EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 2, 2019, 294-331
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Fuzzy Soft Sets over Fully UP-Semigroups∗

Akarachai Satirad1, Aiyared Iampan1,†

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

Abstract. In this paper, we introduce ten types of fuzzy soft sets over fully UP-semigroups, and
investigate the algebraic properties of fuzzy soft sets under the operations of (extended) intersection
and (restricted) union. Further, we discuss the relation between some conditions of fuzzy soft sets
and fuzzy soft UPs-subalgebras (resp., fuzzy soft UPi-subalgebras, fuzzy soft near UPs-filters, fuzzy
soft near UPi-filters, fuzzy soft UPs-filters, fuzzy soft UPi-filters, fuzzy soft UPs-ideals, fuzzy soft
UPi-ideals, fuzzy soft strongly UPs-ideals, fuzzy soft strongly UPi-ideals) of fully UP-semigroups.

2010 Mathematics Subject Classifications: 03G25, 08A72

Key Words and Phrases: UP-algebra, fully UP-semigroup, fuzzy soft set.

1. Introduction and Preliminaries

Several researches introduced a new class of algebras related to logical algebras and
semigroups such as: In 1993, Jun et al. [10] introduced the notion of BCI-semigroups.
In 1998, Jun et al. [13] renamed the BCI-semigroup as the IS-algebra. In 2006, Kim
[14] introduced the notion of KS-semigroups. In 2015, Endam and Vilela [3] introduced
the notion of JB-semigroups. In 2018, Iampan [6] introduced the notion of fully UP-
semigroups.

A fuzzy subset F of a set X is a function from X to a closed interval [0,1]. The
concept of a fuzzy subset of a set was first considered by Zadeh [27] in 1965. The fuzzy set
theories developed by Zadeh and others have found many applications in the domain of
mathematics and elsewhere. After the introduction of the concept of fuzzy sets by Zadeh
[27], several researches were conducted on the generalizations of the notion of fuzzy set
and application to many logical algebras such as: In 1998, Jun et al. [9] applied the notion
of fuzzy sets to BCI-semigroups (it was renamed as an IS-algebra for the convenience of
study), and introduced the concept of fuzzy I-ideals. In 2000, Roh et al. [21] considered
the fuzzification of an associative I-ideal of an IS-algebra. They proved that every fuzzy
associative I-ideal is a fuzzy I-ideal. By giving an appropriate example, they verified that

∗This work was financially supported by the University of Phayao.
†Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3412

Email addresses: akarachai.sa@gmail.com (A. Satirad), aiyared.ia@up.ac.th (A. Iampan)

http://www.ejpam.com 294 c© 2019 EJPAM All rights reserved.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 295

a fuzzy I-ideal may not be a fuzzy associative I-ideal. They gave a condition for a fuzzy
I-ideal to be a fuzzy associative I-ideal, and they investigated some related properties. In
2003, Jun and Kondo [11] proved that some concepts of BCK/BCI-algebras expressed by
a certain formula can be naturally extended to the fuzzy setting and that many results are
obtained immediately with the use of our method. Moreover, they proved that these results
can be extended to fuzzy IS-algebras. In 2003, Jianming and Dajing [8] introduced the
concept of intuitionistic fuzzy associative I-ideals of IS-algebras and they investigated some
related properties. In 2007, Prince Williams and Husain [26] studied fuzzy KS-semigroups.
In 2016, Endam and Manahon [2] introduced the notion of fuzzy JB-semigroups and they
investigated some of its properties. In 2018, Satirad and Iampan [23] introduced the notion
of fuzzy sets in fully UP-semigroups and they investigated some of its properties.

In 1999, to solve complicated problems in economics, engineering, and environment,
we cannot successfully use classical methods because of various uncertainties typical for
those problems. Uncertainties cannot be handled using traditional mathematical tools but
may be dealt with using a wide range of existing theories such as the probability theory,
the theory of (intuitionistic) fuzzy sets, the theory of vague sets, the theory of interval
mathematics, and the theory of rough sets. However, all of these theories have their own
difficulties which are pointed out in [18]. In 2001, Maji et al. [17] introduced the concept
of fuzzy soft sets as a generalization of the standard soft sets, and presented an application
of fuzzy soft sets in a decision making problem. In 2010, Jun et al. [12] applied fuzzy soft
set for dealing with several kinds of theories in BCK/BCI-algebras. The notions of fuzzy
soft BCK/BCI-algebras, (closed) fuzzy soft ideals and fuzzy soft p-ideals are introduced,
and related properties are investigated.

Before we begin our study, we will introduce the definition of a UP-algebra.

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

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

(UP-2) (∀x ∈ A)(0 · x = x),

(UP-3) (∀x ∈ A)(x · 0 = 0), and

(UP-4) (∀x, y ∈ A)(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 [19]).

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

(∀x, y ∈ A)(x ≤ y ⇔ x · y = 0).

Example 1. [24] Let X be a universal set and let Ω ∈ P(X). Let PΩ(X) = {A ∈ P(X) |
Ω ⊆ A}. Define a binary operation · on PΩ(X) by putting A · B = B ∩ (A′ ∪ 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 1 with respect to Ω.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 296

Example 2. [24] Let X be a universal set and let Ω ∈ P(X). Let PΩ(X) = {A ∈ P(X) |
A ⊆ Ω}. Define a binary operation ∗ on PΩ(X) by putting A ∗ B = B ∪ (A′ ∩ 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 the power UP-algebra of type 1 and (P(X), ∗, X) is the
power UP-algebra of type 2.

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

(∀x ∈ A)(x · x = 0), (1.1)

(∀x, y, z ∈ A)(x · y = 0, y · z = 0⇒ x · z = 0), (1.2)

(∀x, y, z ∈ A)(x · y = 0⇒ (z · x) · (z · y) = 0), (1.3)

(∀x, y, z ∈ A)(x · y = 0⇒ (y · z) · (x · z) = 0), (1.4)

(∀x, y ∈ A)(x · (y · x) = 0), (1.5)

(∀x, y ∈ A)((y · x) · x = 0⇔ x = y · x), (1.6)

(∀x, y ∈ A)(x · (y · y) = 0), (1.7)

(∀a, x, y, z ∈ A)((x · (y · z)) · (x · ((a · y) · (a · z))) = 0), (1.8)

(∀a, x, y, z ∈ A)((((a · x) · (a · y)) · z) · ((x · y) · z) = 0), (1.9)

(∀x, y, z ∈ A)(((x · y) · z) · (y · z) = 0), (1.10)

(∀x, y, z ∈ A)(x · y = 0⇒ x · (z · y) = 0), (1.11)

(∀x, y, z ∈ A)(((x · y) · z) · (x · (y · z)) = 0), and (1.12)

(∀a, x, y, z ∈ A)(((x · y) · z) · (y · (a · z)) = 0). (1.13)

Definition 2. [4, 5, 7, 25] A nonempty subset S of a UP-algebra (A, ·, 0) is called

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

(2) a near UP-filter of A if it satisfies the following properties:

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

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

(3) a UP-filter of A if it satisfies the following properties:

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

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

(4) a UP-ideal of A if it satisfies the following properties:

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

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

(5) a strongly UP-ideal of A if it satisfies the following properties:



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 297

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

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

We know 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 A is the only one strongly UP-ideal
of itself.

Definition 3. [15] A nonempty subset S of a semigroup (A, ∗) is called

(1) a subsemigroup of A if (∀x, y ∈ S)(x ∗ y ∈ S).

(2) an ideal of A if (∀x, y ∈ A)(x ∈ A, y ∈ S ⇒ x ∗ y, y ∗ x ∈ S).

Clearly, an ideal is a subsemigroup.

Definition 4. [6] Let A be a nonempty set, · and ∗ are binary operations on A, and 0 is
a fixed element of A (i.e., a nullary operation). An algebra A = (A, ·, ∗, 0) of type (2, 2, 0)
in which (A, ·, 0) is a UP-algebra and (A, ∗) is a semigroup is called a fully UP-semigroup
(in short, an f -UP-semigroup) if the operation “∗” is distributive (on both sides) over the
operation “·”.

Definition 5. [27] A fuzzy set F in a nonempty set U (or a fuzzy subset of U) is described
by its membership function fF. To every point x ∈ U , this function associates a real
number fF(x) in the interval [0, 1]. The number fF(x) is interpreted for the point as a
degree of belonging x to the fuzzy set F, that is, F := {(x, fF(x)) | x ∈ U}. We say that a
fuzzy set F in U is constant if its membership function fF is constant.

Definition 6. [16] Let F and G be fuzzy sets in a nonempty set U . Then F ≤ G is defined
by fF(x) ≤ fG(x) for all x ∈ U .

Definition 7. [15] Let F and G be fuzzy sets in a semigroup A = (A, ∗). Then the
product of F and G, denoted by F ◦ G, is described by their membership function fF and
fG, respectively which defined as follows:

(∀x ∈ A)

(
(fF ◦ fG)(x) =

{
sup{min{fF(y), fG(z)}}x=y∗z if ∃y, z ∈ A such that x = y ∗ z,
0 otherwise.

)
Rosenfeld [22] introduced the notion of fuzzy subsemigroups (resp., fuzzy ideals) of

semigroups as follows:

Definition 8. A fuzzy set F in a semigroup A = (A, ∗) is called

(1) a fuzzy subsemigroup of A if (∀x, y ∈ A)(fF(x ∗ y) ≥ min{fF(x), fF(y)}).

(2) a fuzzy ideal of A if (∀x, y ∈ A)(fF(x ∗ y) ≥ max{fF(x), fF(y)}).



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 298

Clearly, a fuzzy ideal is a fuzzy subsemigroup.

Definition 9. [15] The semigroup A itself is a fuzzy set of A, denoted by A such that
fA(x) = 1 for all x ∈ A.

Lemma 1. [15] Let F be a fuzzy set in a semigroup A = (A, ∗). Then

(1) F is a fuzzy subsemigroup of A if and only if it satisfies the condition

F ◦ F ≤ F. (1.14)

(2) F is a fuzzy ideal of A if and only if it satisfies the condition

A ◦ F ≤ F and F ◦A ≤ F. (1.15)

Definition 10. [16] Let {Fi}i∈I be a nonempty family of fuzzy sets in a nonempty set U
where I is an arbitrary index set. The intersection of Fi, denoted by

⋂
i∈I Fi, is described

by its membership function f⋂
i∈I Fi

which defined as follows:

(∀x ∈ U)(f⋂
i∈I Fi

(x) = inf{fFi(x)}i∈I).

The union of Fi, denoted by
⋃

i∈I Fi, is described by its membership function f⋃
i∈I Fi

which
defined as follows:

(∀x ∈ U)(f⋃
i∈I Fi

(x) = sup{fFi(x)}i∈I)

Somjanta et al. [25] and Guntasow et al. [4] introduced the notion of fuzzy UP-
subalgebras (resp., fuzzy UP-filters, fuzzy UP-ideals, fuzzy strongly UP-ideals) of UP-
algebras as follows:

Definition 11. A fuzzy set F in a UP-algebra A = (A, ·, 0) is called

(1) a fuzzy UP-subalgebra of A if (∀x, y ∈ A)(fF(x · y) ≥ min{fF(x), fF(y)}).

