Indonesian
J
our
nal
of
Electrical
Engineering
and
Computer
Science
V
ol.
23,
No.
3,
September
2021,
pp.
1583
1589
ISSN:
2502-4752,
DOI:
10.11591/ijeecs.v23.i3.pp1583-1589
r
1583
Some
r
esults
on
-single
v
alued
neutr
osophic
subgr
oups
M.
Shazib
Hameed
1
,
Zaheer
Ahmad
2
,
Salman
Mukhtar
3
,
Asad
Ullah
4
1,2
Department
of
Mathematics,
Khw
aja
F
areed
Uni
v
ersity
of
Engineering
&
Information
T
echnology
,
Rahim
Y
ar
Khan,
Punjab
64200,
P
akistan
3
Department
of
Mathematics,
The
Islamia
Uni
v
ersity
of
Baha
w
alpur
,
Punjab
63100,
P
akistan
4
Department
of
Mathematics,
The
Islamia
Uni
v
ersity
of
Baha
w
alpur
,
Rahim
Y
ar
Khan
Campus,
Punjab
64200,
P
akistan
Article
Inf
o
Article
history:
Recei
v
ed
Mar
16,
2021
Re
vised
Jul
9,
2021
Accepted
Jul
13,
2021
K
eyw
ords:
Neutrosophic
-single
v
alued
neutrosophic
set
-single
v
alued
neutrosophic
subgroups
ABSTRA
CT
In
this
study
,
we
de
v
elop
a
no
v
el
structure
-single
v
alued
neutrosophic
set,
whi
ch
is
a
generalization
of
the
intuitionistic
set,
inconsistent
intuitionistic
fuzzy
set,
Pythagorean
fuzzy
set,
spherical
fuzzy
set,
paraconsistent
set,
etc.
Fuzzy
subgroups
play
a
vital
role
in
v
agueness
structure,
it
dif
fer
from
re
gular
subgroups
in
that
it
is
impossible
to
deter
-
mine
which
group
elements
belong
and
which
do
not.
In
this
paper
,
we
in
v
estig
ate
the
concept
of
a
-single
v
alued
neutrosophic
set
and
-single
v
alued
neutrosophic
sub-
groups.
W
e
e
xplore
the
idea
of
-single
v
alued
neutrosophic
set
on
fuzzy
subgroups
and
se
v
e
ral
characterizations
related
to
-single
v
alued
neutrosophic
subgroups
are
suggested.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Muhammad
Shazib
Hameed
Department
of
Mathematics
Khw
aja
F
areed
Uni
v
ersity
of
Engineering
&
Information
T
echnology
Rahim
Y
ar
Khan,
Punjab
64200,
P
akistan
Email:
shazib
.hameed@kfueit.edu.pk
1.
INTR
ODUCTION
In
general,
the
dra
wbacks
of
pre
viously
de
v
el
oped
methods
and
models
are
mitig
ated
by
the
ne
wly
defined
fuzzy
algebraic
structure.
Because
of
the
limitations
of
routine
mathematics,
it
cannot
al
w
ays
be
used.
Certain
daily
systems
ha
v
e
v
ague
and
missing
information.
Methodologies
were
seen
as
an
alternati
v
e
to
dealing
with
these
issues
and
pre
v
enting
fla
ws,
such
as
certainty
,
rough
set,
and
a
fuzzy
set
h
ypothesis.
Unfortunately
,
each
of
these
alternati
v
e
mathematics
has
fla
ws
and
dra
wbacks,
such
as
the
majority
of
terms
lik
e
true,
beautiful,
and
popular
,
which
are
not
readily
identifiable
or
e
v
en
ambiguous.
As
a
result,
the
rules
for
such
terms
can
dif
fer
from
one
person
to
the
ne
xt.
Zadeh
[1]
has
be
gun
an
analysis
of
the
possibility
based
on
the
participation
feature
assigning
a
re
g-
istration
grade
in
[0
;
1]
in
order
to
deal
with
such
unclear
and
uncertain
inform
ation.
Atanasso
v
[2]
suggested
that
intuitionistic
fuzzy
sets
could
be
used
as
a
fuzzy
set
e
xtension
in
lieu
of
the
concepts
of
enrolment
and
non-participation.
Molodtso
v
[3]
coined
the
term
soft
set
to
des
cribe
a
computational
model
for
dealing
with
uncertainties.
Because
of
its
applications
in
a
v
ariety
of
li
v
ely
topics,
the
possibility
of
soft
set
has
g
ained
a
ne
w
destination
for
scientists.
Cris
p
sets
ha
v
e
tw
o
independent
generalizations:
fuzzy
sets
and
soft
sets.
In
the
soft
set
h
ypothesis,
Ali
et
al.
[4]
suggested
se
v
eral
ne
w
operations.
The
y
discussed
e
xtended
and
restricted
union
and
intersection.
Y
ager
[5]-[7]
first
proposed
the
Pythagorean
fuzzy
set.
Fe
w
Pythagorean
fuzzy
data
intrusions
interv
entions
ha
v
e
been
de
v
eloped
and
implemented
by
Peng
et
al.
[8].
Peng
et
al.
look
ed
at
Pythagorean
fuzzy
soft
sets
and
ho
w
the
y
were
implemented
in
[9].
The
v
ariety
of
models
were
in
v
estig
ated
in
[10]-[14].
J
ournal
homepage:
http://ijeecs.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
1584
r
ISSN:
2502-4752
Arockiarani
and
Jenc
y
[15]
studied
the
basic
characteristics
of
fuzzy
neutrosophic
sets
and
also
intro-
duced
the
fuzzy
neutrosophic
topological
spaces.
The
y
also
e
xplored
the
properties
of
the
respecti
v
e
de
v
eloped
spaces
[16].
This
concept
is
e
xtended
for
the
groups
and
v
arious
algebraic
structures
as
gi
v
en
in
[17]-[29].
The
paper
is
arranged
as
follo
ws:
In
Section
2,
we
gi
v
e
some
basic
concepts
related
to
fuzzy
single-v
alued
neutrosophic
sets
(
S
V
N
S
s
)
.
In
Sections
3
and
4,
we
introduce
the
notion
of
-single
v
alued
neutrosophic
sets
(
-
S
V
N
S
s
)
and
-single
v
alued
neutrosophic
subgroups
respecti
v
ely
,
and
also
proposed
se
v
eral
characteriza-
tions
on
-single
v
alued
neutrosophic
subgroups.
