International Journal of Analysis and Applications

Volume 17, Number 5 (2019), 821-837

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-17-2019-821

PROPERTIES OF OPERATIONS FOR FUZZY SOFT SETS OVER FULLY

UP-SEMIGROUPS

AKARACHAI SATIRAD AND AIYARED IAMPAN∗

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

∗Corresponding author: aiyared.ia@up.ac.th

Abstract. The aim of this manuscript is to apply distributivity laws of several fuzzy sets for any fuzzy

sets and study distributivity laws with any fuzzy soft sets. We investigate properties of some operations for

fuzzy soft sets over fully UP-semigroups and their interrelation with respect to different operations such as

“(restricted) union”, “(extended) intersection”, “AND”, and “OR”.

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. [7] introduced the notion of BCI-semigroups. In 2018, Iampan [6] introduced the notion

of fully UP-semigroups.

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 [11]. In 2001, Maji et al. [10] 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

Received 2019-06-02; accepted 2019-07-11; published 2019-09-02.

2010 Mathematics Subject Classification. 03G25; 08A72.

Key words and phrases. fully UP-semigroup; fuzzy soft set; (restricted) union; (extended) intersection; AND; OR.

This work was supported by the Unit of Excellence, University of Phayao.

c©2019 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

821

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-821


Int. J. Anal. Appl. 17 (5) (2019) 822

problem. In 2010, Jun et al. [8] 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. In 2013, Rehman et al. [13] studied some operations

of fuzzy soft sets and give fundamental properties of fuzzy soft sets. They discuss properties of fuzzy soft

sets and their interrelation with respect to different operations such as union, intersection, restricted union

and extended intersection. Then, they illustrate properties of AND and OR operations by giving counter

examples. Also we prove that certain De Morgan’s laws hold in fuzzy soft set theory with respect to different

operations on fuzzy soft sets. In 2019, Satirad and Iampan [16] introduced 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.

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

Definition 1.1. [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 [12]).

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 1.1. [18] Let X be a universal set and let Ω ∈P(X) where P(X) means the power set of X. Let

PΩ(X) = {A ∈P(X) | Ω ⊆ A}. Define a binary operation · on PΩ(X) by putting A·B = B∩(AC∪Ω) for all

A, B ∈PΩ(X) where AC means the complement of a subset A. Then (PΩ(X), ·, Ω) is a UP-algebra and we

shall call it the generalized power UP-algebra of type 1 with respect to Ω. Let PΩ(X) = {A ∈P(X) | A ⊆ Ω}.

Define a binary operation ∗ on PΩ(X) by putting A ∗ B = B ∪ (AC ∩ Ω) for all A, B ∈ PΩ(X). Then

(PΩ(X),∗, Ω) is a UP-algebra and we shall call it the generalized power UP-algebra of type 2 with respect

to Ω. In particular, (P(X), ·,∅) is a UP-algebra and we shall call it the power UP-algebra of type 1, and

(P(X),∗, X) is a UP-algebra and we shall call it the power UP-algebra of type 2.



Int. J. Anal. Appl. 17 (5) (2019) 823

Example 1.2. [3] Let N be the set of all natural numbers with two binary operations ◦ and • defined by

(∀x, y ∈ N)


x◦y =


 y if x < y,0 otherwise




and

(∀x, y ∈ N)


x•y =


 y if x > y or x = 0,0 otherwise


 .

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

For more examples of UP-algebras, see [2, 6, 17, 18].

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 1.2. [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 “·”.



Int. J. Anal. Appl. 17 (5) (2019) 824

Definition 1.3. [20] 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.

Rosenfeld [14] introduced the notion of fuzzy subsemigroups (resp., fuzzy ideals) of semigroups as follows:

Definition 1.4. 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)}).

Clearly, a fuzzy ideal is a fuzzy subsemigroup.

Definition 1.5. [9] 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 ).

Theorem 1.1. Let Fi and F be fuzzy sets in a nonempty set X where I is a nonempty set. Then the

following properties hold:

(1) F ∩ (
⋃

i∈I Fi) =
⋃

i∈I (F ∩ Fi),

(2) (
⋃

i∈I Fi) ∩ F =
⋃

i∈I (Fi ∩ F),

(3) F ∪ (
⋂

i∈I Fi) =
⋂

i∈I (F ∪ Fi), and

(4) (
⋂

i∈I Fi) ∪ F =
⋂

i∈I (Fi ∪ F).

Proof. Let x ∈ X. (1) First, we investigate left hand side of the equality. Assume that
⋃

i∈I Fi = F
∪. Then

F ∩ (
⋃

i∈I Fi) = F ∩ F
∪. Also,

fF∩F∪ (x) = min{fF(x), fF∪ (x)}

= min{fF(x), f⋃i∈I Fi (x)}
= min{fF(x), sup{fFi(x)}i∈I}.



