Fuzzy Soft Groups based on Fuzzy Space
ABDALLAH AL-HUSBAN
Department of Mathematics, Faculty of Science and Technology, Irbid
National University, P.O. Box: 2600 Irbid, JORDAN
Abstract: -In this paper, a new theory of fuzzy soft groups (FSG) based on fuzzy spaces (FS) from
Dib’s point of view is found. The concept of FSG and fuzzy soft subgroup (FSSG) based on FS was
elaborated. The relationship between FSG and FSG is also investigated.
Keywords: - FSG, FSSG, soft set (SS), fuzzy soft set (FSS).
Received: April 12, 2021. Revised: December 26, 2021. Accepted: January 20, 2022. Published: February 18, 2022.
1 Introduction
Explored and discussed the topic of fuzzy sets (F-
sets) [1]. The fuzzy group theory evolved in the
following way: The concept of a fuzzy subgroup of
a group was first described in [2]. The concept of FS
was suggested by Dib [3]. In the ordinary case, this
idea took the place of the universal set concept. Dib
[3] defined fuzzy functions (FF), fuzzy binary
operations (FBO), and fuzzy subspaces (FSB) in FS.
The concepts of fuzzy group, fuzzy subgroup, and
fuzzy group theory were constructed in the stated
study after defining FS and FBO. Begins by
introducing an SS [4]. It is a parameterized family
of universal set subsets and also introduces and
investigates FSS [5]. It is a more general idea that
combines the F-set and the SS. The goal of this
research was to combine two mathematical domains
on the F-set, fuzzy algebra and SS theory, to create a
new algebraic system called FSG based on FS. The
theory of FSG is developed and FSSG is introduced.
Based on the concept of FS, the theory of SS
established by [4] has been applied to fuzzy
subgroups to construct the notions of FSG and
FSSG.
2 Preliminaries
Definition 2.1 ([3]) A FS
, [0,1]()QN
is the
set of all ordered pairs
( , ),q N q Q
( , ) ( , ):Q N q N q Q
where
( , ) {( , ): }.q N q n n N
The ordered pair
( , )qN
is called a fuzzy element
(FE) in the FS.
Definition 2.2 ([3]) A FSB
U
of the FS
( , )QN
is
the collection of all ordered pairs
( , )
q
q
where
0
qU
,
0
UQ
and
.
qN
Definition 2.3 ([3]) Let
( , )QN
( , )YN
and
( , )ZN
be FS. The (FF)
D
from
( , ) ( , ) ,()Q Y QN N NY N
into
( , )ZN
is typified with the ordered pair
(),qy
D J
where
:D Q Y Z
a function and
qy
J
is a family
function
:
xy NJ NN
fulfilling the requirements:
(i)
qy
J
is non-decreasing on
,NN
(ii)
0, )( 00
qy
J
and
1, )( 11
xy
J
.
Definition 2.4 ([3]) A FBO
( , )
qy
DDJ
on the
FS
( , [0,1])QN
is a FF from
( , ) ( , ) ( , )Q N Q N Q N
where
:D Q Q Q
and
:
qy NJ NN
are functions with satisfying
(i)
( , ) 0
qy nJ m
if
0n
and
0m
.
(ii)
qy
J
are onto.
3 FSG over FS
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In this paper, we define (FSG) and (FSSG) and
obtain some related results. Also, we will introduce
FSSG induced by fuzzy soft subset.
Definition 3.1. An FSG typified by
,( , )()NQD
is an FS iff for every,
(( , ( )), ),
()
q q N
Fe
(( , ( )), )
()
y y N
Fe
and
()
, ( ) ,()()
Fe
z z N
the following
requirements are met:
(i) Associative:
(( , ( )), ) (( , ( )), )) (( , ( )), )
( ) ( ) ( )
(( , ( )), ) (( , ( )), ) (( , ( )), )
( ) ( ) ( )
q q N D y y N D z z N
F e F e F e
q q N D y y N F z z N
F e F e F e
It has an identity
()
, ( ) ,()()
Fe
e e N
, for which
( ) ( )
( ) ( )
()
, ( ) , , ( ) ,
, ( ) , , ( ) ,
, ( ) ,,
( ) ( )
(e ) ( )
()
( ) ( )
( ) ( )
()
F e F e
F e F e
Fe
q q N D e e N
x N D q q N
q q N
(ii) Every fuzzy soft element (FSE)
()
, ( ) ,()()
Fe
q q N
has an inverse
() 1
, ( ) ,()()
Fe
q q N
such that
( ) ( )
( ) ( )
()
1
, ( ) , , ( ) ,
1
, ( ) , , ( ) ,
, ( ) , .
( ) ( )
( ) ( )
()
( ) ( )
( ) ( )
()
F e F e
F e F e
Fe
q q N D q q N
q q N D q q N
q q N
Denote
1
(( ) , ),qqNN
, then
()
()
()
, ( ) ,
, ( ) ,
, ( ) ,
()
()
()
()
()
()
Fe
Fe
Fe
qN
yDq q N
qDy Dy
eNe
It follows from (i), (ii), and (iii) which
( , ) Q D
is an
FSG.