(2) a fuzzy UP-filter of A if

(i) (∀x ∈ A)(fF(0) ≥ fF(x)), and

(ii) (∀x, y ∈ A)(fF(y) ≥ min{fF(x · y), fF(x)}).

(3) a fuzzy UP-ideal of A if

(i) (∀x ∈ A)(fF(0) ≥ fF(x)), and

(ii) (∀x, y, z ∈ A)(fF(x · z) ≥ min{fF(x · (y · z)), fF(y)}).

(4) a fuzzy strongly UP-ideal of A if

(i) (∀x ∈ A)(fF(0) ≥ fF(x)), and

(ii) (∀x, y, z ∈ A)(fF(x) ≥ min{fF((z · y) · (z · x)), fF(y)}).



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 299

Now, we introduce the notion of fuzzy near UP-filters of UP-algebras as follows:

Definition 12. A fuzzy set F in a UP-algebra A = (A, ·, 0) is called a fuzzy near UP-filter
of A if

(i) (∀x ∈ A)(fF(0) ≥ fF(x)), and

(ii) (∀x, y ∈ A)(fF(x · y) ≥ fF(y)).

We know that the notion of fuzzy UP-subalgebras is a generalization of fuzzy near UP-
filters, the notion of fuzzy near UP-filters is a generalization of fuzzy UP-filters, the notion
of fuzzy UP-filters is a generalization of fuzzy UP-ideals, and the notion of fuzzy UP-ideals
is a generalization of fuzzy strongly UP-ideals. Moreover, fuzzy strongly UP-ideals and
constant fuzzy sets coincide in UP-algebras.

Satirad and Iampan [23] introduced the notion of fuzzy UPs-subalgebras (resp., fuzzy
UPi-subalgebras, fuzzy UPs-filters, fuzzy UPi-filters, fuzzy UPs-ideals, fuzzy UPi-ideals,
fuzzy strongly UPs-ideals, fuzzy strongly UPi-ideals) of f -UP-semigroups as follows:

Definition 13. [23] A fuzzy set F in an f -UP-semigroup A = (A, ·, ∗, 0) is called

(1) a fuzzy UPs-subalgebra of A if F is a fuzzy UP-subalgebra of (A, ·, 0) and a fuzzy
subsemigroup of (A, ∗).

(2) a fuzzy UPi-subalgebra of A if F is a fuzzy UP-subalgebra of (A, ·, 0) and a fuzzy ideal
of (A, ∗).

(3) a fuzzy UPs-filter of A if F is a fuzzy UP-filter of (A, ·, 0) and a fuzzy subsemigroup
of (A, ∗).

(4) a fuzzy UPi-filter of A if F is a fuzzy UP-filter of (A, ·, 0) and a fuzzy ideal of (A, ∗).

(5) a fuzzy UPs-ideal of A if F is a fuzzy UP-ideal of (A, ·, 0) and a fuzzy subsemigroup
of (A, ∗).

(6) a fuzzy UPi-ideal of A if F is a fuzzy UP-ideal of (A, ·, 0) and a fuzzy ideal of (A, ∗).

(7) a fuzzy strongly UPs-ideal of A if F is a fuzzy strongly UP-ideal of (A, ·, 0) and a
fuzzy subsemigroup of (A, ∗).

(8) a fuzzy strongly UPi-ideal of A if F is a fuzzy strongly UP-ideal of (A, ·, 0) and a
fuzzy ideal of (A, ∗).

Now, we introduce the notion fuzzy near UPs-filters of f -UP-semigroups (resp., fuzzy
near UPi-filters) as follows:

Definition 14. A fuzzy set F in an f -UP-semigroup A = (A, ·, ∗, 0) is called

(1) a fuzzy near UPs-filter of A if F is a fuzzy near UP-filter of (A, ·, 0) and a fuzzy
subsemigroup of (A, ∗).



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 300

(2) a fuzzy near UPi-filter of A if F is a fuzzy near UP-filter of (A, ·, 0) and a fuzzy ideal
of (A, ∗).

Clearly, a fuzzy near UPi-filter is a fuzzy near UPs-filter.

Example 3. Let A = {0, 1, 2, 3} be a set with two binary operations · and ∗ defined by
the following Cayley tables:

· 0 1 2 3

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

∗ 0 1 2 3

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

Then A = (A, ·, ∗, 0) is an f -UP-semigroup. We define a membership function fF as
follows:

fF(0) = 1, fF(1) = 0.4, fF(2) = 0.5, and fF(3) = 0.2.

Then F is a fuzzy near UPs-filter of A. Since fF(2∗3) = fF(1) = 0.4 � 0.5 = max{0.5, 0.2} =
max{fF(2), fF(3)}, we have F is not a fuzzy near UPi-filter of A.

From [7], we can easily prove Theorems 1, 2, 3, and 4.

Theorem 1. Every fuzzy near UPs-filter of an f -UP-semigroup is a fuzzy UPs-subalgebra.

The following example shows that the converse of Theorem 1 is not true.

Example 4. Let A = {0, 1, 2, 3} be a set with two binary operations · and ∗ defined by
the following Cayley tables:

· 0 1 2 3

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

∗ 0 1 2 3

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

Then A = (A, ·, ∗, 0) is an f -UP-semigroup. We define a membership function fF as
follows:

fF(0) = 1, fF(1) = 0.8, fF(2) = 0.9, and fF(3) = 0.7.

Then F is a fuzzy UPs-subalgebra of A. Since fF(1 · 2) = fF(1) = 0.8 � 0.9 = fF(2), we
have F is not a fuzzy near UPs-filter of A.

Theorem 2. Every fuzzy near UPi-filter of an f -UP-semigroup is a fuzzy UPi-subalgebra.

In Example 4, we have F is a fuzzy UPi-subalgebra of A but F is not a fuzzy near
UPi-filter of A.

Theorem 3. Every fuzzy UPs-filter of an f -UP-semigroup is a fuzzy near UPs-filter.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 301

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

Example 5. Let A = {0, 1, 2, 3} be a set with two binary operations · and ∗ defined by
the following Cayley tables:

· 0 1 2 3

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

∗ 0 1 2 3

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

Then A = (A, ·, ∗, 0) is an f -UP-semigroup. We define a membership function fF as
follows:

fF(0) = 1, fF(1) = 0.7, fF(2) = 0.9, and fF(3) = 0.8.

Then F is a fuzzy near UPs-filter of A. Since fF(1) = 0.7 � 0.8 = min{1, 0.8} =
min{fF(0), fF(3)} = min{fF(3 · 1), fF(3)}, we have F is not a fuzzy UPs-filter of A.

Theorem 4. Every fuzzy UPi-filter of an f -UP-semigroup is a fuzzy near UPi-filter.

In Example 5, we have F is a fuzzy near UPi-filter of A but it is not a fuzzy UPi-filter
of A.

Hence, we get the diagram of generalization of fuzzy sets in fully UP-semigroups as
shown in Figure 1.

Figure 1: Fuzzy sets in fully UP-semigroups

Theorem 5. [23] The intersection of any nonempty family of fuzzy UPs-subalgebras (resp.,
fuzzy UPi-subalgebras, fuzzy UPs-filters, fuzzy UPi-filters, fuzzy UPs-ideals, fuzzy UPi-
ideals, fuzzy strongly UPs-ideals, fuzzy strongly UPi-ideals) of an f -UP-semigroup is also
a fuzzy UPs-subalgebra (resp., fuzzy UPi-subalgebra, fuzzy UPs-filter, fuzzy UPi-filter, fuzzy
UPs-ideal, fuzzy UPi-ideal, fuzzy strongly UPs-ideal, fuzzy strongly UPi-ideal).



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 302

Theorem 6. [23] The union of any nonempty family of fuzzy strongly UPs-ideals (resp.,
fuzzy strongly UPi-ideals) of an f -UP-semigroup is also a fuzzy strongly UPs-ideal (resp.,
fuzzy strongly UPi-ideal).

Theorem 7. The intersection of any nonempty family of fuzzy near UPs-filters of an
f -UP-semigroup A = (A, ·, ∗, 0) is also a fuzzy near UPs-filter.

Proof. Let Fi be a fuzzy near UPs-filter of an f -UP-semigroup A = (A, ·, ∗, 0) for all
i ∈ I. Then

f⋂
i∈I Fi

(0) = inf{fFi(0)}i∈I ≥ inf{fFi(x)}i∈I = f⋂
i∈I Fi

(x),

f⋂
i∈I Fi

(x · y) = inf{fFi(x · y)}i∈I ≥ inf{fFi(y)}i∈I = f⋂
i∈I Fi

(y), and

f⋂
i∈I Fi

(x ∗ y) = inf{fFi(x ∗ y)}i∈I
≥ inf{min{fFi(x), fFi(y)}}i∈I
= min{inf{fFi(x)}i∈I , inf{fFi(y)}i∈I}
= min{f⋂

i∈I Fi
(x), f⋂

i∈I Fi
(y)}.

Hence,
⋂

i∈I Fi is a fuzzy near UPs-filter of A.

The following example shows that the union of two fuzzy near UPs-filters of an f -UP-
semigroup is not a fuzzy near UPs-filter.

Example 6. By Cayley tables in Example 3, we know that A = (A, ·, ∗, 0) is an f -UP-
semigroup. We define two membership functions fF1 and fF2 as follows:

A 0 1 2 3

fF1 1 0.7 1 0.5
fF2 1 0.5 0.3 0.8

Then F1 and F2 are fuzzy near UPs-filters of A but F1∪F2 is not a fuzzy near UPs-filter of
A. Indeed, fF1∪F2(3∗2) = fF1∪F2(1) = 0.7 � 0.8 = min{0.8, 1} = min{fF1∪F2(3), fF1∪F2(2)}.

Theorem 8. The intersection of any nonempty family of fuzzy near UPi-filters of an
f -UP-semigroup A = (A, ·, ∗, 0) is also a fuzzy near UPi-filter.

Proof. Let Fi be a fuzzy near UPi-filter of an f -UP-semigroup A = (A, ·, ∗, 0) for all
i ∈ I. Then, by the proof of Theorem 7, we have f⋂

i∈I Fi
(0) ≥ f⋂

i∈I Fi
(x) and f⋂

i∈I Fi
(x·y) ≥

f⋂
i∈I Fi

(y). Thus

f⋂
i∈I Fi

(x ∗ y) = inf{fFi(x ∗ y)}i∈I
≥ inf{max{fFi(x), fFi(y)}}i∈I
≥ max{inf{fFi(x)}i∈I , inf{fFi(y)}i∈I}
= max{f⋂

i∈I Fi
(x), f⋂

i∈I Fi
(y)}.

Hence,
⋂

i∈I Fi is a fuzzy near UPi-filter of A.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 303

Theorem 9. The union of any nonempty family of fuzzy near UPi-filters of an f -UP-
semigroup A = (A, ·, ∗, 0) is also a fuzzy near UPi-filter.

