Some Algebraic Concepts on Discrete Fuzzy
Numbers
1
Thangaraj Beaula,
2Johnson.B
1 PG and Research Department of Mathematics, TBML College, Porayar 609 307. 2. DVC College of Education, Srimushnam 608703.
ABSTRACT
In this paper a new definition of fuzzy ring and field with binary operations on the set of discrete fuzzy numbers are defined. Some properties related to these concepts are investigated.
KEYWORDS
Euclidean distance, Discrete fuzzy number, Fuzzy approximation ambiguity rank.
1 INTRODUCTION
Chang and Zadeh[7] introduced the concept of fuzzy number with the consideration of the properties of probability functions.Since then a lot of mathematicians have been studying on fuzzy number, and have obtained many results [ 6,8,10,12,21,23] .
In 2001 Voxman [20] introduced the concept of discrete fuzzy numbers, the discrete fuzzy number can be used to represent the pixel value in the centre point of a window [8] Also he gave out the canonical representation of discrete fuzzy numbers.
In 1971,Rosenfeld[19] first introduced the concept of fuzzy subgroups, which was the first fuzzifica tion of any algebraic structure.Therefore the notion of different fuzzy algebraic strutures such as fuzzy ideals in rings and semi-rings etc. have been seriously studied by many mathematicians.
In 1982 W.J.Liu[15] introduced the concept of fuzzy ring and fuzzy ideal. In 1985 Ren [18] studied the notions of fuzzy ideal, in the case of Liu and Ren were actually a rational extension of Rosenfeld’s fuzzy group by starting with an ordinary ring and then define a fuzzy sub-ring based on the ordinary operations of the given ring. Based on the notion of fuzzy space which play the role of universal set in ordinary algebra and using fuzzy binary operation K.A Dib[9] obtained a new formulation for fuzzy rings and fuzzy ideals. Since then many mathematicians such as Aktaÿs and Cÿaćgman[5],Malik and Mordeson[16,17], Yuan and Lee[24] have studied about them and more recently in S.Abdullah et al[1-4,13]
In this paper , the basic definitions and concepts related to the topic is discussed. Further a new definition of fuzzy ring and field with binary operations on the set of discrete fuzzy numbers are defined. Some of its characteristic
properties are investigated.
2 PRELIMINARIES
Definition 2.1LetXbe auniverseofdiscourse,afuzzysetisdefinedas A = {(x, Ã (x): x ϵ X }
characterizedbyamembershipfunctionA(x):X[0,1],where A(x)denotesthe degreeofmembershipofthe element x to
the set A.
Definition2.2The-cut, A ofa setAisthe crispsubsetofAwithmembership gradesofatleast.
Thatis,A{x|A(x)}.
Definition 2.3 A fuzzy set on the real line R is a fuzzy number if it at leastpossess the following properties. (i) Amustbe anormalfuzzyset,
(ii)ThealphalevelsA mustbe closed forevery[0,1].
(iii)ThesupportofA,0Amustbe bounded.
number (DFN) if its support is finite (ie) there are 𝑥1, 𝑥2, … … … . . , 𝑥𝑛 ∈ 𝑅 with 𝑥1< 𝑥2<. . … < 𝑥𝑛
such that 𝑠𝑢𝑝𝑝 𝑢 = {𝑥1, 𝑥2, … … … … 𝑥𝑛} , and there are natural numbers 𝑠, 𝑡 with1 ≤ 𝑠 ≤ 𝑡 ≤ 𝑛 such that
i u xi = 1 foranynaturalnumberiwiths ≤ i ≤ t
ii u xi ≤ u xj foreachnaturalnumbersi, jwith 1 ≤ i ≤ j ≤ s
iii u xi ≥ u xj foreachnaturalnumberi, jwith t ≤ i ≤ j ≤ n.
Definition2.5[16](Equal fuzzy sets)Two fuzzy sets
u
andv
are said to be equal (denotedu
v
)if and only if,
x
X
u( )
x
v( )
x
Definition2.6(Euclidean distance)Let
X
( ,
x x x
1 2,
3,... )
x
n andY
( ,
y y y
1 2,
3,...