2.
PRELIMIN
ARIES
Definition
2..1.
[15]
A
S
V
N
S
L
on
the
univer
se
set
S
is
defined
as:
L
=
fh
u;
L
(
u
)
;
L
(
u
)
;
L
(
u
)
i
;
u
2
S
g
wher
e
;
;
:
S
!
[0
;
1]
and
0
L
(
u
)
+
L
(
u
)
+
L
(
u
)
3
:
Definition
2..2.
[15]
Let
S
be
a
non
empty
set,
and
L
=
fh
u;
L
(
u
)
;
L
(
u
)
;
L
(
u
)
ig
;
M
=
fh
u;
M
(
u
)
;
M
(
u
)
;
M
(
u
)
ig
be
S
V
N
S
s
,
then
pr
oceeding
pr
operties
must
satisfy:
(i)
L
M
,
8
u
if
L
(
u
)
M
(
u
)
;
L
(
u
)
M
(
u
)
;
L
(
u
)
M
(
u
)
.
(ii)
L
[
M
=
h
u;
W
(
L
(
u
)
;
M
(
u
))
;
W
(
L
(
u
)
;
M
(
u
))
;
V
(
L
(
u
)
;
M
(
u
))
i
:
(iii)
L
\
M
=
h
u;
V
(
L
(
u
)
;
M
(
u
))
;
V
(
L
(
u
)
;
M
(
u
))
;
W
(
L
(
u
)
;
M
(
u
))
i
:
(iv)
L
n
M
(
u
)
=
h
u;
V
(
L
(
u
)
;
M
(
u
))
;
V
(
L
(
u
)
;
1
M
(
u
))
;
W
(
L
(
u
)
;
M
(
u
))
i
:
Definition
2..3.
[15]
A
S
V
N
S
L
is
called
null
or
empty
S
V
N
S
o
ver
the
univer
se
S
if
L
(
u
)
=
0
;
L
(
u
)
=
0
;
L
(
u
)
=
1
;
8
u
2
S
.
It
is
indicated
with
O
N
.
Definition
2..4.
[15]
A
S
V
N
S
of
L
is
an
absolute
S
V
N
S
o
ver
the
univer
se
of
S
,
if
L
(
u
)
=
1
;
L
(
u
)
=
1
;
L
(
u
)
=
0
;
8
u
2
S
.
It
is
indicated
with
1
N
.
Definition
2..5.
[15]
L
c
is
the
complement
of
S
V
N
S
L
whic
h
is
defined
as
L
c
=
h
u;
L
c
(
u
)
;
L
c
(
u
)
;
L
c
(
u
)
i
wher
e
L
c
(
u
)
=
L
(
u
)
;
L
c
(
u
)
=
1
L
(
u
)
;
L
c
(
u
)
=
L
(
u
)
.
It
is
also
possible
to
describe
the
complement
of
the
S
V
N
S
L
as
L
c
=
1
N
L
.
3.
-SINGLE
V
ALUED
NEUTR
OSOPHIC
SETS
Definition
3..1.
Consider
L
=
fh
u;
L
(
u
)
;
L
(
u
)
;
L
(
u
)
i
;
u
2
S
g
,
then
-
S
V
N
S
L
on
the
discour
se
uni-
ver
se
S
is
defined
as
L
=
fh
L
(
u
)
=
V
f
L
(
u
)
;
g
;
L
(
u
)
=
V
f
L
(
u
)
;
g
;
L
(
u
)
=
W
f
L
(
u
)
;
gi
;
u
2
S
g
and
0
L
(
u
)
+
L
+
L
3
,
wher
e
2
[0
;
1]
,
wher
e
;
;
:
L
!
[0
;
1]
.
Definition
3..2.
Let
S
be
a
non
empty
set,
and
L
=
h
L
(
u
)
=
V
f
L
(
u
)
;
g
;
L
(
u
)
=
V
f
L
(
u
)
;
g
;
L
(
u
)
=
W
f
L
(
u
)
;
gi
;
M
=
h
M
(
u
)
=
V
f
M
(
u
)
;
g
;
M
(
u
)
=
V
f
M
(
u
)
;
g
;
M
(
u
)
=
W
f
M
(
u
)
;
gi
,
then
following
conditions
must
hold
(i)
L
M
,
8
u
if
L
(
u
)
M
(
u
)
;
L
(
u
)
M
(
u
)
;
L
(
u
)
M
(
u
)
.
(ii)
L
[
M
=
h
u;
W
(
L
(
u
)
;
M
(
u
))
;
W
(
L
(
u
)
;
M
(
u
))
;
V
(
L
(
u
)
;
M
(
u
))
i
:
(iii)
L
\
M
=
h
u;
V
(
L
(
u
)
;
M
(
u
))
;
V
(
L
(
u
)
;
M
(
u
))
;
W
(
L
(
u
)
;
M
(
u
))
i
:
(iv)
L
n
M
(
u
)
=
h
u;
V
(
L
(
u
)
;
M
(
u
))
;
V
(
L
(
u
)
;
1
M
(
u
))
;
W
(
L
(
u
)
;
M
(
u
))
i
:
Definition
3..3.
A
-
S
V
N
S
L
is
called
null
or
empty
-
S
V
N
S
o
ver
the
univer
se
S
if
L
(
u
)
=
0
;
L
(
u
)
=
0
;
L
(
u
)
=
1
;
8
u
2
S
.
It
is
indicated
with
O
N
.
Definition
3..4.
A
-
S
V
N
S
of
L
is
an
absolute
-
S
V
N
S
o
ver
the
univer
se
of
S
if
L
(
u
)
=
1
;
L
(
u
)
=
1
;
L
(
u
)
=
0
;
8
u
2
S
.
It
is
indicated
with
1
N
.
Definition
3..5.
L
c
is
the
complement
of
-
S
V
N
S
L
whic
h
is
defined
as
L
c
=
h
u;
L
c
(
u
)
;
L
c
(
u
)
;
L
c
(
u
)
i
wher
e
L
c
(
u
)
=
L
(
u
)
;
L
c
(
u
)
=
1
L
(
u
)
;
L
c
(
u
)
=
L
(
u
)
.
Complement
of
the
-
S
V
N
S
L
is
L
c
=
1
N
L
.