Int. J. Anal. Appl. 17 (5) (2019) 825

Consider the right hand side of the equality. Assume that F ∩ Fi = F∩i for all i ∈ I. Then

f⋃
i∈I F

∩
i

(x) = sup{fF∩
i

(x)}i∈I

= sup{fF∩Fi (x)}i∈I

= sup{min{fF(x), fFi (x)}}i∈I.

It is clear that min{fF(x), sup{fFi(x)}i∈I} = sup{min{fF(x), fFi (x)}}i∈I . Therefore, F∩(
⋃

i∈I Fi) =
⋃

i∈I (F∩

Fi).

(2) By using techniques as in (1), then (2) can be derived.

(3) First, we investigate left hand side of the equality. Assume that
⋂

i∈I Fi = F
∩. Then F ∪ (

⋂
i∈I Fi) =

F ∪ F∩. Also,

fF∪F∩ (x) = max{fF(x), fF∩ (x)}

= max{fF(x), f⋂
i∈I Fi

(x)}

= max{fF(x), inf{fFi(x)}i∈I}.

Consider the right hand side of the equality. Assume that F ∪ Fi = F∪i for all i ∈ I. Then

f⋂
i∈I F

∪
i

(x) = inf{fF∪
i

(x)}i∈I

= inf{fF∪Fi (x)}i∈I

= inf{max{fF(x), fFi (x)}}i∈I.

It is clear that max{fF(x), inf{fFi(x)}i∈I} = inf{max{fF(x), fFi (x)}}i∈I . Therefore, F∪(
⋂

i∈I Fi) =
⋂

i∈I (F∪

Fi).

(4) By using techniques as in (3), then (4) can be derived. �

Somjanta et al. [19], Guntasow et al. [4], and Satirad and Iampan [16] introduced the notion of fuzzy

UP-subalgebras (resp., fuzzy near UP-filters, fuzzy UP-filters, fuzzy UP-ideals, fuzzy strongly UP-ideals) of

UP-algebras as follows:

Definition 1.6. 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 near UP-filter of A if

(i) (∀x ∈ A)(fF(0) ≥ fF(x)), and

(ii) (∀x, y ∈ A)(fF(x ·y) ≥ fF(y)).

(3) a fuzzy UP-filter of A if

(i) (∀x ∈ A)(fF(0) ≥ fF(x)), and



Int. J. Anal. Appl. 17 (5) (2019) 826

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

(4) 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)}).

(5) 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)}).

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 [15,16] introduced the notion of fuzzy UPs-subalgebras (resp., fuzzy UPi-subalgebras,

fuzzy near UPs-filters, fuzzy near UPi-filters, 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 1.7. 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 near UPs-filter of A if F is a fuzzy near UP-filter of (A, ·, 0) and a fuzzy subsemigroup of

(A,∗).

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

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

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

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

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

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

of (A,∗).

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

Theorem 1.2. [15, 16] The intersection of any nonempty family of fuzzy UPs-subalgebras (resp., fuzzy

UPi-subalgebras, fuzzy near UPs-filters, fuzzy near UPi-filters, 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



Int. J. Anal. Appl. 17 (5) (2019) 827

a fuzzy UPs-subalgebra (resp., fuzzy UPi-subalgebra, fuzzy near UPs-filter, fuzzy near UPi-filter, fuzzy UPs-

filter, fuzzy UPi-filter, fuzzy UPs-ideal, fuzzy UPi-ideal, fuzzy strongly UPs-ideal, fuzzy strongly UPi-ideal).

Theorem 1.3. [15, 16] The union of any nonempty family of fuzzy near UPi-filters (resp., fuzzy strongly

UPs-ideals, fuzzy strongly UPi-ideals) of an f-UP-semigroup is also a fuzzy near UPi-filter (resp., fuzzy

strongly UPs-ideal, fuzzy strongly UPi-ideal).

2. 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.

Definition 2.1. 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 2.2. Let (F̃, E1) and (G̃, E2) be two fuzzy soft sets over a common universe U. The union [10]

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 [13] 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 2.3. [10] Let (F̃, E1) and (G̃, E2) be two fuzzy soft sets over a common universe U. The OR

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) H̃[e1, e2] = F̃[e1] ∪ G̃[e2] for all (e1, e2) ∈ E.

Definition 2.4. Let (F̃, E1) and (G̃, E2) be two fuzzy soft sets over a common universe U. The extended

intersection [13] of (F̃, E1) and (G̃, E2) is defined to be the fuzzy soft set (F̃, E1)∩(G̃, E2) = (H̃, E) satisfying

the following conditions:



Int. J. Anal. Appl. 17 (5) (2019) 828

(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 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.

