

Let A = {0, a}. Then A is a UP-ideal (also a UP-filter) of X and

A× T (A) = {(0, 0T ), (0, aT ), (a, 0T ), (a, aT )}.

Since (0, aT ) � (0, cT ) = (0, bT ) = (0, aT ) ∈ A × T (A) and (0, aT ) ∈ A × T (A) but
(0, cT ) /∈ A× T (A). Hence, A× T (A) is not a UP-filter (also not a UP-ideal) of X.

Theorem 8. Let A and B be nonempty subsets of a UP-algebra X and (X× Im(T ),�, 0̃)
be a fuzzy duplex UP-algebra.

(1) If A and B are strong UP-ideals of X, then A × T (B) is a strong UP-ideal of
X × Im(T ).

(2) If A× T (B) is a strong UP-ideal of X × Im(T ), then A is a strong UP-ideal of X.

Proof. (1) Assume that A and B are strong UP-ideals of X. Then A = B = X, so
A× T (B) = X × Im(T ). Hence, A× T (B) is a strong UP-ideal of X × Im(T ).

(2) Assume that A × T (B) is a strong UP-ideal of X × Im(T ). Then A × T (B) =
X × Im(T ), so A = X. Hence, A is a strong UP-ideal of X.



REFERENCES 469

4. Conclusions

In this paper, we have introduced the concept of a fuzzy duplex set base on a UP-
algebra, which is called a fuzzy duplex UP-set, and investigated some related properties.
We have found the necessary conditions that a fuzzy duplex UP-set form a UP-algebra,
which is called a fuzzy duplex UP-algebra. Furthermore, we have studied the relationship
between special subsets of a UP-algebra and the same special subsets of a fuzzy duplex
UP-set and have presented conflicting examples for certain relationships.

Acknowledgements

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

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