Proof. Let Fi be a fuzzy near UPi-filter of an f -UP-semigroup A = (A, ·, ∗, 0) for all
i ∈ I. Then

f⋃
i∈I Fi

(0) = sup{fFi(0)}i∈I ≥ sup{fFi(x)}i∈I = f⋃
i∈I Fi

(x),

f⋃
i∈I Fi

(x · y) = sup{fFi(x · y)}i∈I ≥ sup{fFi(y)}i∈I = f⋃
i∈I Fi

(y), and

f⋃
i∈I Fi

(x ∗ y) = sup{fFi(x ∗ y)}i∈I
≥ sup{max{fFi(x), fFi(y)}}i∈I
= max{sup{fFi(x)}i∈I , sup{fFi(y)}i∈I}
= max{f⋃

i∈I Fi
(x), f⋃

i∈I Fi
(y)}.

Hence,
⋃

i∈I Fi is a fuzzy near UPi-filter of A.

2. Properties of Fuzzy Sets in UP-Algebras

In this section, we shall let A be a UP-algebra A = (A, ·, 0) and find some properties
of fuzzy sets in UP-algebras.

Proposition 1. [25] If F is a fuzzy UP-subalgebra of A, then

(∀x ∈ A)(fF(0) ≥ fF(x)). (2.1)

Proposition 2. [23] If F is a fuzzy UP-filter of A, then

(∀x, y ∈ A)(x ≤ y ⇒ fF(x) ≤ fF(y)). (2.2)

Proposition 3. If F is a fuzzy set in A satisfying the condition

(∀x, y, z ∈ A)(z ≤ x⇒ fF(x · y) ≥ min{fF(z), fF(y)}), (2.3)

then F is a fuzzy UP-subalgebra of A.

Proof. Let x, y ∈ A. By (1.1), we have x ≤ x. It follows from (2.3) that fF(x · y) ≥
min{fF(x), fF(y)}. Hence, F is a fuzzy UP-subalgebra of A.

Theorem 10. If F is a fuzzy set in A satisfying the condition (2.3), then F satisfies the
condition (2.1).

Proof. It is straightforward by Proposition 3.

The following example shows that the converse of Theorem 10 is not true.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 304

Example 7. Let A = {0, 1, 2, 3} be a set with a binary operation · defined by the following
Cayley table:

· 0 1 2 3

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

Then A = (A, ·, 0) is a UP-algebra. We define a membership function fF as follows:

fF(0) = 1, fF(1) = 0.6, fF(2) = 0.2, and fF(3) = 0.9.

Then F satisfies the condition (2.1) but it does not satisfy the condition (2.3). Indeed,
1 ≤ 1 but fF(1 · 3) = fF(2) = 0.2 � 0.6 = min{0.6, 0.9} = min{fF(1), fF(3)}.

It is clear that we have the following proposition.

Proposition 4. If F is a fuzzy set in A satisfying the condition

(∀x, y, z ∈ A)(fF(x · y) ≥ min{fF(z), fF(y)}), (2.4)

then F satisfies the condition (2.3).

The following example shows that the converse of Proposition 4 is not true.

Example 8. Let A = {0, 1, 2, 3} be a set with a binary operation · defined by the following
Cayley table:

· 0 1 2 3

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

Then A = (A, ·, 0) is a UP-algebra. We define a membership function fF as follows:

fF(0) = 1, fF(1) = 0.1, fF(2) = 0.8, and fF(3) = 0.2.

Then F satisfies the condition (2.3) but it does not satisfy the condition (2.4). Indeed,
fF(1 · 2) = fF(3) = 0.2 � 0.8 = min{1, 0.8} = min{fF(0), fF(2)}.

Proposition 5. If F is a fuzzy set in A satisfying the condition (2.2), then F is a fuzzy
near UP-filter of A.

Proof. Let x, y ∈ A. By (UP-3), we have x ≤ 0. It follows from (2.2) that fF(0) ≥ fF(x).
By (1.5), we have y ≤ x · y. It follows from (2.2) that fF(x · y) ≥ fF(y). Hence, F is a fuzzy
near UP-filter of A.

Theorem 11. If F is a fuzzy set in A satisfying the condition (2.2), then F satisfies the
condition (2.4).



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 305

Proof. Let x, y, z ∈ A. By (1.5), we have y ≤ x · y. It follows from (2.2) that
fF(x · y) ≥ fF(y) ≥ min{fF(z), fF(y)}. Hence, F satisfies (2.4).

The following example shows that the converse of Theorem 11 is not true.

Example 9. Let A = {0, 1, 2, 3} be a set with a binary operation · defined by the following
Cayley table:

· 0 1 2 3

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

Then A = (A, ·, 0) is a UP-algebra. We define a membership function fF as follows:

fF(0) = 1, fF(1) = 0.1, fF(2) = 0.7, and fF(3) = 0.8.

Then F satisfies the condition (2.4) but it does not satisfy the condition (2.2). Indeed,
3 ≤ 2 but fF(2) = fF(1) = 0.7 � 0.8 = fF(3).

Theorem 12. If F is a fuzzy UP-subalgebra of A satisfying the condition

(∀x, y ∈ A)(x · y 6= 0⇒ fF(x) ≥ fF(y)), (2.5)

then F is a fuzzy near UP-filter of A.

Proof. Let x, y ∈ A. If x · y = 0, then by (2.1), we have fF(x · y) = fF(0) ≥ fF(y). If
x · y 6= 0, then by (2.5), we have fF(x · y) ≥ min{fF(x), fF(y)} = fF(y). Hence, F is a fuzzy
near UP-filter of A.

Proposition 6. A fuzzy set F in A satisfies the condition

(∀x, y, z ∈ A)(z ≤ x · y ⇒ fF(y) ≥ min{fF(z), fF(x)}) (2.6)

if and only if F is a fuzzy UP-filter of A.

Proof. Let x ∈ A. By (UP-3), we have x ≤ x · 0. It follows from (2.6) that fF(0) ≥
min{fF(x), fF(x)} = fF(x). Let x, y ∈ A. By (1.1), we have x · y ≤ x · y. It follows from
(2.6) that fF(y) ≥ min{fF(x · y), fF(x)}. Hence, F is a fuzzy UP-filter of A.

Conversely, let x, y, z ∈ A be such that z ≤ x · y. Then z · (x · y) = 0, so

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

Thus fF(y) ≥ min{fF(x · y), fF(x)} ≥ min{fF(z), fF(x)}. Hence, F satisfies (2.6).

Theorem 13. If F is a fuzzy set in A satisfying the condition (2.6), then F satisfies the
condition (2.2).



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 306

Proof. Let x, y ∈ A such that x ≤ y. By (1.11), we have x ≤ x · y. It follows from
(2.6) that fF(y) ≥ min{fF(x), fF(x)} = fF(x). Hence, F satisfies (2.2).

The following example shows that the converse of Theorem 13 is not true.

Example 10. Let A = {0, 1, 2, 3} be a set with a binary operation · defined by the following
Cayley table:

· 0 1 2 3

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

Then A = (A, ·, 0) is a UP-algebra. We define a membership function fF as follows:

fF(0) = 0.9, fF(1) = 0.3, fF(2) = 0.6, and fF(3) = 0.2.

Then F satisfies the condition (2.2) but it does not satisfy the condition (2.6). Indeed,
1 ≤ 2 · 3 but fF(3) = 0.2 � 0.3 = min{0.3, 0.6} = min{fF(1), fF(2)}.

Theorem 14. If F is a fuzzy near UP-filter of A satisfying the condition

(∀x, y ∈ A)(fF(x · y) = fF(y)), (2.7)

then F is a fuzzy UP-filter of A.

Proof. Let x, y ∈ A. By (2.7), we have fF(y) ≥ min{fF(y), fF(x)} = min{fF(x ·
y), fF(x)}. Hence, F is a fuzzy UP-filter of A.

Proposition 7. A fuzzy set F in A satisfies the condition

(∀a, x, y, z ∈ A)(a ≤ x · (y · z)⇒ fF(x · z) ≥ min{fF(a), fF(y)}) (2.8)

if and only if F is a fuzzy UP-ideal of A.

Proof. Let x ∈ A. By (UP-3), we have x ≤ x · (x · 0). By (UP-3) and (2.8), we have

fF(0) = fF(x · 0) ≥ min{fF(x), fF(x)} = fF(x).

Let x, y, z ∈ A. By (1.1), we have x · (y · z) ≤ x · (y · z). It follows from (2.8) that

fF(x · z) ≥ min{fF(x · (y · z)), fF(y)}.

Hence, F is a fuzzy UP-ideal of A.

Conversely, let a, x, y, z ∈ A be such that a ≤ x · (y · z). By Proposition 2, we have
fF(a) ≤ fF(x · (y · z)). Thus

fF(x · z) ≥ min{fF(x · (y · z)), fF(y)} ≥ min{fF(a), fF(y)}.

Hence, F satisfies (2.8).



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 307

Proposition 8. If F is a fuzzy UP-ideal of A, then

(∀a, x, y, z ∈ A)(a ≤ x · (y · z)⇒ fF(a · z) ≥ min{fF(x), fF(y)}). (2.9)

Proof. Let a, x, y, z ∈ A be such that a ≤ x · (y · z). Then a · (x · (y · z)) = 0, so

fF(a · (y · z)) ≥ min{fF(a · (x · (y · z))), fF(x)} = min{fF(0), fF(x)} = fF(x).

Thus
fF(a · z) ≥ min{fF(a · (y · z)), fF(y)} ≥ min{fF(x), fF(y)}.

Corollary 1. If F is a fuzzy set in A satisfying the condition (2.8), then F satisfies the
condition (2.9).

Proof. It is straightforward by Propositions 7 and 8.

Theorem 15. Let A be a UP-algebra satisfying the condition

(∀x, y, z ∈ A)(z · (y · x) = y · (z · x)). (2.10)

If F is a fuzzy set in A satisfying the condition (2.9), then F satisfies the condition (2.8).

Proof. Let a, x, y, z ∈ A such that a ≤ x · (y ·z). By (2.10), we have 0 = a · (x · (y ·z)) =
x · (a · (y · z)), that is, x ≤ a · (y · z). It follows from (2.9) that fF(x · z) ≥ min{fF(a), fF(y)}.
Hence, F satisfies (2.8).

Theorem 16. If F is a fuzzy set in A satisfying the condition (2.9), then F satisfies the
condition (2.6).

Proof. Let x, y, z ∈ A be such that z ≤ x · y. By (1.1) and (1.3), we have 0 = z · z ≤
z · (x · y). By (UP-2) and (2.9), we have fF(y) = fF(0 · y) ≥ min{fF(z), fF(x)}. Hence, F
satisfies (2.6).

Corollary 2. If F is a fuzzy set in A satisfying the condition (2.8), then F satisfies the
condition (2.6).

Proof. It is straightforward by Corollary 1 and Theorem 16.

The following example shows that the converse of Theorem 16 is not true.

Example 11. Let A = {0, 1, 2, 3} be a set with a binary operation · defined by the following
Cayley table:

· 0 1 2 3

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



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 308

Then A = (A, ·, 0) is a UP-algebra. We define a membership function fF as follows:

fF(0) = 1, fF(1) = 0.9, fF(2) = 0.1, and fF(3) = 0.1.

Then F satisfies the condition (2.6) but it does not satisfy the condition (2.9). Indeed,
3 ≤ 1 · (1 · 2) but fF(3 · 2) = fF(2) = 0.1 � 0.9 = fF(1) = min{fF(1), fF(1)}.

The following example shows that fuzzy set in a UP-algebra which satisfies the condi-
tion (2.8) is not constant.