Definition 2.5. [10] Let (F̃, E1) and (G̃, E2) be two fuzzy soft sets over a common universe U. The AND

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) H̃[e1, e2] = F̃[e1] ∩ G̃[e2] for all (e1, e2) ∈ E.

Definition 2.6. 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.

We can call fuzzy soft sets that 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 ) based on a statistic or 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 A if

fuzzy soft sets satisfy statement in Definition 2.6.

We will introduce the notions of the restricted union, the union, the intersection, the extended intersection,

the AND, and the OR of any fuzzy soft sets and apply to f-UP-semigroups.

Definition 2.7. Let {(F̃i, Ei) | i ∈ I} be a nonempty family of fuzzy soft sets over a common universe

U where I is an arbitrary index set. The restricted union of (F̃i, Ei) is defined to be the fuzzy soft set

di∈I (F̃i, Ei) = (F̃, E) satisfying the following conditions:
(i) E =

⋂
i∈I Ei 6= ∅ and

(ii) F̃[e] =
⋃

i∈I F̃i[e] for all e ∈ E.



Int. J. Anal. Appl. 17 (5) (2019) 829

Theorem 2.1. The restricted union of family of fuzzy soft near UPi-filters of A is also a fuzzy soft near

UPi-filter.

Proof. Let (F̃i, Ei) be a fuzzy soft near UPi-filters of A for all i ∈ I. Assume that di∈I (F̃i, Ei) = (F̃, E)
be the restricted union of (F̃i, Ei) for all i ∈ I. Then E =

⋂
i∈I Ei 6= ∅. Let e ∈ E. By Theorem 1.3, we

have F̃[e] =
⋃

i∈I F̃i[e] is a fuzzy near UPi-filter of A. Therefore, (F̃, E) is an e-fuzzy soft near UPi-filter of

A. But since e is an arbitrary statistic of E, we have (F̃, E) is a fuzzy soft near UPi-filter of A. �

In the same way as Theorem 2.1, we can use Theorem 1.3 to prove that the restricted union of family

of fuzzy soft strongly UPs-ideals (resp., fuzzy soft strongly UPi-ideals) of A is also a fuzzy soft strongly

UPs-ideal (resp., fuzzy soft strongly UPi-ideal).

Definition 2.8. Let {(F̃i, Ei) | i ∈ I} be a nonempty family of fuzzy soft sets over a common universe U

where I is an arbitrary index set. The union of (F̃i, Ei) is defined to be the fuzzy soft set
⋃

i∈I (F̃i, Ei) = (F̃, E)

satisfying the following conditions:

(i) E =
⋃

i∈I Ei and

(ii) F̃[e] =
⋃

j∈J F̃j[e] for all e ∈ E with e ∈
⋂

j∈J Ej −
⋃

k∈I−J Ek where ∅ 6= J ⊆ I.

Theorem 2.2. The union of family of fuzzy soft near UPi-filters of A is also a fuzzy soft near UPi-filter.

Proof. Let (F̃i, Ei) be a fuzzy soft near UPi-filters of A for all i ∈ I. Assume that
⋂

i∈I (F̃i, Ei) = (F̃, E) be

the union of (F̃i, Ei) for all i ∈ I. Then E =
⋃

i∈I Ei. Let e ∈ E.

Case 1: |J| = |I|. By Theorem 2.1, we have F̃[e] =
⋂

i∈I F̃i[e] is a fuzzy near UPi-filter of A.

Case 2: |J| = 1, that is, J is a singleton set. Then F̃[e] =
⋂

j∈{j} F̃j[e] = F̃j[e] is a fuzzy near UPi-filter

of A.

Case 3: 1 < |J| < |I|. Then F̃[e] =
⋂

j∈J F̃j[e]. Since e ∈ Ej for all j ∈ J and e /∈ Ek for some k ∈ I − J

and by same Case 1, we have F̃[e] is a fuzzy near UPi-filter of A.

Therefore, (F̃, E) is an e-fuzzy soft near UPi-filter of A. But since e is an arbitrary statistic of E, we have

(F̃, E) is a fuzzy soft near UPi-filter of A. �

In the same way as Theorem 2.2, we can prove that the union of family of fuzzy soft strongly UPs-ideals

(resp., fuzzy soft strongly UPi-ideals) of A is also a fuzzy soft strongly UPs-ideal (resp., fuzzy soft strongly

UPi-ideal).

In [16], we show that the union of two fuzzy soft UPs-subalgebras (resp., fuzzy soft UPi-subalgebras, fuzzy

soft near UPs-filters, fuzzy soft UPs-filters, fuzzy soft UPi-filters, fuzzy soft UPs-ideals, fuzzy soft UPi-ideals)

of A is not fuzzy soft UPs-subalgebra (resp., fuzzy soft UPi-subalgebra, fuzzy soft near UPs-filter, fuzzy soft

UPs-filter, fuzzy soft UPi-filter, fuzzy soft UPs-ideal, fuzzy soft UPi-ideal).