Theorem 3.1 Associated to each FSG
, (,)()Q N D
there is a fuzzy group
,( , )()NQD
and they are isomorphic to each
other by the correspondence
()
, ( ) ( , ).,()()
Fe
qqN q N
Definition 3.2 A FSG
)(,,()Q N D
is said to be a
commutative FSG if
( ) ( )
( ) ( )
, ( ) , , ( ) ,
, ( ) , , ( ) ,
( ) ( )
( ) ( )
( ) ( )
( ) ( )
F e F e
F e F e
q D y
y y q
NN
NND
qy
q
for all FSE
()
, ( ) ,()()
Fe
q q N
and
()
, ( ) ,()()
Fe
y y N
of the FS
( , ).QN
Example 3.2 Consider the set
4{0,1,2,3},C
{0,1,2}.E
Define FBO
( , )
qy
D D J
over the
FS
4
( , [0,1])N
set as follow:
4
( ,y) ,D q q y
where
4
refers to
addition modulo 4,
( , )
xy nmJnm
and
4
: , F(e): [0,1]
A
F N C N
.
0 1 2 3
,,
() 1 1 1 1
0 1 2 3
() 0.5 0 0 0
0 1 2 3
,,
() 0.8 0.8 0.8 0.8
0 1 2 3
,
() 0.2 0 0.2 0
( , ) (0) , },
(1) { , , , },
(2) { , },
(3) { , , }
{{
}.
Fe
Fe
Fe
Fe
FC
The element of FSS
0 1 2 3
,,
() 1 1 1 1
(0) , }.{
Fe
Thus
4
( , ), )ND
is an FSG.
Theorem 3.2 The statements are correct for any
FSG:
(1) For fuzzy soft identity, the element is unique.
(2) The inverse of each FSE
()
, ( ) , ( , ),( ) ( )()
Fe Qq NDqN
is
unique.
1 1 1 1
()
()
( ) , ( ) ,
, ( ) ,
()
()
()
( ).
qD e
qD e
q q N
q q N
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Definition 3.3. Let
, [0,1])( , )( Q N D
be an FSG
and let
0
, ): ({}
q
U q q U
be an FSB of
, . ,()Q N U D
is called an FSSG of the FSG
)( , ,()Q N D
if:
(i)
D
is closed on the FSB U, i.e.,
, , )
,
,
( ) (
()
( )()
q
qy
Dy
q q y y
qy
qDy
qDy J
D
(ii)
( , )UD
meets the requirements of
an FSG
Theorem 3.3
(,)UD
is an FSSG of the FSG
, [0,1]( ),()Q N D
if and only if:
(1)
0
(), UD
is a subgroup of
,, QD
(2)
( , ) ( ).
qy q y xD y q q y y
JJ
Proof. If (1) and (2) are satisfied, then
(a)
D
is closed on the FSB
.U
Let
, ), ( , ) .( qy
q y U
Then
,,
, ,
( ) ( )
( ( ))
.(),
qy
q y q y
qD y
q D y
qDy J
qDy U
(b)
(), U D
satisfies the FSG. Let
(), ,
q
q
(),, ),.(
yz
y z U
Then
(
( )
)
(( ) ( )( ( )
( ) ( )
(( )
()
( ) ( , )
(
1 , , ,
, ,
) ,
( ),
,
, , ,) (( ) ( ).
q y z
qD y z
qD y Dz
qD yDz
q yDz
q y z
yDz
b q D y D z
qDy D z
qDy Dz
qD yDz
qD
q D y D z
b2 , , ,
,
,
,
( ) ( ) ( )
()
)
( ( ),) .
(
q e qDe
q
eDq
ee
D e qDe
q
eD
q
q
e D q
1
11
1
1
1
11
1
1
b3 Each , has an inverse ( , ),
sin
()
()
()
(
ce
, ( , ) ( , )
( , )
,
( , ) , .)
qq
qq qDq
q Dq
e
q
q
q
q D q qDq
q Dq
q
q
e
Dq
We can deduce from (a) and (b) that
(),UD
is an
FSSG of FSG
,)(( ).,Q N D
Conversely if
(,)UD
is an FSSG of FSG
)( ,,()Q N D
Then (i) is
restricted by associativity. The following is also
valid:
) ( )
.
(q qy y qy q
q
y
Dy
JJ
Theorem 3.4
();UD
is an FSSG of the FSG
, [0,1]( ),()Q N D
iff:
(i)
10,qDy U
for every
0
, ;q y U
(ii)
( , ) ( ).
q y q y qD y q q y y
JJ
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Theorem 3.5 let
0( ), , ( ) ( ( ), )H C D C DH
and
( ( ), )HCD
are FSSG of the FSG
( , ), )( Q N D
if;
(i)
0
(, )CD
is a subgroup of
,QD
where
0: ( ) 0 ;{}C q Q C q
(ii)
0
( ), (y) ( ) , , .()
xy
J C q C C qd y q y C
Proof. We prove the result for
0)(,( )H C D
and
the rest would be done in the same way.
Let
0
( , ) ()
qq
qH
and let be
0) )(,(q
HD
an
FSSG of the FSG
( , ),( ).QN D
Then (i) holds by
Again by Theorem 3.2 we have:
( , ) (0, ), (0,
(0, )
)
qy q y qy q y
qD y
qF y
JJ
That is, (0, ), (0, ) (0, ), (0, )
.
qy q y qy q
qDy
y
u
JJ
Conversely, assume that conditions (i) and (ii) hold,
then:
(0, ),(0, ) (0, ),(0, )
(0, 0), (0, ), ( ,
.
0)
qy q y qy q y
qy q
qD y
y q y
JJ
JJ
Similarly, we can prove that
0, , 0,( ) ( )
qy q y qD y
J
which proves by
Theorem 3.2 that
0)(,( )H C D
is fuzzy subgroups
of
( , ),( ).Q N D
4 Conclusion
In this paper, we have applied the fuzzy group study
started by Dib (1994) to the context of FSG.
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