Example 12. Let A = {0, 1, 2, 3} be a set with a binary operation · defined by the following
Cayley table:

· 0 1 2 3

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

Then A = (A, ·, 0) is a UP-algebra. We define a membership function fF as follows:

fF(0) = 0.7, fF(1) = 0.5, fF(2) = 0.4, and fF(3) = 0.4.

Then F satisfies the condition (2.8) but it is not constant.

Theorem 17. If F is a fuzzy UP-filter of A satisfying the condition

(∀x, y, z ∈ A)(fF(y · (x · z)) = fF(x · (y · z))), (2.11)

then F is a fuzzy UP-ideal of A.

Proof. Let x, y, z ∈ A. By (2.11), we have

fF(x · z) ≥ min{fF(y · (x · z)), fF(y)} = min{fF(x · (y · z)), fF(y)}.

Hence, F is a fuzzy UP-ideal of A.

Proposition 9. A fuzzy set F in A satisfies the condition

(∀a, x, y, z ∈ A)(a ≤ (z · y) · (z · x)⇒ fF(x) ≥ min{fF(a), fF(y)}) (2.12)

if and only if F is a fuzzy strongly UP-ideal of A.

Proof. Let x ∈ A. By (UP-3), we have x ≤ 0 = x · 0 = (0 · x) · (0 · 0). By (2.12), we
have fF(0) ≥ min{fF(x), fF(x)} = fF(x). Let x, y, z ∈ A. By (1.1), we have (z · y) · (z ·x) ≤
(z · y) · (z · x). By (2.12), we have fF(x) ≥ min{fF((z · y) · (z · x)), fF(y)}. Hence, F is a
fuzzy strongly UP-ideal of A.

The converse is obvious because F is constant.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 309

Theorem 18. If F is a fuzzy set in A satisfying the condition

(∀x, y, z ∈ A)(z ≤ x · y ⇒ fF(z) ≥ min{fF(x), fF(y)}), (2.13)

then F satisfies the condition (2.3).

Proof. Let x, y, z ∈ A be such that z ≤ x. By (1.4), we have x · y ≤ z · y. By (2.13),
we have fF(x · y) ≥ min{fF(z), fF(y)}. Hence, F satisfies (2.3).

Proposition 10. A fuzzy set F in A satisfies the condition (2.13) if and only if F is a
fuzzy strongly UP-ideal of A.

Proof. Let x ∈ A. By (UP-3), we have x ≤ 0 = 0 · 0. By (2.13), we have fF(x) ≥
min{fF(0), fF(0)} = fF(0). By Theorem 18 and Proposition 3, we have fF(0) ≥ fF(x). Thus
fF(x) = fF(0) for all x ∈ A, so F is constant. Hence, F a fuzzy strongly UP-ideal of A.

The converse is obvious because F is constant.

Theorem 19. If F is a fuzzy set in A satisfying the condition

(∀x, y, z ∈ A)(z ≤ x · y ⇒ fF(z) ≥ fF(y)), (2.14)

then F satisfies the condition (2.3).

Proof. Let x, y, z ∈ A be such that z ≤ x. By (1.4), we have x · y ≤ z · y. It follows
from (2.14) that fF(x · y) ≥ fF(y) ≥ min{fF(z), fF(y)}. Hence, F satisfies (2.3).

Proposition 11. A fuzzy set F in A satisfies the condition (2.14) if and only if F is a
fuzzy strongly UP-ideal of A.

Proof. Let x ∈ A. By (UP-3), we have x ≤ 0 = 0 · 0. By (2.14), we have fF(x) ≥ fF(0).
By Theorem 19 and Proposition 3, we have fF(0) ≥ fF(x). Thus fF(x) = fF(0) for all
x ∈ A, so F is constant. Hence, F is a fuzzy strongly UP-ideal of A.

The converse is obvious because F is constant.

We have provided various important properties of fuzzy sets in various types in UP-
algebras which will be used in the next section. We get the diagram of the properties of
fuzzy sets in UP-algebras as shown in Figure 2.

3. Fuzzy Soft Sets over Fully UP-Semigroups

From now on, we shall let A be an f -UP-semigroup A = (A, ·, ∗, 0) and P be a set of
parameters. Let F(A) denotes the set of all fuzzy sets in A. A subset E of P is called a
set of statistics.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 310

Figure 2: Properties of fuzzy sets in UP-algebras

Definition 15. Let E ⊆ P . A pair (F̃, E) is called a fuzzy soft set over A if F̃ is a
mapping given by F̃ : E → F(A), that is, a fuzzy soft set is a statistic family of fuzzy sets
in A. In general, for every e ∈ E, F̃[e] := {(x, f

F̃[e]
(x)) | x ∈ A} is a fuzzy set in A and it

is called a fuzzy value set of statistic e.

Definition 16. Let (F̃, E1) and (G̃, E2) be two fuzzy soft sets over a common universe U .
The union [17] of (F̃, E1) and (G̃, E2) is defined to be the fuzzy soft set (F̃, E1)∪ (G̃, E2) =
(H̃, E) satisfying the following conditions:

(i) E = E1 ∪ E2 and

(ii) for all e ∈ E,

H̃[e] =


F̃[e] if e ∈ E1 \ E2

G̃[e] if e ∈ E2 \ E1

F̃[e] ∪ G̃[e] if e ∈ E1 ∩ E2.

The restricted union [20] of (F̃, E1) and (G̃, E2) is defined to be the fuzzy soft set (F̃, E1)d
(G̃, E2) = (H̃, E) satisfying the following conditions:

(i) E = E1 ∩ E2 6= ∅ and

(ii) H̃[e] = F̃[e] ∪ G̃[e] for all e ∈ E.

Definition 17. [20] Let (F̃, E1) and (G̃, E2) be two fuzzy soft sets over a common universe
U . The extended intersection of (F̃, E1) and (G̃, E2) is defined to be the fuzzy soft set
(F̃, E1) ∩ (G̃, E2) = (H̃, E) satisfying the following conditions:

(i) E = E1 ∪ E2 and



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 311

(ii) for all e ∈ E,

H̃[e] =


F̃[e] if e ∈ E1 \ E2

G̃[e] if e ∈ E2 \ E1

F̃[e] ∩ G̃[e] if e ∈ E1 ∩ E2.

The intersection [1] of (F̃, E1) and (G̃, E2) is defined to be the fuzzy soft set (F̃, E1) e
(G̃, E2) = (H̃, E) satisfying the following conditions:

(i) E = E1 ∩ E2 6= ∅ and

(ii) H̃[e] = F̃[e] ∩ G̃[e] for all e ∈ E.

3.1. Fuzzy Soft UPs-Subalgebras

Definition 18. A fuzzy soft set (F̃, E) over A is called a fuzzy soft UPs-subalgebra based
on e ∈ E (we shortly call an e-fuzzy soft UPs-subalgebra) of A if a fuzzy set F̃[e] in A is a
fuzzy UPs-subalgebra of A. If (F̃, E) is an e-fuzzy soft UPs-subalgebra of A for all e ∈ E,
we say that (F̃, E) is a fuzzy soft UPs-subalgebra of A.

In the next theorem, we give necessary condition for fuzzy soft UPs-subalgebras of
f -UP-semigroups.

Theorem 20. If (F̃, E) is a fuzzy soft set over A such that for all e ∈ E, a fuzzy set F̃[e]
in A satisfies the conditions (2.3) and (1.14), then (F̃, E) is a fuzzy soft UPs-subalgebra
of A.

Proof. It is straightforward by Proposition 3 and Lemma 1 (1).

The proof of the following theorem can be verified easily.

Theorem 21. If (F̃, E) is a fuzzy soft UPs-subalgebra of A and ∅ 6= E∗ ⊆ E, then
(F̃|E∗ , E∗) is a fuzzy soft UPs-subalgebra of A.

The following example shows that there exists a nonempty subset E∗ of E such that
(F̃|E∗ , E∗) is a fuzzy soft UPs-subalgebra of A, but (F̃, E) is not a fuzzy soft UPs-subalgebra
of A.

Example 13. Let A be the set of four series of the iPhone, that is,

A = {5, 6, 7, X}.

Define two binary operations · and ∗ on A as the following Cayley tables:

· X 7 6 5

X X 7 6 5
7 X X 6 5
6 X 7 X 5
5 X 7 6 X

∗ X 7 6 5

X X X X X
7 X X X X
6 X X X 7
5 X X 7 X



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 312

Then A = (A, ·, ∗,X) is an f -UP-semigroup. Let (F̃, E) be a fuzzy soft set over A where

E := {price, beauty, specifications, stability}

with F̃[price], F̃[beauty], F̃[specifications], and F̃[stability] are fuzzy sets in A defined as
follows:

F̃ X 7 6 5

price 0.8 0.3 0.7 0.1
beauty 0.5 0.3 0.2 0.4

specifications 0.9 0.8 0.5 0.6
stability 1 0.4 0.7 0.6

Then F̃[stability] is not a fuzzy UPs-subalgebra of A. Indeed,

f
F̃[stability]

(5 ∗ 6) = f
F̃[stability]

(7) = 0.4 � 0.6 = min{0.6, 0.7} =

min{f
F̃[stability]

(5), f
F̃[stability]

(6)}.

Hence, (F̃, E) is not a fuzzy soft UPs-subalgebra of A. We take

E∗ := {price, beauty, specifications}.

Thus (F̃|E∗ , E∗) is a fuzzy soft UPs-subalgebra of A.

Theorem 22. The extended intersection of two fuzzy soft UPs-subalgebras of A is also a
fuzzy soft UPs-subalgebra. Moreover, the intersection of two fuzzy soft UPs-subalgebras of
A is also a fuzzy soft UPs-subalgebra.

Proof. Let (F̃, E1) and (G̃, E2) be two fuzzy soft UPs-subalgebras of A. Assume that
(F̃, E1) ∩ (G̃, E2) = (H̃, E) with E = E1 ∪ E2. Let e ∈ E.

Case 1: e ∈ E1 \ E2 (resp., e ∈ E2 \ E1). Then H̃[e] = F̃[e] (resp., H̃[e] = G̃[e]) is a
fuzzy soft UPs-subalgebra of A.

Case 2: e ∈ E1 ∩ E2. By Theorem 5, we have H̃[e] = F̃[e] ∩ G̃[e] is a fuzzy soft
UPs-subalgebra.

Thus (H̃, E) is an e-fuzzy soft UPs-subalgebra of A for all e ∈ E. Hence, (H̃, E) is a
fuzzy soft UPs-subalgebra of A.

Theorem 23. The union of two fuzzy soft UPs-subalgebras of A is also a fuzzy soft UPs-
subalgebra if sets of statistics of two fuzzy soft UPs-subalgebras are disjoint.

Proof. Let (F̃, E1) and (G̃, E2) be two fuzzy soft UPs-subalgebras of A such that
E1 ∩E2 = ∅. Assume that (F̃, E1)∪ (G̃, E2) = (H̃, E) with E = E1 ∪E2. Let e ∈ E. Since
E1 ∩ E2 = ∅, we have e ∈ E1 \ E2 or e ∈ E2 \ E1.