Int. J. Anal. Appl. 17 (5) (2019) 830

Definition 2.9. Let {(F̃i, Ei) | i ∈ I} be a nonempty family of fuzzy soft sets over a common universe U

where I is an arbitrary index set. The intersection of (F̃i, Ei) is defined to be the fuzzy soft setei∈I (F̃i, Ei) =
(F̃, E) satisfying the following conditions:

(i) E =
⋂

i∈I Ei 6= ∅ and

(ii) F̃[e] =
⋂

i∈I F̃i[e] for all e ∈ E.

Theorem 2.3. The intersection of family of fuzzy soft UPs-subalgebras of A is also a fuzzy soft UPs-

subalgebra.

Proof. Let (F̃i, Ei) be a fuzzy soft UPs-subalgebras of A for all i ∈ I. Assume that ei∈I (F̃i, Ei) = (F̃, E)
is the intersection of (F̃i, Ei) for all i ∈ I. Then E =

⋂
i∈I Ei 6= ∅. Let e ∈ E. By Theorem 1.2, we have

F̃[e] =
⋂

i∈I F̃i[e] is a fuzzy UPs-subalgebra of A. Therefore, (F̃, E) is an e-fuzzy soft UPs-subalgebra of A.

But since e is an arbitrary statistic of E, we have (F̃, E) is a fuzzy soft UPs-subalgebra of A. �

In the same way as Theorem 2.3, we can use Theorem 1.2 to prove that the intersection of family of fuzzy

soft UPi-subalgebras (resp., 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 A is also a fuzzy soft UPi-subalgebra (resp., fuzzy soft near UPs-filter, fuzzy soft near

UPi-filter, fuzzy soft UPs-filter, fuzzy soft UPi-filter, fuzzy soft UPs-ideal, fuzzy soft UPi-ideal, fuzzy soft

strongly UPs-ideal, fuzzy soft strongly UPi-ideal).

Definition 2.10. Let {(F̃i, Ei) | i ∈ I} be a nonempty family of fuzzy soft sets over a common universe

U where I is an arbitrary index set. The extended intersection of (F̃i, Ei) is defined to be the fuzzy soft set⋂
i∈I (F̃i, Ei) = (F̃, E) satisfying the following conditions:

(i) E =
⋃

i∈I Ei and

(ii) F̃[e] =
⋂

j∈J F̃j[e] for all e ∈ E with e ∈
⋂

j∈J Ej −
⋃

k∈I−J Ek where ∅ 6= J ⊆ I.

Theorem 2.4. The extended intersection of family of fuzzy soft UPs-subalgebras of A is also a fuzzy soft

UPs-subalgebra.

Proof. Let (F̃i, Ei) be a fuzzy soft UPs-subalgebras of A for all i ∈ I. Assume that
⋂

i∈I (F̃i, Ei) = (F̃, E) is

the extended intersection of (F̃i, Ei) for all i ∈ I. Then E =
⋃

i∈I Ei. Let e ∈ E.

Case 1: |J| = |I|. By Theorem 2.3, we have F̃[e] =
⋂

i∈I F̃i[e] is a fuzzy UPs-subalgebra of A.

Case 2: |J| = 1, that is, J is a singleton set. Then F̃[e] =
⋂

j∈{j} F̃j[e] = F̃j[e] is a fuzzy UPs-subalgebra

of A.

Case 3: 1 < |J| < |I|. Then F̃[e] =
⋂

j∈J F̃j[e]. Since e ∈ Ej for all j ∈ J and e /∈ Ek for some k ∈ I − J

and by same Case 1, we have F̃[e] is a fuzzy UPs-subalgebra of A.



Int. J. Anal. Appl. 17 (5) (2019) 831

Therefore, (F̃, E) is an e-fuzzy soft UPs-subalgebra of A. But since e is an arbitrary statistic of E, we

have (F̃, E) is a fuzzy soft UPs-subalgebra of A. �

In the same way as Theorem 2.4, we can prove that the extended intersection of family of fuzzy soft UPi-

subalgebras (resp., 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 A is also a fuzzy soft UPi-subalgebra (resp., fuzzy soft near UPs-filter, fuzzy soft near UPi-

filter, fuzzy soft UPs-filter, fuzzy soft UPi-filter, fuzzy soft UPs-ideal, fuzzy soft UPi-ideal, fuzzy soft strongly

UPs-ideal, fuzzy soft strongly UPi-ideal).