Definition
3..6.
Let
S
and
T
be
two
non-empty
set,
Define
a
function
g
:
S
!
T
.
(i)
If
M
=
fh
v
;
M
(
v
)
;
M
(
v
)
;
M
(
v
)
i
:
v
in
T
g
be
a
-
S
V
N
S
in
T
,
then
g
1
(
M
)
is
a
pr
e-ima
g
e
of
M
under
g
be
a
-
S
V
N
S
in
S
as
descri
bed
g
1
(
M
)
=
fh
u;
g
1
(
M
(
u
))
;
g
1
(
M
(
u
))
;
g
1
(
M
(
u
))
i
:
u
in
S
g
wher
e
g
1
(
M
(
u
))
=
M
(
g
(
u
))
.
(ii)
If
L
=
fh
u;
L
(
u
)
;
L
(
u
)
;
L
(
u
)
i
:
u
in
S
g
be
a
-
S
V
N
S
in
S
then
under
g
the
ima
g
e
of
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
23,
No.
3,
September
2021
:
1583
–
1589
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
1585
L
is
denoted
by
g
(
L
)
,
is
t
he
-
S
V
N
S
in
T
as
described
g
(
L
)
=
fh
v
;
g
(
L
(
v
))
;
g
(
L
(
v
))
;
g
s
(
L
(
v
))
i
:
v
in
T
g
g
(
L
(
v
))
=
8
<
:
sup
u
2
g
1
(
v
)
L
(
u
)
;
if
g
1
(
v
)
6
=
0
N
0
;
otherwise
g
(
L
(
v
))
=
8
<
:
sup
u
2
g
1
(
v
)
L
(
u
)
;
if
g
1
(
v
)
6
=
0
N
0
;
otherwise
g
s
(
L
(
v
))
=
8
<
:
inf
u
2
g
1
(
v
)
L
(
u
)
;
if
g
1
(
v
)
6
=
0
N
1
;
otherwise
and
g
s
(
L
(
v
))
=
(1
g
(1
L
))
v
.
Definition
3..7.
Consider
L
is
a
-
S
V
N
S
in
gr
oup
(
S
;
:
)
.
Then
L
is
said
to
be
-single
valued
neutr
osophic
gr
oup
(in
short,
-
S
V
N
G
)
in
S
if
it
fulfill
these
two
conditions:
(i)
L
(
uv
)
L
(
u
)
^
L
(
v
)
;
L
(
uv
)
L
(
u
)
^
L
(
v
)
and
L
(
uv
)
L
(
u
)
^
L
(
v
)
(ii)
L
(
u
1
)
L
(
u
)
;
L
(
u
1
)
L
(
u
)
;
L
(
u
1
)
L
(
u
)
.
Definition
3..8.
let
L
and
M
be
two
-
S
V
N
S
s
in
S
wher
e
(
S
;
:
)
be
a
gr
oupoid,
Then
the
-single
valued
neutr
osophic
pr
oduct
of
L
and
M
,
L
M
is
defined
as
follows:
for
any
u
2
S
,
L
M
(
u
)
=
(
W
v
w
=
u
[
L
(
v
)
^
M
(
w
)]
;
for
eac
h
(
v
;
w
)
2
S
S
w
ith
v
w
=
u;
0
;
otherwise
L
M
(
u
)
=
(
W
v
w
=
u
[
L
(
v
)
^
M
(
w
)]
;
for
eac
h
(
v
;
w
)
2
S
S
w
ith
v
w
=
u;
0
;
otherwise
L
M
(
u
)
=
(
V
v
w
=
u
[
L
(
v
)
^
M
(
w
)]
;
for
eac
h
(
v
;
w
)
2
S
S
w
ith
v
w
=
u;
1
;
otherwise
:
Definition
3..9.
Consider
L
2
-
S
V
N
S
(
G
)
and
G
be
a
gr
oupoid.
Then
L
is
called:
(1)
-single
val-
ued
neutr
osophic
left
ideal
(
-
S
V
N
LI
)
of
G
if
for
some
u;
v
2
G;
L
(
uv
)
L
(
v
)
.
(i.e
.,)
L
(
uv
)
L
(
v
)
;
L
(
uv
)
L
(
v
)
;
and
L
(
uv
)
L
(
v
)
(2)
-single
valued
neutr
osophic
right
ideal
(
-
S
V
N
R
I
)
of
G
if
for
some
u;
v
2
G;
L
(
uv
)
L
(
u
)
.
(i.e
.,)
L
(
uv
)
L
(
u
)
;
L
(
uv
)
L
(
u
)
;
and
L
(
uv
)
L
(
u
)
(3)
-single
valued
neutr
osophic
ideal
(
-
S
V
N
I
)
of
G
if
it
is
-
S
V
N
LI
as
well
as
-
S
V
N
R
I
Clearly
,
L
is
a
-
S
V
N
I
of
G
,
for
any
u;
v
2
G
,
L
(
uv
)
L
(
u
)
_
L
(
v
)
;
L
(
uv
)
L
(
u
)
_
L
(
v
)
;
and
L
(
uv
)
L
(
u
)
^
L
(
v
)
:
Furthermor
e
,
a
-
S
V
N
I
(r
espectively
-
S
V
N
LI
,
-
S
V
N
R
I
)
is
a
single
valued
-
neutr
osophic
subgr
oupoid
-
S
V
N
S
GP
of
G
.
Remember
for
e
ver
y
-
S
V
N
S
G
P
L
of
G
we
g
et
L
(
u
n
)
L
(
u
)
;
L
(
u
n
)
L
(
u
)
;
and
L
(
u
n
)
L
(
u
)
for
e
very
u
2
G
,
while
u
n
is
any
composite
of
u
0
s
.
The
collection
of
all
-
S
V
N
S
GP
s
with
G
will
be
denoted
as
-
S
V
N
S
G
P
(
G
)
.
Definition
3..10.
Let
(
G;
:
)
be
a
gr
oupoid
and
assume
O
N
6
=
L
2
-
S
V
N
S
(
G
)
Then
L
is
called
a
-single
valued
neutr
osophic
subgr
oupoid
in
G
(
-
S
V
N
S
GP
in
G
)
if
L
L
L
.
Definition
3..11.