Case 1: e ∈ E1 \ E2. Then H̃[e] = F̃[e] is a fuzzy soft UPs-subalgebra of A.
Case 2: e ∈ E2 \ E1. Then H̃[e] = G̃[e] is a fuzzy soft UPs-subalgebra of A.
Thus (H̃, E) is an e-fuzzy soft UPs-subalgebra of A for all e ∈ E. Hence, (H̃, E) is a

fuzzy soft UPs-subalgebra of A.

The following example shows that Theorem 23 is not valid if sets of statistics of two
fuzzy soft UPs-subalgebras are not disjoint.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 313

Example 14. By Cayley tables in Example 13, we know that A = (A, ·, ∗,X) is an f -UP-
semigroup. Let (G̃1, E1) and (G̃2, E2) be two fuzzy soft sets over A where

E1 := {price, beauty, specifications} and E2 := {price, stability}

with G̃1[price], G̃1[beauty], G̃1[specifications], G̃2[price], and G̃2[stability] are fuzzy sets in A
defined as follows:

G̃1 X 7 6 5

price 0.9 0.7 0.9 0.2
beauty 1 0.8 0.3 0.2

specifications 0.6 0.5 0.3 0.4

G̃2 X 7 6 5

price 0.9 0.3 0.2 0.8
stability 0.7 0.2 0.5 0.2

Then (G̃1, E1) and (G̃2, E2) are two fuzzy soft UPs-subalgebras of A. Since price ∈ E1∩E2,
we have

(f
G̃1[price]∪G̃2[price]

)(6 ∗ 5) = (f
G̃1[price]∪G̃2[price]

)(7)

= 0.7

� 0.8

= min{0.9, 0.8}
= min{(f

G̃1[price]∪G̃2[price]
)(6), (f

G̃1[price]∪G̃2[price]
)(5)}.

Thus G̃1[price]∪ G̃2[price] is not a fuzzy UPs-subalgebra of A, that is, (G̃1, E1)∪ (G̃2, E2)
is not a price-fuzzy soft UPs-subalgebra of A. Hence, (G̃1, E1) ∪ (G̃2, E2) is not a fuzzy
soft UPs-subalgebra of A. Moreover, (G̃1, E1)d (G̃2, E2) is not a fuzzy soft UPs-subalgebra
of A.

3.2. Fuzzy Soft UPi-Subalgebras

Definition 19. A fuzzy soft set (F̃, E) over A is called a fuzzy soft UPi-subalgebra based
on e ∈ E (we shortly call an e-fuzzy soft UPi-subalgebra) of A if a fuzzy set F̃[e] in A is a
fuzzy UPi-subalgebra of A. If (F̃, E) is an e-fuzzy soft UPi-subalgebra of A for all e ∈ E,
we say that (F̃, E) is a fuzzy soft UPi-subalgebra of A.

In the next theorem, we give necessary condition for fuzzy soft UPi-subalgebras of
f -UP-semigroups.

Theorem 24. If (F̃, E) is a fuzzy soft set over A such that for all e ∈ E, a fuzzy set F̃[e]
in A satisfies the conditions (2.3) and (1.15), then (F̃, E) is a fuzzy soft UPi-subalgebra
of A.

Proof. It is straightforward by Proposition 3 and Lemma 1 (2).

From Figure 1, we have the following theorem.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 314

Theorem 25. Every e-fuzzy soft UPi-subalgebra of A is an e-fuzzy soft UPs-subalgebra.
Moreover, every fuzzy soft UPi-subalgebra of A is a fuzzy soft UPs-subalgebra.

The following example shows that the converse of Theorem 25 is not true.

Example 15. In Example 13, we know that (F̃, E) is a price-fuzzy soft UPs-subalgebra of
A but F̃[price] is not a fuzzy UPi-subalgebra of A. Indeed,

f
F̃[price]

(6 ∗ 5) = f
F̃[price]

(7) = 0.3 � 0.7 = max{0.7, 0.1} = max{f
F̃[price]

(6), f
F̃[price]

(5)}.

Hence, (F̃, E) is not a price-fuzzy soft UPi-subalgebra of A.

The proof of the following theorem can be verified easily.

Theorem 26. If (F̃, E) is a fuzzy soft UPi-subalgebra of A and ∅ 6= E∗ ⊆ E, then
(F̃|E∗ , E∗) is a fuzzy soft UPi-subalgebra of A.

The following two theorems can be deduced in the same way as Theorems 22 and 23.

Theorem 27. The extended intersection of two fuzzy soft UPi-subalgebras of A is also a
fuzzy soft UPi-subalgebra. Moreover, the intersection of two fuzzy soft UPi-subalgebras of
A is also a fuzzy soft UPi-subalgebra.

Theorem 28. The union of two fuzzy soft UPi-subalgebras of A is also a fuzzy soft UPi-
subalgebra if sets of statistics of two fuzzy soft UPi-subalgebras are disjoint.

The following example shows that Theorem 28 is not valid if sets of statistics of two
fuzzy soft UPi-subalgebras are not disjoint.

Example 16. Let A be the set of four types of a music, that is,

A = {pop, rock, classic, disco}.

Define two binary operations · and ∗ on A as the following Cayley tables:

· pop rock disco classic

pop pop rock disco classic
rock pop pop disco disco
disco pop rock pop disco

classic pop rock pop pop

∗ pop rock disco classic

pop pop pop pop pop
rock pop pop pop pop
disco pop pop pop pop

classic pop pop pop pop

Then A = (A, ·, ∗, pop) is an f -UP-semigroup. Let (G̃1, E1) and (G̃2, E2) be two fuzzy soft
sets over A where

E1 := {sorrow, modernity} and E2 := {modernity, enjoyment}



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 315

with G̃1[sorrow], G̃1[modernity], G̃2[modernity], and G̃2[enjoyment] are fuzzy sets in A de-
fined as follows:

G̃1 pop rock disco classic

sorrow 0.7 0.7 0.5 0.5
modernity 0.9 0.8 0.3 0.3

G̃2 pop rock disco classic

modernity 0.8 0.3 0.4 0.5
enjoyment 1 0.9 0.1 0.1

Then (G̃1, E1) and (G̃2, E2) are two fuzzy soft UPi-subalgebras of A. Since modernity ∈
E1 ∩ E2, we have

(f
G̃1[modernity]∪G̃2[modernity]

)(rock · classic)

= (f
G̃1[modernity]∪G̃2[modernity]

)(disco)

= 0.4

� 0.5

= min{0.8, 0.5}
= min{(f

G̃1[modernity]∪G̃2[modernity]
)(rock), (f

G̃1[modernity]∪G̃2[modernity]
)(classic)}.

Thus G̃1[modernity]∪ G̃2[modernity] is not a fuzzy UPi-subalgebra of A, that is, (G̃1, E1)∪
(G̃2, E2) is not a modernity-fuzzy soft UPi-subalgebra of A. Hence, (G̃1, E1) ∪ (G̃2, E2) is
not a fuzzy soft UPi-subalgebra of A. Moreover, (G̃1, E1) d (G̃2, E2) is not a fuzzy soft
UPi-subalgebra of A.

3.3. Fuzzy Soft Near UPs-Filters

Definition 20. A fuzzy soft set (F̃, E) over A is called a fuzzy soft near UPs-filter based
on e ∈ E (we shortly call an e-fuzzy soft near UPs-filter) of A if a fuzzy set F̃[e] in A is a
fuzzy near UPs-filter of A. If (F̃, E) is an e-fuzzy soft near UPs-filter of A for all e ∈ E,
we say that (F̃, E) is a fuzzy soft near UPs-filter of A.

In the next theorem, we give necessary condition for fuzzy soft near UPs-filters of
f -UP-semigroups.

Theorem 29. If (F̃, E) is a fuzzy soft set over A such that for all e ∈ E, a fuzzy set F̃[e]
in A satisfies the conditions (2.2) and (1.14), then (F̃, E) is a fuzzy soft near UPs-filter
of A.

Proof. It is straightforward by Proposition 5 and Lemma 1 (1).

From Figure 1, we have the following theorem.

Theorem 30. Every e-fuzzy soft near UPs-filter of A is an e-fuzzy soft UPs-subalgebra.
Moreover, every fuzzy soft near UPs-filter of A is a fuzzy soft UPs-subalgebra.

The following example shows that the converse of Theorem 30 is not true.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 316

Example 17. Let A be a set of four foods, that is,

A = {apple, banana, meat, rice}.

Define two binary operations · and ∗ on A as the following Cayley tables:

· rice apple banana meat

rice rice apple banana meat
apple rice rice apple meat

banana rice rice rice meat
meat rice apple apple rice

∗ rice apple banana meat

rice rice rice rice rice
apple rice rice rice rice

banana rice rice rice rice
meat rice rice rice apple

Then A = (A, ·, ∗, rice) is an f -UP-semigroup. Let (F̃, E) be a fuzzy soft set over A where

E := {pig, monkey, chicken}

with F̃[pig], F̃[monkey], and F̃[chicken] are fuzzy sets in A defined as follows:

F̃ rice apple banana meat

pig 1 0.8 0.9 0.3
monkey 0.8 0.4 0.8 0.3
chicken 0.7 0.4 0.3 0.2

Then (F̃, E) is a pig-fuzzy soft UPs-subalgebra of A. But (F̃, E) is not a pig-fuzzy soft near
UPs-filter of A since

f
F̃[pig]

(meat · banana) = f
F̃[pig]

(apple)

= 0.8

� 0.9

= f
F̃[pig]

(banana),

that is, F̃[pig] is not a fuzzy near UPs-filter of A.

In the next theorem, we give necessary condition for fuzzy soft UPs-subalgebras as
fuzzy soft near UPs-filters of f -UP-semigroups.

Theorem 31. If (F̃, E) is a fuzzy soft UPs-subalgebra of A such that for all e ∈ E, a
fuzzy set F̃[e] in A satisfies the condition (2.5), then (F̃, E) is a fuzzy soft near UPs-filter
of A.

Proof. It is straightforward by Theorem 12.

The proof of the following theorem can be verified easily.

Theorem 32. If (F̃, E) is a fuzzy soft near UPs-filter of A and ∅ 6= E∗ ⊆ E, then
(F̃|E∗ , E∗) is a fuzzy soft near UPs-filter of A.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 317

The following two theorems can be deduced in the same way as Theorems 22 and 23.

Theorem 33. The extended intersection of two fuzzy soft near UPs-filters of A is also a
fuzzy soft near UPs-filter. Moreover, the intersection of two fuzzy soft near UPs-filters of
A is also a fuzzy soft near UPs-filter.

Theorem 34. The union of two fuzzy soft near UPs-filters of A is also a fuzzy soft near
UPs-filter if sets of statistics of two fuzzy soft near UPs-filters are disjoint.

The following example shows that Theorem 34 is not valid if sets of statistics of two
fuzzy soft near UPs-filters are not disjoint.

Example 18. In Example 14, we have (G̃1, E1) and (G̃2, E2) are two fuzzy soft near
UPs-filters of A. Since price ∈ E1 ∩ E2, we have

(f
G̃1[price]∪G̃2[price]

)(6 ∗ 5) = (f
G̃1[price]∪G̃2[price]

)(7)

= 0.7

� 0.8

= min{0.9, 0.8}
= min{(f

G̃1[price]∪G̃2[price]
)(6), (f

G̃1[price]∪G̃2[price]
)(5)}.