Definition 2.11. Let {(F̃i, Ei) | i ∈ I} be a nonempty family of fuzzy soft sets over a common universe U

where I is an arbitrary index set. The AND of (F̃i, Ei) is defined to be the fuzzy soft set
∧

i∈I (F̃i, Ei) = (F̃, E)

satisfying the following conditions:

(i) E =
∏

i∈I Ei and

(ii) F̃[(ei)i∈I ] =
⋂

i∈I F̃i[ei] for all (ei)i∈I ∈ E.

Theorem 2.5. The AND of family of fuzzy soft UPs-subalgebras of A is also a fuzzy soft UPs-subalgebra.

Proof. Let (F̃i, Ei) be a fuzzy soft UPs-subalgebras of A for all i ∈ I. By means of Definition 2.11, we

assume that
∧

i∈I (F̃i, Ei) = (F̃, E) such that E =
∏

i∈I Ei and F̃[(ei)i∈I ] =
⋂

i∈I F̃i[ei] for all (ei)i∈I ∈ E.

Assume that e = (ei)i∈I ∈ E and let x, y ∈ A. Then

f
F̃[e]

(x ·y) = f⋂
i∈I F̃i[ei]

(x ·y)

= inf{f
F̃i[ei]

(x ·y)}i∈I

≥ inf{min{f
F̃i[ei]

(x), f
F̃i[ei]

(y)}}i∈I

= min{inf{f
F̃i[ei]

(x)}i∈I, inf{fF̃i[ei](y)}i∈I}

= min{f⋂
i∈I F̃i[ei]

(x), f⋂
i∈I F̃i[ei]

(y)}

= min{f
F̃[e]

(x), f
F̃[e]

(y)}, and

f
F̃[e]

(x∗y) = f⋂
i∈I F̃i[ei]

(x∗y)

= inf{f
F̃i[ei]

(x∗y)}i∈I

≥ inf{min{f
F̃i[ei]

(x), f
F̃i[ei]

(y)}}i∈I

= min{inf{f
F̃i[ei]

(x)}i∈I, inf{fF̃i[ei](y)}i∈I}

= min{f⋂
i∈I F̃i[ei]

(x), f⋂
i∈I F̃i[ei]

(y)}

= min{f
F̃[e]

(x), f
F̃[e]

(y)}.



Int. J. Anal. Appl. 17 (5) (2019) 832

Therefore, F̃[e] is a fuzzy UPs-subalgebra of A, that is, (F̃, E) is an e-fuzzy soft UPs-subalgebra of A. But

since e is an arbitrary statistic of E, we have (F̃, E) is a fuzzy soft UPs-subalgebra of A. �

In the same way as Theorem 2.5, we can use Theorem 1.2 to prove that the AND of family of fuzzy

soft UPi-subalgebras (resp., 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 A is also a fuzzy soft UPi-subalgebra (resp., fuzzy soft near UPs-filter, fuzzy soft near

UPi-filter, fuzzy soft UPs-filter, fuzzy soft UPi-filter, fuzzy soft UPs-ideal, fuzzy soft UPi-ideal, fuzzy soft

strongly UPs-ideal, fuzzy soft strongly UPi-ideal).

Definition 2.12. Let {(F̃i, Ei) | i ∈ I} be a nonempty family of fuzzy soft sets over a common universe U

where I is an arbitrary index set. The OR of (F̃i, Ei) is defined to be the fuzzy soft set
∨

i∈I (F̃i, Ei) = (F̃, E)

satisfying the following conditions:

(i) E =
∏

i∈I Ei and

(ii) F̃[(ei)i∈I ] =
⋃

i∈I F̃i[ei] for all (ei)i∈I ∈ E.

Theorem 2.6. The OR of family of fuzzy soft near UPi-filters of A is also a fuzzy soft near UPi-filter.

Proof. Let (F̃i, Ei) be a fuzzy soft near UPi-filters of A for all i ∈ I. By means of Definition 2.12, we assume

that
∨

i∈I (F̃i, Ei) = (F̃, E) such that E =
∏

i∈I Ei and F̃[(ei)i∈I ] =
⋃

i∈I F̃i[ei] for all (ei)i∈I ∈ E. Assume

that e = (ei)i∈I ∈ E and let x, y ∈ A. Then

f
F̃[e]

(0) = f⋃
i∈I F̃i[ei]

(0)

= sup{f
F̃i[ei]

(0)}i∈I

≥ sup{f
F̃i[ei]

(x)}i∈I

= f⋃
i∈I F̃i[ei]

(x)

= f
F̃[e]

(x),

f
F̃[e]

(x ·y) = f⋃
i∈I F̃i[ei]

(x ·y)

= sup{f
F̃i[ei]

(x ·y)}i∈I

≥ sup{f
F̃i[ei]

(y)}i∈I

= f⋃
i∈I F̃i[ei]