Let
(
G;
:
)
be
a
gr
oupoid
and
consider
L
2
-
S
V
N
S
(
G
)
.
Then
L
is
said
to
be
-
S
V
N
S
G
P
in
G
,
if
for
e
very
u;
v
2
G
,
L
(
uv
)
L
(
u
)
_
L
(
v
)
;
L
(
uv
)
L
(
u
)
_
L
(
v
)
;
and
L
(
uv
)
L
(
u
)
^
L
(
v
)
:
Clearly
0
N
and
1
N
ar
e
both
-
S
V
N
S
GP
s
of
G
.
Definition
3..12.
Let
L
2
-
S
V
N
S
(
G
)
.
If
for
any
2
P
(
G
)
,
9
a
t
0
2
suc
h
that
L
(
t
0
)
=
S
t
2
(
L
(
t
))
t
0
2
suc
h
that
L
(
t
0
)
=
S
t
2
(
L
(
t
))
.i.e
.,
L
(
t
0
)
=
W
t
2
(
L
(
t
))
;
L
(
t
0
)
=
W
t
2
(
L
(
t
))
;
L
(
t
0
)
=
V
t
2
(
L
(
t
))
,
wher
e
P
(
G
)
denote
the
power
set
of
G
.
then
we
called
L
have
a
sup-pr
operty
.
Definition
3..13.
Let
L
be
a
-
S
V
N
S
in
S
and
let
$
;
;
2
with
$
+
+
3
.
Then
the
set
S
(
$
;
;
)
L
=
f
u
2
S
:
L
(
u
)
C
(
$
;
;
)
(
u
)
g
=
f
i
2
S
:
L
(
u
)
;
L
;
L
(
u
)
g
is
called
a
(
$
;
;
)
-
le
vel
subset
of
L
.
Some
r
esults
on
-single
valued
neutr
osophic
subgr
oups
(M.
Shazib
Hameed)
Evaluation Warning : The document was created with Spire.PDF for Python.
1586
r
ISSN:
2502-4752
4.
-SINGLE
V
ALUED
NEUTR
OSOPHIC
SUBGR
OUPS
Definition
4..1.
Consider
L
2
-
S
V
N
S
G
P
(
G
)
and
assume
G
be
a
gr
oup.
Then
L
is
said
to
be
-
single
valued
neutr
osophic
subgr
oup
(
-
S
V
N
S
G
)
of
G
if
L
(
u
1
)
L
(
u
)
.
i.e
.
L
(
u
1
)
L
(
u
)
;
L
(
u
1
)
L
(
u
)
;
and
L
(
u
1
)
L
(
u
)
;
8
u
2
G
.
Pr
oposition
4..2.
Let
f
L
g
2
-
S
V
N
S
G
(
G
)
.
Then
T
2
L
2
-
S
V
N
S
G
(
G
)
:
Pr
oposition
4..3.
Let
L
and
M
be
any
two
-
S
V
N
S
G
s
of
a
gr
oup
G
.
Then
these
ar
e
equivalent
conditions:
(1)
L
M
2
-
S
V
N
S
G
(
G
)
(2)
L
M
=
M
L
.
Pr
oof
.
Proof
is
ob
vious.
Pr
oposition
4..4.
Let
L
2
-
S
V
N
S
G
(
G
)
.
Then
L
(
u
1
)
=
L
(
u
)
;
i.e
.
L
(
u
1
)
=
L
(
u
)
;
L
(
u
1
)
=
L
(
u
)
;
L
(
u
1
)
=
L
(
u
)
and
L
(
u
)
L
(
e
)
i.e
.
L
(
u
)
L
(
e
)
;
L
(
u
)
L
(
e
)
;
L
(
u
)
L
(
e
)
for
e
very
u
2
G
,
wher
e
e
signify
the
identity
element
in
G
.
Pr
oof
.
Suppose
u
2
G
.
So
L
(
u
)
=
L
((
u
1
)
1
)
L
(
u
1
)
,
8
u
2
G
.
L
(
u
)
=
L
((
u
1
)
1
)
L
(
u
1
)
,
8
u
2
G
.
L
(
u
)
=
L
((
u
1
)
1
)
L
(
u
1
)
,
8
u
2
G
.
Since
L
2
-
S
V
N
S
G
(
G
)
,
L
(
u
1
)
L
(
u
)
,
L
(
u
1
)
L
(
u
)
and
L
(
u
1
)
L
(
u
)
for
e
v
ery
u
2
G
.
Hence
L
(
u
1
)
=
L
(
u
)
,
L
(
u
1
)
=
L
(
u
)
,
L
(
u
1
)
=
L
(
u
)
.(i.e.,)
L
(
u
1
)
=
L
(
u
)
Also,
L
(
e
)
=
L
(
uu
1
)
L
(
u
)
^
L
(
u
1
)
=
L
(
u
)
,
L
(
e
)
=
L
(
uu
1
)
L
(
u
)
^
L
(
u
1
)
=
L
(
u
)
L
(
e
)
=
L
(
uu
1
)
L
(
u
)
^
L
(
u
1
)
=
L
(
u
)
Hence
L
(
u
)
L
(
e
)
;
L
(
u
)
L
(
e
)
;
L
(
u
)
L
(
e
)
8
u
2
G
.
(i.e.,)
L
(
u
)
L
(
e
)
.
Pr
oposition
4..5.
If
L
2
-
S
V
N
S
G
(
G
)
,
then
G
L
=
f
u
2
G
:
L
(
u
)
=
L
(
e
)
;
i:e:;
L
(
u
)
=
L
(
e
)
;
L
(
u
)
=
L
(
e
)
;
L
(
u
)
=
L
(
e
)
g
is
a
subgr
oup
of
G
.
Pr
oof
.
Let
u;
v
2
G
L
.
Then
L
(
u
)
=
L
(
e
)
;
L
(
u
)
=
L
(
e
)
;
L
(
u
)
=
L
(
e
)
and
L
(
v
)
=
L
(
e
)
;
L
(
v
)
=
L
(
e
)
;
L
(
v
)
=
L
(
e
)
.
Thus
L
(
uv
1
)
L
(
u
)
^
L
(
v
1
)
=
L
(
u
)
^
L
(
v
)
by
proposition
4..4
=
L
(
e
)
^
L
(
e
)
=
L
(
e
)
Similarly
L
(
uv
1
)
L
(
e
)
.