y
n)
are two points in Euclidean n-space, then the distance from X to Y or from Y to X is given by1 2 2 1
(X, Y)
(
)
n
i i
i
d
x
y
3.AMBIGUITY RANKS ON FUZZY GROUP OF DISCRETE FUZZY NUMBERS
Definition 3.1
For A⊂ R denote max A = max { x : x ∈ A}, andmin A = min {x : x ∈ A},Let u be a discrete fuzzy number a fuzzy set
on Rwithits support
u
0 finiteFor any r ∈ [0,1] and x0 ∈
0
u
denote𝑢 (r) = max
u
r,u
(r) = min
u
rforr≤1where
u
r= {x∕ u(x) ≥r}Let us define some sets as follows
(i)
u
─
r ≤ 1= 𝑥 ∈
0
u
≤ 𝑥u
(𝑟)(ii)
u
─
r ≤ 1= 𝑥 ∈
0
u
∕ 𝑥 ≥u
𝑟(iii) [
u
𝑟 , 𝑢 (r)] r ≤ 1= 𝑥 ∈
0
u
u
(𝑟) ≤ 𝑥 ≤u
𝑟(iv) av
r
u
=( )
( )
2
u r
u r
(v)
u x
( )
=
1 1 1 1( )
(
)( )
[ ( ), ( )]
( )
r r r ru
x
if
x
u
u r
x
if x
u r u r
u
x
if
x
u
(i) For every𝑥 ∈
u
─
r ≤ 1, 𝑦 ∈
v
─
r ≤ 1,if
d x u r
( , ( ))
d y v r
( , ( ))
then
u
r1( )
x
=
v
r1(y)
(ii)
d u r u r
( ( ), ( ))
d v r v r
( ( ), ( ))
then(
u r
)( )
x
(
v r
)( )
y
for
x
[ ( ), ( )],
u r u r
y
[ ( ), ( )]
v r v r
(iii) For every𝑥 ∈
u
r ≤ 1, 𝑦 ∈
v
r ≤ 1,ifd x u r
( , ( ))
d y v r
( , ( ))
then
u
r1( )
x
=
v
r1(y)
Definition 3.3Two discrete fuzzy number
u
andv
are said to be equal (denotedu
v
)if and only iffor every
1 1
,
r( )
r( )
x
X u
x
v
x
Example 3.4Consider the discrete fuzzy number
0.2 0.6 0.7 0.8 0.9 0.4
(
,
,
,
,
,
)
4
5
6
7
8
9
u
,(
0.2 0.6 0.7 0.8 0.9 0.4
,
,
,
,
,
)
1
2
3
4
5
6
v
By using definition 3.2,
u
v
Definition 3.5 (Fuzzy approximation)Let
u v
,
be two discrete fuzzy number thenu
v
if and only if av
r
u
= av
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
4
5
6
7
8
9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Definition 3.6 (Fuzzy group) A fuzzy group is an ordered pair
(G, )
whereG
is a set of discrete fuzzynumberand
is a binary operation onG
satisfying the following properties(i) For every
u v w G
, ,
,(
u v
)
w
u
(
v w
)
(ii) For every
u v
,
G
ifu
v
thenu v
u
(or)u v
v
(iii) For every
u
G
there existse
u
G
;e
u
u
andu e
ue
uu
u
(iv) For every
u
G
there exists 𝑢 ∈−1G
;𝑢−1
u
andu
𝑢−1= 𝑢−1
u
=e
uDefinition 3.7 A fuzzy group
( , )
G
is said to be abelian ifu v
v u
for allu
andv
inG
.A fuzzy group issaid to be non-abelian if it is not abelian.
Definition 3.8(Fuzzy ring)A fuzzy ring is a triplet
(G, , )
whereG
is a set of the discrete fuzzy numbersand
are binary operations onG
satisfying the following properties.(i) For every
u v w G
, ,
,(
u v
)
w
u
(
v w
)
(ii) For every
u v
,
G
ifu
v
thenu v
u
(or)u v
v
(iii) For every
u
G
there existse
u
G
;e
u
u
andu e
ue
uu
u
(iv) For every
u
G
there exists 𝑢 ∈−1G
;𝑢−1
u
andu
𝑢−1= 𝑢−1
u
=u
e
(v) For every
u v
,
G
,u v
v u
(vi) For every
u v w G
, ,
,(
u v
)
w
u
(
v w
)
(vii) For every
u v
,
G
ifu
v
thenu v
u
(or)u v
v
(viii) For every
u v w G
, ,
,u
(
v w
)
(
u v
) (
u w
)
and(
v w
)
u
(
v u
) (
w u
)
Definition 3.9 (Fuzzy commutative ring)A fuzzy ring
(G, , )
is said to be commutative ifu v
v u
for all,
u v
inG
Definition3.10 (Fuzzy division ring)A fuzzy ring
(G, , )
is said tobe division if its non zero(non the neutralelements related
binary operation)elements form a fuzzy group under