Thus G̃1[price] ∪ G̃2[price] is not a fuzzy near UPs-filter of A, that is, (G̃1, E1) ∪ (G̃2, E2)
is not a price-fuzzy soft near UPs-filter of A. Hence, (G̃1, E1) ∪ (G̃2, E2) is not a fuzzy
soft near UPs-filter of A. Moreover, (G̃1, E1)d (G̃2, E2) is not a fuzzy soft near UPs-filter
of A.

3.4. Fuzzy Soft Near UPi-Filters

Definition 21. A fuzzy soft set (F̃, E) over A is called a fuzzy soft near UPi-filter based
on e ∈ E (we shortly call an e-fuzzy soft near UPi-filter) of A if a fuzzy set F̃[e] in A is a
fuzzy near UPi-filter of A. If (F̃, E) is an e-fuzzy soft near UPi-filter of A for all e ∈ E,
we say that (F̃, E) is a fuzzy soft near UPi-filter of A.

In the next theorem, we give necessary condition for fuzzy soft near UPi-filters of
f -UP-semigroups.

Theorem 35. If (F̃, E) is a fuzzy soft set over A such that for all e ∈ E, a fuzzy set F̃[e]
in A satisfies the conditions (2.2) and (1.15), then (F̃, E) is a fuzzy soft near UPi-filter of
A.

Proof. It is straightforward by Proposition 5 and Lemma 1 (2).

From Figure 1, we have the following two theorems.

Theorem 36. Every e-fuzzy soft near UPi-filter of A is an e-fuzzy soft near UPs-filter.
Moreover, every fuzzy soft near UPi-filter of A is a fuzzy soft near UPs-filter.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 318

Theorem 37. Every e-fuzzy soft near UPi-filter of A is an e-fuzzy soft UPi-subalgebra.
Moreover, every fuzzy soft near UPi-filter of A is a fuzzy soft UPi-subalgebra.

The following two examples show that the converse of Theorems 36 and 37 is not true.

Example 19. In Example 13, we know that (F̃, E) is a price-fuzzy soft near UPs-filter of
A but F̃[price] is not a fuzzy near UPi-filter of A. Indeed,

f
F̃[price]

(6 ∗ 5) = f
F̃[price]

(7) = 0.3 � 0.7 = max{0.7, 0.1} = max{f
F̃[price]

(6), f
F̃[price]

(5)}.

Hence, (F̃, E) is not a price-fuzzy soft near UPi-filter of A.

Example 20. In Example 17, we know that (F̃, E) is a monkey-fuzzy soft UPi-subalgebra
of A but F̃[monkey] is not a fuzzy near UPi-filter of A. Indeed,

f
F̃[monkey]

(apple · banana) = f
F̃[monkey]

(apple) = 0.4 � 0.8 = f
F̃[monkey]

(banana).

Hence, (F̃, E) is not a monkey-fuzzy soft near UPi-filter of A.

In the next theorem, we give necessary condition for fuzzy soft UPi-subalgebras as
fuzzy soft near UPi-filters of f -UP-semigroups.

Theorem 38. If (F̃, E) is a fuzzy soft UPi-subalgebra of A such that for all e ∈ E, a fuzzy
set F̃[e] in A satisfies the condition (2.5), then (F̃, E) is a fuzzy soft near UPi-filter of A.

Proof. It is straightforward by Theorem 12.

The proof of the following theorem can be verified easily.

Theorem 39. If (F̃, E) is a fuzzy soft near UPi-filter of A and ∅ 6= E∗ ⊆ E, then
(F̃|E∗ , E∗) is a fuzzy soft near UPi-filter of A.

By using Theorem 9, we can obtain the following two theorems in the same way as
Theorems 22 and 23.

Theorem 40. The extended intersection of two fuzzy soft near UPi-filters of A is also a
fuzzy soft near UPi-filter. Moreover, the intersection of two fuzzy soft near UPi-filters of
A is also a fuzzy soft near UPi-filter.

Theorem 41. The union of two fuzzy soft near UPi-filters of A is also a fuzzy soft near
UPi-filter. Moreover, the restricted union of two fuzzy soft near UPi-filters of A is also a
fuzzy soft near UPi-filter.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 319

3.5. Fuzzy Soft UPs-Filters

Definition 22. A fuzzy soft set (F̃, E) over A is called a fuzzy soft UPs-filter based on
e ∈ E (we shortly call an e-fuzzy soft UPs-filter) of A if a fuzzy set F̃[e] in A is a fuzzy
UPs-filter of A. If (F̃, E) is an e-fuzzy soft UPs-filter of A for all e ∈ E, we say that (F̃, E)
is a fuzzy soft UPs-filter of A.

In the next theorem, we give necessary condition for fuzzy soft UPs-filters of f -UP-
semigroups.

Theorem 42. If (F̃, E) is a fuzzy soft set over A such that for all e ∈ E, a fuzzy set F̃[e]
in A satisfies the conditions (2.6) and (1.14), then (F̃, E) is a fuzzy soft UPs-filter of A.

Proof. It is straightforward by Proposition 6 and Lemma 1 (1).

From Figure 1, we have the following theorem.

Theorem 43. Every e-fuzzy soft UPs-filter of A is an e-fuzzy soft near UPs-filter. More-
over, every fuzzy soft UPs-filter of A is a fuzzy soft near UPs-filter.

The following example shows that the converse of Theorem 43 is not true.

Example 21. Let A be a set of four coffees, that is,

A = {Mocha(M), Americano(A), Cappuccino(C), Latte(L)}.

Define two binary operations · and ∗ on A as the following Cayley tables:

· L A M C

L L A M C
A L L M C
M L L L C
C L L L L

· L A M C

L L L L L
A L L L L
M L L L L
C L L L M

Then A = (A, ·, ∗,Latte) is an f -UP-semigroup. Let (F̃, E) be a fuzzy soft set over A
where

E := {sweetness, strong, aroma}

with F̃[sweetness], F̃[strong], and F̃[aroma] are fuzzy sets in A defined as follows:

F̃ L A M C

sweetness 0.8 0.1 0.6 0.6
strong 0.7 0.7 0.6 0.5
aroma 0.5 0.3 0.4 0.1

Then (F̃, E) is a sweetness-fuzzy soft near UPs-filter of A but F̃[sweetness] is not a fuzzy
UPs-filter of A. Indeed,



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 320

f
F̃[sweetness]

(A) = 0.1 � 0.6 = min{0.8, 0.6} = min{f
F̃[sweetness]

(L), f
F̃[sweetness]

(M)} =

min{f
F̃[sweetness]

(M ·A), f
F̃[sweetness]

(M)}

Hence, (F̃, E) is not a sweetness-fuzzy soft UPs-filter of A.

In the next theorem, we give necessary condition for fuzzy soft near UPs-filters as fuzzy
soft UPs-filters of f -UP-semigroups.

Theorem 44. If (F̃, E) is a fuzzy soft near UPs-filter of A such that for all e ∈ E, a fuzzy
set F̃[e] in A satisfies the condition (2.7), then (F̃, E) is a fuzzy soft UPs-filter of A.

Proof. It is straightforward by Theorem 14.

The proof of the following theorem can be verified easily.

Theorem 45. If (F̃, E) is a fuzzy soft UPs-filter of A and ∅ 6= E∗ ⊆ E, then (F̃|E∗ , E∗)
is a fuzzy soft UPs-filter of A.

The following two theorems can be deduced in the same way as Theorems 22 and 23.

Theorem 46. The extended intersection of two fuzzy soft UPs-filters of A is also a fuzzy
soft UPs-filter. Moreover, the intersection of two fuzzy soft UPs-filters of A is also a fuzzy
soft UPs-filter.

Theorem 47. The union of two fuzzy soft UPs-filters of A is also a fuzzy soft UPs-filter
if sets of statistics of two fuzzy soft UPs-filters are disjoint.

The following example shows that Theorem 47 is not valid if sets of statistics of two
fuzzy soft UPs-filters are not disjoint.

Example 22. In Example 14, we have (G̃1, E1) and (G̃2, E2) are two fuzzy soft UPs-filters
of A. Since price ∈ E1 ∩ E2, we have

(f
G̃1[price]∪G̃2[price]

)(6 ∗ 5) = (f
G̃1[price]∪G̃2[price]

)(7) = 0.7 � 0.8 = min{0.9, 0.8} =

min{(f
G̃1[price]∪G̃2[price]

)(6), (f
G̃1[price]∪G̃2[price]

)(5)}.

Thus G̃1[price]∪ G̃2[price] is not a fuzzy UPs-filter of A, that is, (G̃1, E1)∪ (G̃2, E2) is not
a price-fuzzy soft UPs-filter of A. Hence, (G̃1, E1)∪ (G̃2, E2) is not a fuzzy soft UPs-filter
of A. Moreover, (G̃1, E1) d (G̃2, E2) is not a fuzzy soft UPs-filter of A.

3.6. Fuzzy Soft UPi-Filters

Definition 23. A fuzzy soft set (F̃, E) over A is called a fuzzy soft UPi-filter based on
e ∈ E (we shortly call an e-fuzzy soft UPi-filter) of A if a fuzzy set F̃[e] in A is a fuzzy
UPi-filter of A. If (F̃, E) is an e-fuzzy soft UPi-filter of A for all e ∈ E, we say that (F̃, E)
is a fuzzy soft UPi-filter of A.

In the next theorem, we give necessary condition for fuzzy soft UPi-filters of f -UP-
semigroups.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 321

Theorem 48. If (F̃, E) is a fuzzy soft set over A such that for all e ∈ E, a fuzzy set F̃[e]
in A satisfies the conditions (2.6) and (1.15), then (F̃, E) is a fuzzy soft UPi-filter of A.

Proof. It is straightforward by Proposition 6 and Lemma 1 (2).

From Figure 1, we have the following two theorems.

Theorem 49. Every e-fuzzy soft UPi-filter of A is an e-fuzzy soft UPs-filter. Moreover,
every fuzzy soft UPi-filter of A is a fuzzy soft UPs-filter.

Theorem 50. Every e-fuzzy soft UPi-filter of A is an e-fuzzy soft near UPi-filter. More-
over, every fuzzy soft UPi-filter of A is a fuzzy soft near UPi-filter.

The following two examples show that the converse of Theorems 49 and 50 is not true.

Example 23. In Example 13, we know that (F̃, E) is a beauty-fuzzy soft UPs-filter of A
but F̃[beauty] is not a fuzzy UPi-filter of A. Indeed,

f
F̃[beauty]

(6 ∗ 5) = f
F̃[beauty]

(7) = 0.3 � 0.4 = max{0.2, 0.4} = max{f
F̃[beauty]

(6), f
F̃[beauty]

(5)}.

Hence, (F̃, E) is not a beauty-fuzzy soft UPi-filter of A.

Example 24. In Example 21, we know that (F̃, E) is a aroma-fuzzy soft near UPi-filter
of A but F̃[aroma] is not a fuzzy UPi-filter of A. Indeed,

f
F̃[aroma]

(A) = 0.3 � 0.4 = min{0.5, 0.4} = min{f
F̃[aroma]

(L), f
F̃[aroma]

(M)} =

min{f
F̃[aroma]

(M ·A), f
F̃[aroma]

(M)}.

Hence, (F̃, E) is not a aroma-fuzzy soft UPi-filter of A.