L
(
uv
1
)
L
(
u
)
_
L
(
v
1
)
=
L
(
u
)
_
L
(
v
)
by
proposition
4..4
=
L
(
e
)
_
L
(
e
)
=
L
(
e
)
.
Also,
by
proposition
4..4,
L
(
uv
1
)
L
(
e
)
;
L
(
uv
1
)
L
(
e
)
;
L
(
uv
1
)
L
(
e
)
.
So,
L
(
uv
1
)
=
L
(
e
)
;
L
(
uv
1
)
=
L
(
e
)
;
L
(
uv
1
)
=
L
(
e
)
.
.(i.e.,)
L
(
uv
1
)
=
L
(
e
)
.
Thus
uv
1
2
G
L
.
Hence
G
L
is
a
subgroup
of
G
.
Pr
oposition
4..6.
Let
L
2
-
S
V
N
S
G
(
G
)
:
If
L
(
uv
1
)
=
L
(
e
)
.(i.e
.,)
L
(
uv
1
)
=
L
(
e
)
;
L
(
uv
1
)
=
L
(
e
)
;
L
(
uv
1
)
=
L
(
e
)
for
any
u;
v
2
G
,
then
L
(
u
)
=
L
(
v
)
(i.e
.,)
L
(
u
)
=
L
(
v
)
;
L
(
u
)
=
L
(
v
)
;
L
(
u
)
=
L
(
v
)
Pr
oof
.
Let
u;
v
2
G
L
.
Then
L
(
u
)
=
L
((
uv
1
)
v
)
L
(
uv
1
)
^
L
(
v
)
=
L
(
e
)
^
L
(
v
)
=
L
(
v
)
Also,
by
proposi
tion
4..4
L
(
u
1
)
=
L
(
u
)
,
then
we
ha
v
e
L
(
uv
1
)
=
L
((
v
u
1
)
1
)
=
L
(
v
u
1
)
and
thus
L
(
v
)
=
L
((
v
u
1
)
u
)
L
(
v
u
1
)
^
L
(
u
)
=
L
(
uv
1
)
^
L
(
u
)
=
L
(
e
)
^
L
(
u
)
=
L
(
u
)
.
So
L
(
u
)
=
L
(
v
)
.
Similarly
,
we
ha
v
e
L
(
u
)
=
L
(
v
)
;
L
(
u
)
=
L
(
v
)
.
Pr
oposition
4..7.
L
2
-
S
V
N
S
G
(
G
)
if
and
only
if
L
(
uv
1
)
L
(
u
)
^
L
(
v
)
;
L
(
uv
1
)
L
(
u
)
^
L
(
v
)
;
L
(
uv
1
)
L
(
u
)
_
L
(
v
)
for
any
u;
v
2
G
.
Pr
oof
.
Using
Definition
4..1
and
proposition
4..4
we
get
the
proof.
Pr
oposition
4..8.
The
gr
oup
G
cannot
be
the
union
of
two
pr
oper
-
S
V
N
S
G
s
.
Pr
oof
.
Let
L
and
M
are
proper
-
S
V
N
S
Gs
of
a
group
G
whene
v
er
L
[
M
=
1
N
;
L
6
=
1
N
and
M
6
=
1
N
.
L
[
M
=
1
N
)
L
_
M
=
1
;
L
_
M
=
1
;
L
^
M
=
0
.
Then
L
=
1
or
M
=
1
;
L
=
1
or
M
=
1
;
L
=
0
or
M
=
0
Since
L
6
=
1
N
and
M
6
=
1
N
,
L
6
=
1
or
L
6
=
1
or
L
6
=
0
and
M
6
=
1
or
M
6
=
1
or
M
6
=
0
.
In
either
cases,
we
get
the
contradiction.
Pr
oposition
4..9.
If
L
is
a
-
S
V
N
S
G
P
of
a
gr
oup
G
then
it
is
-
S
V
N
S
G
of
G
.
Pr
oof
.
Suppose
u
2
G
.
Also
G
has
a
order
finite,
Assume
order
of
u
is
n
(finite).
)
u
n
=
e
,whereas
e
indicate
identity
of
G
.
Thus
u
1
=
u
n
1
.
Since
L
is
a
-
S
V
N
S
GP
of
a
group
G
,
Thus
L
(
u
1
)
=
L
(
u
n
1
)
=
L
(
u
n
2
u
)
L
(
u
)
L
(
u
1
)
=
L
(
u
n
1
)
=
L
(
u
n
2
u
)
L
(
u
)
;
L
(
u
1
)
=
L
(
u
n
1
)
=
L
(
u
n
2
i
)
L
(
u
)
.
Hence
L
is
a
-
S
V
N
S
G
of
G
.
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
23,
No.
3,
September
2021
:
1583
–
1589
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
1587
Pr
oposition
4..10.
Suppose
L
be
a
-
S
V
N
S
G
of
a
gr
oup
G
and
let
u
2
G
.
Then
L
(
uv
)
=
L
(
v
)
,i.e
.
L
(
uv
)
=
L
(
u
)
;
L
(
uv
)
=
L
(
u
)
;
L
(
uv
)
=
L
(
u
)
8
v
2
G
,
L
(
u
)
=
L
(
e
)
.
i.e
.
L
(
u
)
=
L
(
e
)
;
L
(
u
)
=
L
(
e
)
;
L
(
u
)
=
L
(
e
)
,
wher
e
identity
of
G
is
e
.
Pr
oof
.
Suppose
L
(
uv
)
=
L
(
v
)
for
e
v
ery
v
2
G
.
Then
ob
viously
L
(
u
)
=
L
(
e
)
.
Con
v
ersely
,
considering
L
(
u
)
=
L
(
e
)
.
Then
by
Proposition
4..4
L
(
v
)
L
(
u
)
;
L
(
v
)
L
(
u
)
;
L
(
v
)
L
(
u
)
8
v
2
G
.
Since
L
is
a
-
S
V
N
S
G
of
G
,
then
L
(
uv
)
L
(
u
)
^
L
(
v
)
;
L
(
uv
)
L
(
u
)
^
L
(
v
)
;
L
(
uv
)
L
(
u
)
_
L
(
v
)
.
Thus
L
(
uv
)
L
(
v
)
;
L
(
uv
)
L
(
v
)
;
L
(
uv
)
L
(
v
)
8
v
2
G
.