(y)

= f
F̃[e]

(y), and



Int. J. Anal. Appl. 17 (5) (2019) 833

f
F̃[e]

(x∗y) = f⋃
i∈I F̃i[ei]

(x∗y)

= sup{f
F̃i[ei]

(x∗y)}i∈I

≥ sup{max{f
F̃i[ei]

(x), f
F̃i[ei]

(y)}}i∈I

= max{sup{f
F̃i[ei]

(x)}i∈I, sup{fF̃i[ei](y)}i∈I}

= max{f⋂
i∈I F̃i[ei]

(x), f⋂
i∈I F̃i[ei]

(y)}

= max{f
F̃[e]

(x), f
F̃[e]

(y)}.

Therefore, F̃[e] is a fuzzy near UPi-filter of A, that is, (F̃, E) is an e-fuzzy soft near UPi-filter of A. But

since e is an arbitrary statistic of E, we have (F̃, E) is a fuzzy soft near UPi-filter of A. �

In the same way as Theorem 2.6, we can use Theorem 1.3 to prove that the OR of family of fuzzy soft

strongly UPs-ideals (resp., fuzzy soft strongly UPi-ideals) of A is also a fuzzy soft strongly UPs-ideal (resp.,

fuzzy soft strongly UPi-ideal).

The following example shows that the OR of two fuzzy soft UPs-subalgebras of A is not fuzzy soft UPs-

subalgebra.

Example 2.1. 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

Then A = (A, ·,∗, X) is an f-UP-semigroup. Let (F̃1, E1) and (F̃2, E2) be two fuzzy soft sets over A where

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

with F̃1[price], F̃1[beauty], F̃1[specifications], F̃2[price], and F̃2[stability] are fuzzy sets in A defined as follows:

F̃1 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

F̃2 X 7 6 5

price 0.9 0.3 0.2 0.8

stability 0.7 0.2 0.5 0.2



Int. J. Anal. Appl. 17 (5) (2019) 834

Then (F̃1, E1) and (F̃2, E2) are two fuzzy soft UPs-subalgebras of A. Since (price, price) ∈ E1 ×E2, we have

(f
F̃1[price]∪F̃2[price]

)(5 ∗ 6) = (f
F̃1[price]∪F̃2[price]

)(7)

= 0.7

� 0.8

= min{0.8, 0.9}

= min{(f
F̃1[price]∪F̃2[price]

)(5), (f
F̃1[price]∪F̃2[price]

)(6)}.

Thus F̃1[price]∪F̃2[price] is not a fuzzy UPs-subalgebra of A, that is, (F̃1, E1)∪(F̃2, E2) is not a (price, price)-

fuzzy soft UPs-subalgebra of A. Hence, (F̃1, E1)∪(F̃2, E2) is not a fuzzy soft UPs-subalgebra of A. Moreover,

(F̃1, E1) ∨ (F̃2, E2) is not a fuzzy soft UPs-subalgebra of A.

We can apply those examples in [16] to check that the OR of two fuzzy soft UPi-subalgebras (resp.,

fuzzy soft near UPs-filters, fuzzy soft UPs-filters, fuzzy soft UPi-filters, fuzzy soft UPs-ideals, fuzzy soft

UPi-ideals) of A is not fuzzy soft UPi-subalgebra (resp., fuzzy soft near UPs-filter, fuzzy soft UPs-filter,

fuzzy soft UPi-filter, fuzzy soft UPs-ideal, fuzzy soft UPi-ideal).

We prove that certain distributive laws hold in fuzzy soft set theory with respect to the restricted union,

the union, the intersection, and the extended intersection on any fuzzy soft sets.

Theorem 2.7. Let (F̃i, Ei) and (F̃, E) be fuzzy soft sets over a common universe U where I is a nonempty

set. Then the following properties hold:

(1) (F̃, E) e (
⋃

i∈I (F̃i, Ei)) =
⋃

i∈I ((F̃, E) e (F̃i, Ei)),

(2) (
⋃

i∈I (F̃i, Ei)) e (F̃, E) =
⋃

i∈I ((F̃i, Ei) e (F̃, E)),

(3) (F̃, E) d (
⋂

i∈I (F̃i, Ei)) =
⋂

i∈I ((F̃, E) d (F̃i, Ei)),

(4) (
⋂

i∈I (F̃i, Ei)) d (F̃, E) = (F̃i, Ei)) d
⋂

i∈I ((F̃, E),

(5) (F̃, E) ∩ (di∈I (F̃i, Ei)) =di∈I ((F̃, E) ∩ (F̃i, Ei)),
(6) (di∈I (F̃i, Ei)) ∩ (F̃, E) =di∈I ((F̃i, Ei) ∩ (F̃, E)),
(7) (F̃, E) ∪ (ei∈I (F̃i, Ei)) =ei∈I ((F̃, E) ∪ (F̃i, Ei)),
(8) (ei∈I (F̃i, Ei)) ∪ (F̃, E) =ei∈I ((F̃i, Ei) ∪ (F̃, E)),
(9) (F̃, E) e (di∈I (F̃i, Ei)) =di∈I ((F̃, E) e (F̃i, Ei)),