In the next theorem, we give necessary condition for fuzzy soft near UPi-filters as fuzzy
soft UPi-filters of f -UP-semigroups.

Theorem 51. If (F̃, E) is a fuzzy soft near UPi-filter of A such that for all e ∈ E, a fuzzy
set F̃[e] in A satisfies the condition (2.7), then (F̃, E) is a fuzzy soft UPi-filter of A.

Proof. It is straightforward by Theorem 14.

The proof of the following theorem can be verified easily.

Theorem 52. If (F̃, E) is a fuzzy soft UPi-filter of A and ∅ 6= E∗ ⊆ E, then (F̃|E∗ , E∗)
is a fuzzy soft UPi-filter of A.

The following two theorems can be deduced in the same way as Theorems 22 and 23.

Theorem 53. The extended intersection of two fuzzy soft UPi-filters of A is also a fuzzy
soft UPi-filter. Moreover, the intersection of two fuzzy soft UPi-filters of A is also a fuzzy
soft UPi-filter.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 322

Theorem 54. The union of two fuzzy soft UPi-filters of A is also a fuzzy soft UPi-filter
if sets of statistics of two fuzzy soft UPi-filters are disjoint.

The following example shows that Theorem 54 is not valid if sets of statistics of two
fuzzy soft UPi-filters are not disjoint.

Example 25. Let A be a set of four colors, that is,

A = {blue, green, cyan, black}.

Define two binary operations · and ∗ on A as the following Cayley tables:

· black cyan blue green

black black cyan blue green
cyan black black blue blue
blue black cyan black cyan

green black black black black

∗ black cyan blue green

black black black black black
cyan black black black black
blue black black black black

green black black black black

Then A = (A, ·, ∗, black) is an f -UP-semigroup. Let (G̃1, E1) and (G̃2, E2) be two fuzzy
soft sets over A where

E1 := {endurance, beauty} and E2 := {endurance, warmth}

with G̃1[endurance], G̃1[beauty], G̃2[endurance], and G̃2[warmth] are fuzzy sets in A defined
as follows:

G̃1 black cyan blue green

endurance 1 0.5 0.7 0.5
beauty 0.4 0.3 0.2 0.2

G̃2 black cyan blue green

endurance 1 0.6 0.5 0.5
warmth 0.9 0.4 0.5 0.4

Then (G̃1, E1) and (G̃2, E2) are two fuzzy soft UPi-filters of A. Since endurance ∈ E1∩E2,
we have

(f
G̃1[endurance]∪G̃2[endurance]

)(green) = 0.5 � 0.6 = min{0.6, 0.7} =

min{(f
G̃1[endurance]∪G̃2[endurance]

)(cyan), (f
G̃1[endurance]∪G̃2[endurance]

)(blue)} =

min{(f
G̃1[endurance]∪G̃2[endurance]

)(blue · green), (f
G̃1[endurance]∪G̃2[endurance]

)(blue)}.

Thus G̃1[endurance] ∪ G̃2[endurance] is not a fuzzy UPi-filter of A, that is, (G̃1, E1) ∪
(G̃2, E2) is not a endurance-fuzzy soft UPi-filter of A. Hence, (G̃1, E1) ∪ (G̃2, E2) is not
a fuzzy soft UPi-filter of A. Moreover, (G̃1, E1) d (G̃2, E2) is not a fuzzy soft UPi-filter of
A.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 323

3.7. Fuzzy Soft UPs-Ideals

Definition 24. A fuzzy soft set (F̃, E) over A is called a fuzzy soft UPs-ideal based on
e ∈ E (we shortly call an e-fuzzy soft UPs-ideal) of A if a fuzzy set F̃[e] in A is a fuzzy
UPs-ideal of A. If (F̃, E) is an e-fuzzy soft UPs-ideal of A for all e ∈ E, we say that (F̃, E)
is a fuzzy soft UPs-ideal of A.

In the next theorem and corollary, we give necessary condition for fuzzy soft UPs-ideals
of f -UP-semigroups.

Theorem 55. If (F̃, E) is a fuzzy soft set over A such that for all e ∈ E, a fuzzy set F̃[e]
in A satisfies the conditions (2.8) and (1.14), then (F̃, E) is a fuzzy soft UPs-ideal of A.

Proof. It is straightforward by Proposition 7 and Lemma 1 (1).

Corollary 3. Let A be an f -UP-semigroup satisfying the condition (2.10). If (F̃, E) is a
fuzzy soft set over A such that for all e ∈ E, a fuzzy set F̃[e] in A satisfies the conditions
(2.9) and (1.14), then (F̃, E) is a fuzzy soft UPs-ideal of A.

Proof. It is straightforward by Theorems 55 and 15.

From Figure 1, we have the following theorem.

Theorem 56. Every e-fuzzy soft UPs-ideal of A is an e-fuzzy soft UPs-filter. Moreover,
every fuzzy soft UPs-ideal of A is a fuzzy soft UPs-filter.

The following example shows that the converse of Theorem 56 is not true.

Example 26. By Cayley tables in Example 16, we know that A = (A, ·, ∗, pop) is an
f -UP-semigroup. Let (F̃, E) be a fuzzy soft set over A where

E := {sorrow, relaxation, enjoyment}

with F̃[sorrow], F̃[modernity], and F̃[enjoyment] are fuzzy sets in A defined as follows:

F̃ pop rock disco classic

sorrow 0.6 0.2 0.1 0.1
modernity 1 0.5 0.5 0.5
enjoyment 0.7 0.5 0.2 0.2

Then (F̃, E) is a sorrow-fuzzy soft UPs-filter of A but F̃[sorrow] is not a fuzzy UPs-ideal
of A. Indeed,

f
F̃[sorrow]

(disco · classic) = f
F̃[sorrow]

(disco) = 0.1 � 0.2 = min{0.6, 0.2} =

min{f
F̃[sorrow]

(pop), f
F̃[sorrow]

(rock)} = min{f
F̃[sorrow]

(disco · (rock · classic)), f
F̃[sorrow]

(rock)}.

Hence, (F̃, E) is not a sorrow-fuzzy soft UPs-ideal of A.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 324

In the next theorem, we give necessary condition for fuzzy soft UPs-filters as fuzzy soft
UPs-ideals of f -UP-semigroups.

Theorem 57. If (F̃, E) is a fuzzy soft UPs-filter of A such that for all e ∈ E, a fuzzy set
F̃[e] in A satisfies the condition (2.11), then (F̃, E) is a fuzzy soft UPs-ideal of A.

Proof. It is straightforward by Theorem 17.

The proof of the following theorem can be verified easily.

Theorem 58. If (F̃, E) is a fuzzy soft UPs-ideal of A and ∅ 6= E∗ ⊆ E, then (F̃|E∗ , E∗)
is a fuzzy soft UPs-ideal of A.

The following two theorems can be deduced in the same way as Theorems 22 and 23.

Theorem 59. The extended intersection of two fuzzy soft UPs-ideals of A is also a fuzzy
soft UPs-ideal. Moreover, the intersection of two fuzzy soft UPs-ideals of A is also a fuzzy
soft UPs-ideal.

Theorem 60. The union of two fuzzy soft UPs-ideals of A is also a fuzzy soft UPs-ideal
if sets of statistics of two fuzzy soft UPs-ideals are disjoint.

The following example shows that Theorem 60 is not valid if sets of statistics of two
fuzzy soft UPs-ideals are not disjoint.

Example 27. In Example 14, we have (G̃1, E1) and (G̃2, E2) are two fuzzy soft UPs-ideals
of A. Since price ∈ E1 ∩ E2, we have

(f
G̃1[price]∪G̃2[price]

)(6 ∗ 5) = (f
G̃1[price]∪G̃2[price]

)(7) = 0.7 � 0.8 = min{0.9, 0.8} =

min{(f
G̃1[price]∪G̃2[price]

)(6), (f
G̃1[price]∪G̃2[price]

)(5)}.

Thus G̃1[price]∪ G̃2[price] is not a fuzzy UPs-ideal of A, that is, (G̃1, E1)∪ (G̃2, E2) is not
a price-fuzzy soft UPs-ideal of A. Hence, (G̃1, E1)∪ (G̃2, E2) is not a fuzzy soft UPs-ideal
of A. Moreover, (G̃1, E1) d (G̃2, E2) is not a fuzzy soft UPs-ideal of A.

3.8. Fuzzy Soft UPi-Ideals

Definition 25. A fuzzy soft set (F̃, E) over A is called a fuzzy soft UPi-ideal based on
e ∈ E (we shortly call an e-fuzzy soft UPi-ideal) of A if a fuzzy set F̃[e] in A is a fuzzy
UPi-ideal of A. If (F̃, E) is an e-fuzzy soft UPi-ideal of A for all e ∈ E, we say that (F̃, E)
is a fuzzy soft UPi-ideal of A.

In the next theorem and corollary, we give necessary condition for fuzzy soft UPi-ideals
of f -UP-semigroups.

Theorem 61. If (F̃, E) is a fuzzy soft set over A such that for all e ∈ E, a fuzzy set F̃[e]
in A satisfies the conditions (2.8) and (1.15), then (F̃, E) is a fuzzy soft UPi-ideal of A.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 325

Proof. It is straightforward by Proposition 7 and Lemma 1 (2).

Corollary 4. Let A be an f -UP-semigroup satisfying the condition (2.10). If (F̃, E) is a
fuzzy soft set over A such that for all e ∈ E, a fuzzy set F̃[e] in A satisfies the conditions
(2.9) and (1.15), then (F̃, E) is a fuzzy soft UPi-ideal of A.

Proof. It is straightforward by Theorems 61 and 15.

From Figure 1, we have the following two theorems.

Theorem 62. Every e-fuzzy soft UPi-ideal of A is an e-fuzzy soft UPs-ideal. Moreover,
every fuzzy soft UPi-ideal of A is a fuzzy soft UPs-ideal.

Theorem 63. Every e-fuzzy soft UPi-ideal of A is an e-fuzzy soft UPi-filter. Moreover,
every fuzzy soft UPi-ideal of A is a fuzzy soft UPi-filter.

The following two examples show that the converse of Theorems 62 and 63 is not true.

Example 28. In Example 13, we know that (F̃, E) is a price-fuzzy soft UPs-ideal of A
but F̃[price] is not a fuzzy UPi-ideal of A. Indeed,

f
F̃[price]

(5 ∗ 6) = f
F̃[price]

(7) = 0.3 � 0.7 = max{0.1, 0.7} = max{f
F̃[price]

(5), f
F̃[price]

(6)}.

Hence, (F̃, E) is not a price-fuzzy soft UPi-ideal of A.

Example 29. In Example 26, we know that (F̃, E) is a enjoyment-fuzzy soft UPi-filter of
A but F̃[enjoyment] is not a fuzzy UPi-ideal of A. Indeed,

f
F̃[enjoyment]

(disco · classic) = f
F̃[enjoyment]

(disco) = 0.2 � 0.5 = min{0.7, 0.5} =

min{f
F̃[enjoyment]

(pop), f
F̃[enjoyment]

(rock)} =

min{f
F̃[enjoyment]

(disco · (rock · classic)), f
F̃[enjoyment]

(rock)}.

Hence, (F̃, E) is not a enjoyment-fuzzy soft UPi-ideal of A.