On
the
other
hand,
by
Proposit
ion
4..4
L
(
v
)
=
L
(
u
1
uv
)
L
(
u
)
^
L
(
uv
)
;
L
(
v
)
L
(
u
)
^
L
(
uv
)
;
L
(
v
)
L
(
u
)
_
L
(
uv
)
.
Since
L
(
u
)
L
(
v
)
;
L
(
u
)
L
(
v
)
;
L
(
u
)
L
(
v
)
8
v
2
G
L
(
u
)
^
L
(
uv
)
=
L
(
uv
)
;
L
(
u
)
^
L
(
uv
)
=
L
(
uv
)
;
L
(
u
)
_
L
(
uv
)
=
L
(
uv
)
.
So
L
(
v
)
L
(
uv
)
;
L
(
v
)
L
(
uv
)
;
L
(
v
)
L
(
uv
)
8
v
2
G
.
Hence
L
(
uv
)
=
L
(
v
)
;
L
(
uv
)
=
L
(
v
)
;
L
(
uv
)
=
L
(
v
)
;
8
v
2
G
.
Pr
oposition
4..11.
Let
define
a
gr
oup
homomorphism
g
:
G
!
G
0
,
wher
eas
L
2
-
S
V
N
S
G
(
G
)
,
M
2
-
S
V
N
S
G
(
G
0
)
.
Then
these
conditions
must
be
satisfy:
(i)
L
contain
the
sup-pr
operty
)
g
(
L
)
2
-
S
V
N
G
(
G
0
)
.
(ii)
g
1
(
M
)
2
-
S
V
N
S
G
(
G
)
.
Pr
oof
.
(i)
By
Proposition,
Assume
g
r
ou
poi
d
homomorphism
g
:
G
!
G
00
also
consider
L
2
-
S
V
N
S
(
G
)
contain
the
sup
property
.
(1)
L
2
-
S
V
N
S
G
P
(
G
)
)
g
(
L
)
2
-
S
V
N
S
GP
(
G
00
)
.
(2)
If
L
is
a
-
S
V
N
I
(
-
S
V
N
LI
;
-
S
V
N
R
I
)
of
G
,
then
g
(
L
)
is
a
-
S
V
N
I
(
-
S
V
N
LI
;
-
S
V
N
R
I
)
of
G
00
.
Since
g
(
L
)
2
-
S
V
N
S
GP
(
G
)
,
it
is
enough
indicate
that
g
(
L
)
(
v
1
)
g
(
L
)
(
v
)
;
g
(
L
)
(
v
1
)
g
(
L
)
(
v
)
;
g
(
L
)
(
v
1
)
g
(
L
)
(
v
)
,
8
v
2
g
(
G
)
.
Let
v
2
g
(
G
)
.
Then
6
=
g
1
(
v
)
G
.
S
ince
L
has
the
sup-property
,
9
u
0
2
g
1
(
v
)
for
that
L
(
u
0
)
=
W
t
2
g
1
(
v
)
L
(
t
)
;
L
(
u
0
)
=
W
t
2
g
1
(
v
)
L
(
t
)
;
L
(
u
0
)
=
V
t
2
g
1
(
v
)
L
(
t
)
;
g
(
L
)
(
v
1
)
=
g
(
L
)(
v
1
)
=
W
t
2
g
1
(
v
1
)
L
(
t
)
L
(
u
1
0
)
L
(
u
0
)
;
g
(
L
)
(
v
)
:
g
(
L
)
(
v
1
)
=
g
(
L
)(
v
1
)
=
W
t
2
g
1
(
v
1
)
L
(
t
)
L
(
u
1
0
)
L
(
u
0
)
;
g
(
L
)
(
v
)
:
g
(
L
)
(
v
1
)
=
g
(
L
)(
v
1
)
=
V
t
2
g
1
(
v
1
)
L
(
t
)
L
(
u
1
0
)
L
(
u
0
)
;
g
(
L
)
(
v
)
.
Hence
g
(
L
)
2
-
S
V
N
S
G
(
G
)
.
(ii)
By
Proposition
in
[18],
we
ha
v
e
a
groupoid
homomorphism
g
:
G
!
G
00
and
suppose
M
2
-
S
V
N
S
(
G
00
)
(1)
If
M
2
-
S
V
N
S
G
P
(
G
00
)
,
then
g
1
(
M
)
2
-
S
V
N
S
GP
(
G
)
.
(2)
If
M
is
a
-
S
V
N
I
(
-
S
V
N
LI
;
-
S
V
N
R
I
)
of
G
00
then
g
1
(
M
)
is
a
-
S
V
N
I
(
-
S
V
N
LI
;
-
S
V
N
R
I
)
of
G
.
Since
g
1
(
M
)
2
-
S
V
N
S
G
P
(
G
)
,
It
is
adequate
to
e
xpress
g
1
(
M
)(
u
1
)
g
1
(
M
)(
u
)
8
u
2
G
.
Let
u
2
G
.
Then
g
1
(
M
)
(
u
1
)
=
g
1
(
M
)(
u
1
)
=
M
(
g
((
u
1
))
=
M
(((
g
(
u
))
1
)
M
(
g
(
u
))
=
g
1
(
M
)
(
u
)
,
g
1
(
M
)
(
u
1
)
=
g
1
(
M
)(
u
1
)
=
M
(
g
((
u
1
))
=
M
(((
g
(
u
))
1
)
M
(
g
(
u
))
=
g
1
(
M
)
(
u
)
,
g
1
(
M
)
(
u
1
)
=
g
1
(
M
)(
u
1
)
=
M
(
g
((
u
1
))
=
M
(((
g
(
u
))
1
)
M
(
g
(
u
))
=
g
1
(
M
)
(
u
)
.
Hence
g
1
(
M
)
2
-
S
V
N
S
G
(
G
)
.
Pr
oposition
4..12.
Let
L
be
a
-
S
V
N
S
G
of
a
gr
oup
G
.
Then
for
e
very
(
$
;
;
)
2
3
with
(
$
;
;
)
L
(
e
)
,
(i.e
.,)
$
L
(
e
)
;
L
(
e
)
;
L
(
e
)
;
G
(
$
;
;
)
L
is
a
subgr
oup
of
G
,
wher
e
e
r
epr
esent
the
identity
of
G
.
Pr
oof
.