(10) (di∈I (F̃i, Ei)) e (F̃, E) =di∈I ((F̃i, Ei) e (F̃, E)),
(11) (F̃, E) d (ei∈I (F̃i, Ei)) =ei∈I ((F̃, E) d (F̃i, Ei)), and
(12) (ei∈I (F̃i, Ei)) d (F̃, E) =ei∈I ((F̃i, Ei) d (F̃, E)).

Proof. (1) First, we investigate left hand side of the equality. Suppose that
⋃

i∈I (F̃i, Ei) = (G̃, E
U ) is

the union of (F̃i, Ei) for all i ∈ I. Then EU =
⋃

i∈I Ei and for any e ∈ E
U , G̃[e] =

⋃
j∈J F̃j[e] with



Int. J. Anal. Appl. 17 (5) (2019) 835

e ∈
⋂

j∈J Ej−
⋃

k∈I−J Ek where ∅ 6= J ⊆ I. Thus (F̃, E)e(
⋃

i∈I (F̃i, Ei)) = (F̃, E)e(G̃, E
U ) = (H̃, EUI ). For

any e ∈ EUI = E∩EU 6= ∅, H̃[e] = F̃[e]∩G̃[e] where E∩EU = E∩(
⋃

i∈I Ei) =
⋃

i∈I (E∩Ei). By considering

G̃ as piecewise defined function, we have H̃[e] = F̃[e]∩(
⋃

j∈J F̃j[e]) with e ∈
⋂

j∈J (E∩Ej)−
⋃

k∈I−J (E∩Ek)

where ∅ 6= J ⊆ I.

Consider the right hand side of the equality. Suppose that (F̃, E) e (F̃i, Ei) = (̃Ii, E
I
i ) is the intersection

of (F̃, E) and (F̃i, Ei) for all i ∈ I. Then EIi = E ∩ Ei 6= ∅ and for any e ∈ E
I
i , Ĩi[e] = F̃[e] ∩ F̃i[e].

Now,
⋃

i∈I ((F̃, E) e (F̃i, Ei)) =
⋃

i∈I (̃Ii, E
I
i ) = (J̃, E

IU ), where EIU =
⋃

i∈I E
I
i =

⋃
i∈I (E ∩ Ei). For any

e ∈ EIU , J̃[e] =
⋃

j∈J Ĩj[e] with e ∈
⋂

j∈J E
I
j −

⋃
k∈I−J E

I
k where ∅ 6= J ⊆ I. Considering Ĩi as piecewise

functions for all i ∈ I, we have J̃[e] =
⋃

j∈J (F̃[e] ∩ F̃j[e]) with e ∈
⋂

j∈J (E ∩ Ej) −
⋃

k∈I−J (E ∩ Ek)

where ∅ 6= J ⊆ I. By Theorem 1.1 (1), it is clear that H̃ and J̃ are same set-valued mapping. Hence,

(F̃, E) e (
⋃

i∈I (F̃i, Ei)) =
⋃

i∈I ((F̃, E) e (F̃i, Ei)).

(2) By using techniques as in (1) and by Theorem 1.1 (2), then (2) can be derived.

(3) By using techniques as in (1) and by Theorem 1.1 (3), then (3) can be derived.

(4) By using techniques as in (1) and by Theorem 1.1 (4), then (4) can be derived.

(5) First, we investigate left hand side of the equality. Suppose that di∈I (F̃i, Ei) = (G̃, ERU ) is the
restricted union of (F̃i, Ei) for all i ∈ I. Then ERU =

⋂
i∈I Ei 6= ∅ and for any e ∈ E

RU , G̃[e] =
⋃

i∈I F̃i[e].

Thus (F̃, E) ∩ (di∈I (F̃i, Ei)) = (F̃, E) ∩ (G̃, ERU ) = (H̃, ERUEI ). For any e ∈ ERUEI = E ∪ERU , we have

H̃[e] =




F̃[e] if e ∈ E \ERU

G̃[e] if e ∈ ERU \E

F̃[e] ∩ G̃[e] if e ∈ E ∩ERU.

By taking into account the definition of G̃ along with H̃, we can write

H̃[e] =




F̃[e] if e ∈ E \ (
⋂

i∈I Ei)⋃
i∈I F̃i[e] if e ∈ (

⋂
i∈I Ei) \E

F̃[e] ∩ (
⋃

i∈I F̃i[e]) if e ∈ E ∩ (
⋂

i∈I Ei).