In the next theorem, we give necessary condition for fuzzy soft UPi-filters as fuzzy soft
UPi-ideals of f -UP-semigroups.

Theorem 64. If (F̃, E) is a fuzzy soft UPi-filter of A such that for all e ∈ E, a fuzzy set
F̃[e] in A satisfies the condition (2.11), then (F̃, E) is a fuzzy soft UPi-ideal of A.

Proof. It is straightforward by Theorem 17.

The proof of the following theorem can be verified easily.

Theorem 65. If (F̃, E) is a fuzzy soft UPi-ideal of A and ∅ 6= E∗ ⊆ E, then (F̃|E∗ , E∗)
is a fuzzy soft UPi-ideal of A.

The following two theorems can be deduced in the same way as Theorems 22 and 23.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 326

Theorem 66. The extended intersection of two fuzzy soft UPi-ideals of A is also a fuzzy
soft UPi-ideal. Moreover, the intersection of two fuzzy soft UPi-ideals of A is also a fuzzy
soft UPi-ideal.

Theorem 67. The union of two fuzzy soft UPi-ideals of A is also a fuzzy soft UPi-ideal
if sets of statistics of two fuzzy soft UPi-ideals are disjoint.

The following example shows that the converse of Theorem 67 is not true.

Example 30. In Example 25, we have (G̃1, E1) and (G̃2, E2) are two fuzzy soft UPi-ideals
of A. Since endurance ∈ E1 ∩ E2, we have

(f
G̃1[endurance]∪G̃2[endurance]

)(black · green) = (f
G̃1[endurance]∪G̃2[endurance]

)(green) = 0.5 �
0.6 = min{0.6, 0.7} =

min{(f
G̃1[endurance]∪G̃2[endurance]

)(cyan), (f
G̃1[endurance]∪G̃2[endurance]

)(blue)} =

min{(f
G̃1[endurance]∪G̃2[endurance]

)(black · (blue · green)), (f
G̃1[endurance]∪G̃2[endurance]

)(blue)}.

Thus G̃1[endurance] ∪ G̃2[endurance] is not a fuzzy UPi-ideal of A, that is, (G̃1, E1) ∪
(G̃2, E2) is not a endurance-fuzzy soft UPi-ideal of A. Hence, (G̃1, E1) ∪ (G̃2, E2) is not
a fuzzy soft UPi-ideal of A. Moreover, (G̃1, E1) d (G̃2, E2) is not a fuzzy soft UPi-ideal of
A.

3.9. Fuzzy Soft Strongly UPs-Ideals

Definition 26. A fuzzy soft set (F̃, E) over A is called a fuzzy soft strongly UPs-ideal
based on e ∈ E (we shortly call an e-fuzzy soft strongly UPs-ideal) of A if a fuzzy set F̃[e]
in A is a fuzzy strongly UPs-ideal of A. If (F̃, E) is an e-fuzzy soft strongly UPs-ideal of
A for all e ∈ E, we say that (F̃, E) is a fuzzy soft strongly UPs-ideal of A.

Definition 27. A fuzzy soft set (F̃, E) over A is called a constant fuzzy soft set based
on e ∈ E (we shortly call an e-constant fuzzy soft set) of A if a fuzzy set F̃[e] in A is
constant. If (F̃, E) is an e-constant fuzzy soft set over A for all e ∈ E, we say that (F̃, E)
is a constant fuzzy soft set over A.

From Figure 1, we have the following two theorem.

Theorem 68. Every e-fuzzy soft strongly UPs-ideal of A is an e-fuzzy soft UPs-ideal.
Moreover, every fuzzy soft strongly UPs-ideal of A is a fuzzy soft UPs-ideal.

Theorem 69. e-fuzzy soft strongly UPs-ideals and e-constant fuzzy soft sets coincide in
A. Moreover, fuzzy soft strongly UPs-ideals and constant fuzzy soft sets coincide in A.

In the next theorem, we give necessary condition for fuzzy soft strongly UPs-ideals of
f -UP-semigroups.

Theorem 70. If (F̃, E) is a fuzzy soft set over A such that for all e ∈ E, a fuzzy set F̃[e]
in A satisfies the conditions (2.12) (or (2.13) or (2.14)) and (1.14), then (F̃, E) is a fuzzy
soft strongly UPs-ideal of A.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 327

Proof. It is straightforward by Propositions 9 (or 10 or 11) and Lemma 1 (1).

The following example shows that the converse of Theorem 68 is not true.

Example 31. Let A be a set of four brands of a pick-up truck, that is,

A = {Toyota Hilux (TH ), Mitsubishi Triton(MT ), Ford Ranger(FR), Isuzu D-Max (ID)}.

Define two binary operations · and ∗ on A as the following Cayley tables:

· MT FR ID TH

MT MT FR ID TH
FR MT MT ID TH
ID MT FR MT TH
TH MT FR ID MT

∗ MT FR ID TH

MT MT MT MT MT
FR MT FR MT MT
ID MT MT ID MT
TH MT TH MT MT

Then A = (A, ·, ∗,Mitsubishi Triton) is an f -UP-semigroup. Let (F̃, E) be a fuzzy soft set
over A where

E := {displacement, horse power, torque}
with F̃[displacement], F̃[horse power], and F̃[torque] are fuzzy sets in A defined as follows:

F̃ MT FR ID TH

displacement 1 0.6 0.4 0.7
horse power 0.9 0.6 0.5 0.5

torque 0.9 0.7 0.6 0.5

Then (F̃, E) is a torque-fuzzy soft UPs-ideal of A but F̃[torque] is not a fuzzy strongly
UPs-ideal of A. Indeed,

f
F̃[torque]

(ID) = 0.6 � 0.7 = min{0.9, 0.7} = min{f
F̃[torque]

(MT), f
F̃[torque]

(FR)} =

min{f
F̃[torque]

((ID · FR) · (ID · ID)), f
F̃[torque]

(FR)}.

Hence, (F̃, E) is not a torque-fuzzy soft strongly UPs-ideal of A.

The proof of the following theorem can be verified easily.

Theorem 71. If (F̃, E) is a fuzzy soft strongly UPs-ideal of A and ∅ 6= E∗ ⊆ E, then
(F̃|E∗ , E∗) is a fuzzy soft strongly UPs-ideal of A.

By using Theorem 6, we can obtain the following two theorems in the same way as
Theorems 22 and 23.

Theorem 72. The extended intersection of two fuzzy soft strongly UPs-ideals of A is
also a fuzzy soft strongly UPs-ideal. Moreover, the intersection of two fuzzy soft strongly
UPs-ideals of A is also a fuzzy soft strongly UPs-ideal.

Theorem 73. The union of two fuzzy soft strongly UPs-ideals is also a fuzzy soft strongly
UPs-ideal. Moreover, the restricted union of two fuzzy soft strongly UPs-ideals of A is also
a fuzzy soft strongly UPs-ideal.



A. Satirad, A. Iampan / Eur. J. Pure Appl. Math, 12 (2) (2019), 294-331 328

3.10. Fuzzy Soft Strongly UPi-Ideals

Definition 28. A fuzzy soft set (F̃, E) over A is called a fuzzy soft strongly UPi-ideal
based on e ∈ E (we shortly call an e-fuzzy soft strongly UPi-ideal) of A if a fuzzy set F̃[e]
in A is a fuzzy strongly UPi-ideal of A. If (F̃, E) is an e-fuzzy soft strongly UPi-ideal of
A for all e ∈ E, we say that (F̃, E) is a fuzzy soft strongly UPi-ideal of A.

From Figure 1, we have the following two theorem.

Theorem 74. Every e-fuzzy soft strongly UPi-ideal of A is an e-fuzzy soft UPi-ideal.
Moreover, every fuzzy soft strongly UPi-ideal of A is a fuzzy soft UPi-ideal.

Theorem 75. e-fuzzy soft strongly UPi-ideals and e-constant fuzzy soft sets coincide in
A. Moreover, fuzzy soft strongly UPi-ideals and constant fuzzy soft sets coincide in A.

Corollary 5. e-fuzzy soft strongly UPs-ideals, e-fuzzy soft strongly UPi-ideals, and e-
constant fuzzy soft sets coincide in A. Moreover, fuzzy soft strongly UPs-ideals, fuzzy soft
strongly UPi-ideals and constant fuzzy soft sets coincide in A.

Proof. It is straightforward by Theorems 69 and 75.

In the next theorem, we give necessary condition for fuzzy soft strongly UPi-ideals of
f -UP-semigroups.

Theorem 76. If (F̃, E) is a fuzzy soft set over A such that for all e ∈ E, a fuzzy set F̃[e]
in A satisfies the conditions (2.12) (or (2.13) or (2.14)) and (1.15), then (F̃, E) is a fuzzy
soft strongly UPi-ideal of A.

Proof. It is straightforward by Proposition 9 (or 10 or 11) and Lemma 1 (2).

The following example shows that the converse of Theorem 74 is not true.

Example 32. In Example 31, we know that (F̃, E) is a displacement-fuzzy soft UPi-ideal
of A but F̃[displacement] is not a fuzzy strongly UPi-ideal of A. Indeed,

f
F̃[displacement]

(ID) = 0.4 � 0.6 = min{1, 0.6} =

min{f
F̃[displacement]

(MT), f
F̃[displacement]

(FR)} =

min{f
F̃[displacement]

((ID · FR) · (ID · ID)), f
F̃[displacement]

(FR)}.

Hence, (F̃, E) is not a displacement-fuzzy soft strongly UPi-ideal of A.

The proof of the following theorem can be verified easily.

Theorem 77. If (F̃, E) is a fuzzy soft strongly UPi-ideal of A and ∅ 6= E∗ ⊆ E, then
(F̃|E∗ , E∗) is a fuzzy soft strongly UPi-ideal of A.

By using Theorem 6, we can obtain the following two theorems in the same way as
Theorems 22 and 23.



REFERENCES 329

Theorem 78. The extended intersection of two fuzzy soft strongly UPi-ideals of A is
also a fuzzy soft strongly UPi-ideal. Moreover, the intersection of two fuzzy soft strongly
UPi-ideals of A is also a fuzzy soft strongly UPi-ideal.

Theorem 79. The union of two fuzzy soft strongly UPi-ideals of A is also a fuzzy soft
strongly UPs-ideal. Moreover, the restricted union of two fuzzy soft strongly UPi-ideals of
A is also a fuzzy soft strongly UPi-ideal.

4. Conclusions

In this paper, we have introduced the notions of fuzzy soft UPs-subalgebras, fuzzy
soft UPi-subalgebras, fuzzy soft near UPs-filters, fuzzy soft near UPi-filters, fuzzy soft
UPs-filters, fuzzy soft UPi-filters, fuzzy soft UPs-ideals, fuzzy soft UPi-ideals, fuzzy soft
strongly UPs-ideals, and fuzzy soft strongly UPi-ideals of fully UP-semigroups and the
conditions for fuzzy soft sets over fully UP-semigroups, proved its generalizations and
investigated some of its important properties. Then, we get the diagram of generalization
of fuzzy soft sets over fully UP-semigroups as shown in Figure 3.

Figure 3: Fuzzy soft sets over fully UP-semigroups

Acknowledgements

The authors wish to express their sincere thanks to the referees for the valuable sug-
gestions which lead to an improvement of this paper.

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