Clearly
,
G
(
$
;
;
)
L
6
=
:
Let
u;
v
2
G
(
$
;
;
)
L
:
Then
L
(
u
)
(
$
;
;
)
and
L
(
v
)
(
$
;
;
)
.
(i.e.,)
L
(
u
)
$
;
L
(
u
)
;
L
(
u
)
and
L
(
v
)
$
;
L
(
v
)
;
L
(
v
)
.
Since
L
2
-
S
V
N
S
G
(
G
)
,
L
(
uv
)
L
(
u
)
^
L
(
v
)
$
;
L
(
uv
)
L
(
u
)
^
L
(
v
)
;
L
(
uv
)
L
(
u
)
_
L
(
v
)
.
Thus
L
(
uv
)
(
$
;
;
)
.
So
uv
2
G
(
$
;
;
)
L
.
On
the
other
hand
,
L
(
u
1
)
L
(
u
)
$
;
L
(
u
1
)
L
(
u
)
;
L
(
u
1
)
L
(
u
)
.
Thus
L
(
u
1
)
(
$
;
;
)
:
So
u
1
2
G
(
$
;
;
)
L
.
Hence
G
(
$
;
;
)
L
is
a
subgroup
of
G
.
Pr
oposition
4..13.
Assume
L
be
a
-
S
V
N
S
in
a
gr
oup
G
suc
h
that
G
(
$
;
;
)
L
is
a
subgr
oup
of
G
for
eac
h
(
$
;
;
)
2
3
with
(
$
;
;
)
L
(
e
)
.
Then
L
is
a
-
S
V
N
S
G
of
a
gr
oup
G
.
Pr
oof
.
F
or
an
y
u;
v
2
G
,
let
L
(
u
)
=
(
t
1
;
s
1
;
r
1
)
and
let
L
(
v
)
=
(
t
2
;
s
2
;
r
2
)
.
Then
clearly
,
u
2
G
(
t
1
;s
1
;r
1
)
L
and
v
2
G
(
t
2
;s
2
;r
2
)
L
.
Suppose
t
1
<
t
2
;
s
1
<
s
2
and
r
1
>
r
2
.
Then
G
(
t
2
;s
2
;r
2
)
L
G
(
t
1
;s
1
;r
1
)
L
.
Thus
v
2
G
(
t
1
;s
1
;r
1
)
L
.
Since
G
(
t
1
;s
1
;r
1
)
L
is
a
subgroup
of
G
,
uv
2
G
(
t
1
;s
1
;r
1
)
L
.
Then
L
(
uv
)
=
(
t
1
;
s
1
;
r
1
)
.
(i.e.,)
Some
r
esults
on
-single
valued
neutr
osophic
subgr
oups
(M.
Shazib
Hameed)
Evaluation Warning : The document was created with Spire.PDF for Python.
1588
r
ISSN:
2502-4752
L
(
uv
)
t
1
;
L
(
uv
)
s
1
;
L
(
uv
)
r
1
:
So
L
(
uv
)
L
(
u
)
^
L
(
v
)
;
L
(
uv
)
L
(
u
)
^
L
(
v
)
;
L
(
uv
)
L
(
u
)
_
L
(
v
)
F
or
each
u
2
G
.
let
L
(
uv
)
=
(
$
;
;
)
.
Then
u
2
G
(
$
;
;
)
L
.
Since
G
(
$
;
;
)
L
is
a
subgroup
of
G
,
u
1
2
G
(
$
;
;
)
L
.
So
L
(
u
1
)
(
$
;
;
)
.
(i.e.,)
L
(
u
1
)
L
(
u
)
;
L
(
u
1
)
L
(
u
)
;
L
(
u
1
)
L
(
u
)
.
Hence
L
is
a
-
S
V
N
S
G
of
a
group
G
.
Pr
oposition
4..14.
Let
L
be
a
-
S
V
N
S
in
S
,
suppose
(
$
1
;
1
;
1
)
;
(
$
2
;
2
;
2
)
2
I
m
(
L
)
.
If
$
1
<
$
2
;
1
<
2
;
1
<
2
then
L
(
$
1
;
1
;
1
)
L
(
$
2
;
2
;
2
)
.
Pr
oposition
4..15.
Let
L
be
a
-
S
V
N
S
in
a
gr
oup
G
.Then
L
is
a
-
S
V
N
S
G
of
G
,
L
(
$
;
;
)
is
a
subgr
oup
of
G
for
e
very
(
$
;
;
)
2
I
m
(
L
)
.
Definition
4..16.
Let
L
be
a
-
S
V
N
S
G
of
gr
oup
G
and
consider
(
$
;
;
)
2
I
m
(
L
)
.
Then
subgr
oup
L
(
$
;
;
)
is
known
a
(
$
;
;
)
-le
vel
subgr
oup
of
L
.
Lemma
4..17.
Assume
L
be
any
-
S
V
N
S
in
S
.
Then
L
(
u
)
=
W
f
$
:
u
2
L
(
$
;
;
)
g
;
L
(
u
)
=
W
f
:
u
2
L
(
$
;
;
)
g
;
L
(
u
)
=
V
f
:
u
2
L
(
$
;
;
)
g
.
wher
e
u
2
S
and
(
$
;
;
)
2
3
with
$
+
+
3
.
Pr
oof
.
Let
=
W
f
$
:
u
2
L
(
$
;
;
)
g
;
=
W
f
:
u
2
L
(
$
;
;
)
g
;
=
^f
:
u
2
L
(
$
;
;
)
g
and
let
>
0
be
arbitrary
.
Then
<
W
f
$
:
u
2
L
(
$
;
;
)
g
;
<
W
f
:
u
2
L
(
$
;
;
)
g
;
+
>
^f
$
:
u
2
L
(
$
;
;
)
g
.
Thus
9
(
$
;
;
)
2
with
$
+
+
3
such
that
u
2
L
(
$
;
;
)
;
<
$
;
<
;
+
>
.
Since
u
2
L
(
$
;
;
)
;
L
(
u
)
$
;
L
(
u
)
;
L
(
u
)
.
Thus
L
(
u
)
;
L
(
u
)
;
L
(
u
)
.
Since
>
0
is
arbitrary
,
L
(
u
)
;
L
(
u
)
;
L
(
u
)
W
e
sho
w
that
L
(
u
)
;
L
(
u
)
;
L
(
u
)
.