Consider the right hand side of the equality. Suppose that (F̃, E) ∩ (F̃i, Ei) = (̃Ii, EEIi ) is the extended

intersection of (F̃, E) and (F̃i, Ei) for all i ∈ I. Then for any e ∈ EEIi = E ∪Ei, we have

Ĩi[e] =




F̃[e] if e ∈ E \Ei

F̃i[e] if e ∈ Ei \E

F̃[e] ∩ F̃i[e] if e ∈ E ∩Ei.

Now, di∈I ((F̃, E) ∩ (F̃i, Ei)) = di∈I (̃Ii, EEIi ) = (J̃, EEIRU ) where EEIRU =
⋂

i∈I E
I
i =

⋂
i∈I (E ∪ Ei) =

E∪(
⋂

i∈I Ei) 6= ∅. For any e ∈ E
EIRU , J̃[e] =

⋃
i∈I Ĩi[e]. By taking into account the properties of operations



Int. J. Anal. Appl. 17 (5) (2019) 836

in set theory and considering Ĩi as piecewise defined functions for all i ∈ I, we have

J̃[e] =




⋃
i∈I F̃[e] if e ∈ E \ (

⋂
i∈I Ei)⋃

i∈I F̃i[e] if e ∈ (
⋂

i∈I Ei) \E⋃
i∈I (F̃[e] ∩ F̃i[e]) if e ∈ E ∩ (

⋂
i∈I Ei).

And so

J̃[e] =




F̃[e] if e ∈ E \ (
⋂

i∈I Ei)⋃
i∈I F̃i[e] if e ∈ (

⋂
i∈I Ei) \E⋃

i∈I (F̃[e] ∩ F̃i[e]) if e ∈ E ∩ (
⋂

i∈I Ei).

By Theorem 1.1 (1), it is clear that H̃ and J̃ are same set-valued mapping. Hence, (F̃, E)∩(di∈I (F̃i, Ei)) =
di∈I ((F̃, E) ∩ (F̃i, Ei)).

(6) By using techniques as in (5) and by Theorem 1.1 (2), then (6) can be derived.

(7) By using techniques as in (5) and by Theorem 1.1 (3), then (7) can be derived.

(8) By using techniques as in (5) and by Theorem 1.1 (4), then (8) can be derived.

(9) First, we investigate left hand side of the equality. Suppose that di∈I (F̃i, Ei) = (G̃, ERU ) is the
restricted union of (F̃i, Ei) for all i ∈ I. Then ERU =

⋂
i∈I Ei 6= ∅ and for any e ∈ E

RU , G̃[e] =
⋃

i∈I F̃i[e].

Thus (F̃, E) e (di∈I (F̃i, Ei)) = (F̃, E) e (G̃, ERU ) = (H̃, ERUI ). For any e ∈ ERUI = E ∩ ERU = E ∩
(
⋂

i∈I Ei) 6= ∅, we have H̃[e] = F̃[e] ∩ G̃[e] = F̃[e] ∩ (
⋃

i∈I F̃i[e]).

Consider the right hand side of the equality. Suppose that (F̃, E) e (F̃i, Ei) = (̃Ii, E
I
i ) is the intersection

of (F̃, E) and (F̃i, Ei) for all i ∈ I. Then EIi = E ∩ Ei 6= ∅ and for any e ∈ E
I
i , Ĩi[e] = F̃[e] ∩ F̃i[e].

Now, di∈I ((F̃, E) e (F̃i, Ei)) = di∈I (̃Ii, EIi ) = (J̃, EIRU ), where EIRU =
⋂

i∈I E
I
i =

⋂
i∈I (E ∩ Ei) 6= ∅.

For any e ∈ EIRU , J̃[e] =
⋃

j∈J Ĩj[e] =
⋃

j∈J (F̃[e] ∩ F̃i[e]). Since
⋂

i∈I (E ∩ Ei) = E ∩ (
⋂

i∈I Ei), we

have EIRU = ERUI . By Theorem 1.1 (1), it is clear that H̃ and J̃ are same set-valued mapping. Hence,

(F̃, E) e (di∈I (F̃i, Ei)) = di∈I ((F̃, E) e (F̃i, Ei)).
(10) By using techniques as in (9) and by Theorem 1.1 (2), then (10) can be derived.

(11) By using techniques as in (9) and by Theorem 1.1 (3), then (11) can be derived.

(12) By using techniques as in (9) and by Theorem 1.1 (4), then (12) can be derived. �

Acknowledgment

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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	1. Introduction and Preliminaries
	2. Fuzzy Soft Sets over Fully UP-Semigroups
	Acknowledgment
	References