Let
L
(
u
)
=
t
1
;
L
(
u
)
=
t
2
;
L
(
u
)
=
t
3
.
Then
t
1
+
t
2
+
t
3
3
.
Thus
u
2
L
(
t
1
;t
2
;t
3
)
.
So
t
1
2
f
$
:
u
2
L
(
$
;
;
)
g
;
t
2
2
f
:
u
2
L
(
$
;
;
)
g
;
t
3
2
f
:
u
2
L
(
$
;
;
)
g
.
Thus
t
1
W
f
$
:
u
2
L
(
$
;
;
)
g
;
t
2
W
f
:
u
2
L
(
$
;
;
)
g
;
t
3
V
f
:
u
2
L
(
$
;
;
)
g
.
(i.e.,)
L
(
u
)
;
L
(
u
)
;
L
(
u
)
.
W
e
should
indicate
by
(
L
)
the
-
S
V
N
S
G
generated
by
the
fuzzy
-neutrosophic
set
L
in
G
.
Similarly
(
L
(
$
;
;
)
)
for
the
le
v
el
subset
L
(
$
;
;
)
.
Lemma
4..18.
Let
G
be
a
gr
oup
of
or
der
finite
.
Suppose
9
is
a
-
S
V
N
S
G
L
of
G
that
meets
these
condi-
tions:
for
any
u;
v
2
G
,
(i)
L
(
u
)
=
L
(
v
)
)
(
u
)
=
(
v
)
.
(ii)
L
(
u
)
>
L
(
v
)
;
L
(
u
)
>
L
(
v
)
;
L
(
u
)
<
L
(
v
)
)
(
u
)
(
v
)
:
Then
G
is
a
cyclic.
Pr
oof
.
Consider
L
is
constant
on
G
.
So
L
(
u
)
=
L
(
v
)
)
(
u
)
=
(
v
)
.
By
(u),
(
u
)
=
(
v
)
.
)
G
=
(
u
)
.
Assume
L
is
not
constant
on
G
.
Assume
I
m
(
L
)
=
f
(
t
0
;
s
0
;
r
0
)
;
(
t
1
;
s
1
;
r
1
)
;
:::;
(
t
n
;
s
n
;
r
n
)
g
,
where
t
0
>
t
1
>
:::
>
t
n
;
s
0
>
s
1
>
:::
>
s
n
;
r
0
<
r
1
<
:::
<
r
n
:
Using
proposition
4..14,
4..15,
we
attain
the
chain
of
le
v
el
subgroups
of
L
:
L
(
t
0
;s
0
;r
0
)
L
(
t
1
;s
1
;r
1
)
::::L
(
t
n
;s
n
;r
n
)
=
G
Let
u
2
G
L
(
t
n
1
;s
n
1
;r
n
1
)
.
W
e
ha
v
e
to
sho
w
G
=
(
u
)
.
Let
g
2
G
L
(
t
n
1
;s
n
1
;r
n
1
)
.
Since
t
0
>
t
1
>
:::
>
t
n
;
s
0
>
s
1
>
:::
>
s
n
;
t
0
<
t
1
<
:::
<
t
n
;
L
(
g
)
=
L
(
u
)
=
L
(
t
n
1
;s
n
1
;r
n
1
)
:
By
(u),
(
g
)
=
(
u
)
.
Thus
G
L
(
t
n
1
;s
n
1
;r
n
1
)
(
u
)
.
No
w
assume
g
2
L
(
t
n
1
;s
n
1
;r
n
1
)
:
Then
L
(
g
)
t
n
1
>
t
n
=
L
(
u
)
;
L
(
g
)
s
n
1
>
s
n
=
L
(
u
)
;
L
(
g
)
r
n
1
<
r
n
=
L
(
u
)
.
By
(ii),
(
g
)
(
u
)
.
Thus
L
(
t
n
1
;s
n
1
;r
n
1
)
(
u
)
.
So
G
=
(
u
)
.
So
in
each
case,
G
is
c
yclic.
Lemma
4..19.
Suppose
p
n
be
the
or
der
of
gr
oup
G
suc
h
that
p
is
prime
.
Then
9
a
-
S
V
N
S
G
L
of
G
Complying
with
the
following
conditions:
e
very
u;
v
2
G
,
(i)
L
(
u
)
=
L
(
v
)
)
(
u
)
=
(
v
)
(ii)
L
(
u
)
>
L
(
v
)
;
L
(
u
)
>
L
(
v
)
;
L
(
u
)
<
L
(
v
)
)
(
u
)
(
v
)
:
Pr
oof
.
Assume
chain
of
follo
wing
subgroup
of
G
:
(
e
)
=
G
0
G
1
:::
G
n
1
G
n
=
G
,
where
G
u
;
the
collection
of
subgroup
of
G
and
generated
by
element
with
order
p
u
;
i
=
0
;
1
;
:::n
whereas
e
is
the
identity
of
G
.
W
e
construct
a
comple
x
mappi
n
g
L
=
(
L
;
L
;
L
)
:
G
!
3
as:
for
e
v
ery
u
2
G
,
L
(
e
)
=
(
t
0
;
s
0
;
r
0
)
and
L
(
u
)
=
(
t
u
;
s
u
;
r
u
)
if
u
2
G
u
G
i
1
for
an
y
i
=
1
;
2
;
:::n
,
where
t
u
;
s
u
;
r
u
2
such
that
t
u
+
s
u
+
r
u
3
,
t
0
>
t
1
>
:::
>
t
n
;
s
0
>
s
1
>
:::
>
s
n
;
r
0
<
r
1
<
:::
<
r
n
:
W
e
can
then
easily
v
erify
that
L
is
a
-
S
V
N
S
G
of
G
sustaining
both
conditions.
From
Lemma
4..18
and
Lemma
4..19,
get
necessary
result.
5.
CONCLUSION
In
this
article,
we
gi
v
e
the
notion
of
-
S
V
N
S
s
and
subgroups.
W
e
in
v
estig
ate
se
v
eral
operations
and
algebraic
properties
related
to
these
ideas.
In
future
w
ork,
researchers
may
e
xtend
this
idea
in
topological
spaces,
rings,
ideals,
fields,
and
v
ector
spaces.
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
23,
No.
3,
September
2021
:
1583
–
1589
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
1589
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