binary operation. Definition3.11 (Fuzzy field)A fuzzy field is a fuzzy commutative division ring.Definition3.12(Fuzzy approximation ambiguity rank) Let
G
be a fuzzy group then, ambiguity rank foru v
,
defined withar
1uv
(
l
11 ,vu ,
r
11 ,u v ,
lr
u v ,,
l
12 ,u v ,
r
12 ,u v )
as follows11 ,u v
l
1 1
[sup{ (
r( ),av
r) /
(0,1]}
d
d u
u
,
1 1
sup{ (
d
v
r( ),av
v
r) /
(0,1]}
]11 ,u v
r
1 1
[sup{ (
r( ),av
r) /
(0,1]}
d
d u
u
,
1
1
,
u v
lr
sup{ ( [ ]( ), [ ]( )) /
d u r x v r y
x
[ ( ), ( )],
u r u r
y
[ ( ), ( )]
v r v r
}suchthat( ( ), ( ))
d u r u r
d v r v r
( ( ), ( ))
12 ,u v
l
sup{
u
r1( ),
x
v
r1( )
y
/∀ 𝑥 ∈
u
─
r ≤ 1, 𝑦 ∈
v
─
r ≤ 1;
d x av u
( ,
[ ] )
r
d y av v
( ,
[ ] )}
r12 ,u v
r
sup{
u
r1( ),
x
v
r1( )
y
/∀ 𝑥 ∈
u
r ≤ 1, 𝑦 ∈
v
r ≤ 1;d x av u
( ,
[ ] )
r
d y av v
( ,
[ ] )}
rDefinition 3.13(Ambiguity rank of fuzzy group)Let
G
be a fuzzy group then, ambiguity rank forG
is defined withG
ar1
(
l
11G,
r
11G,
lr
*G,
l
12G, r
12G)
as follows11 ,
11G
sup{
u v/ ,
}
l
l
u v
G
11 ,
11G
sup{r
u v/ ,
}
r
u v
G
* ,
*G
sup{
u v/ ,
}
lr
lr
u v
G
12 ,
12G
sup{
u v/ ,
}
l
l
u v
G
12 ,
12G
sup{r
u v/ ,
}
r
u v
G
Lemma3.14
Let
G
be a fuzzy group with the ambiguity rankG
ar1.Then,11G
,
11G,
*G,
12G,r
12G0
l
r
lr
l
ProofBy definition
l
11G
sup{
l
11 ,u v / ,
u v
G
}
=M(say)>0⇒
l
11G
0
Similarly
r
11G,
lr
*G,
l
12G, r
12G
0
Lemma3.15
Let
G
be a fuzzy group with the ambiguity rankG
ar1.Then,
(
l
12G,
r
12 ,Glr
*G)
(0, 0, 0)
(
l
11G,
r
11G,
lr
*G)
(0, 0, 0)
Proof
Suppose for the sake of contradiction that
11 11 ,
(
l
G,
r
G)
(0,0)
,The following three modes can beinvestigated
(i)
11G
0
(ii)
11G
0
l
andr
11G
0
(iii)
11G
0
l
andr
11G
0
Case(i)
If
11G
0
l
then there existsu v
,
inG
1
1
sup{ (
( ),av
) /
(0,1]}
r r
d u
u
1 1
sup{ (
( ),av
) /
(0,1]}
r r
d
v
v
Without loss of generality ,we may assume that
1
1
sup{ (
d
u
r( ),av
u
r) /
(0,1]}
>
1 1
sup{ (
d
v
r( ),av
v
r) /
(0,1]}
Define, x =
1
1
sup{ (
d
u
r( ),av
u
r) /
(0,1]}
y =
1
1
sup{ (
d
v
r( ),av
v
r) /
(0,1]}
From definition3.13, there exists
s
support( );
u s
x
y
so,
t
s
there existst
ysuch that
d t av u
( ,
r)
d t av v
( ,
y
r),
1 1
( )
r
u
t
>0,
11
( )
y0
r
v
t
Therefore,0<
12 1
( )
G ru
t
l
namelyl
12G
0
This is a contradiction.Similar to the above, the cases (ii),(iii) leadsto
l
12G
0
andl
12G,
r
12 ,G
(0,0)
respectively, which are inconsistent..Lemma3.16
Let
( ,*)
G
be a fuzzy group with ambiguity rankG
ar1
(0, 0, 0, 0, 0).
Then(i)
u v
,
G
;u
v
(ii)
u v
,
G
;u v
u
(or)u v
v
Proof
Suppose
u v
,
ϵG
with11 11 *
(
l
G,
r
G,
lr
G)
(0, 0, 0)
Then
1
1
sup{ (
d u
r( ),av
u
r) /
(0,1]}
=
1 1
sup{ (
d
v
r( ),av
v
r) /
(0,1]}
⟹
sup{ (
d
u
1r 1( ),av
u
r) /
(0,1]}
=
1 1
sup{ (
d
v
r( ),av
v
r) /
(0,1]}
So|{
1
( ) /
( )
r
x
u r
u
x
>0}|= |{x
v r
( ) /
v
r1( )
x
>0}|,|{
1
( ) /
r( )
x
u r
u
x
>0}|= |{
1
( ) /
r( )
x
v r
v
x
>0}|Therefore, the prerequisite for
u
v
Also from
12 12 , *
(
l
G,
r
Glr
G)
(0,0,0)
For every
x
u r
( ), y
v(r)
;d x av u
( ,
[ ] )
r
d y av v
( ,
[ ] )
r⟹
d u
[
r1( ),
x
v
r1(y)]
=0⟹
u
r1( )
x
(
v
r1(y)
……….
1
And For every
x u r
( ), y
v(r)
;d x av u
( ,
[ ] )
r
d y av v
( ,
[ ] )
r⟹
d u
[
r1( ),
x
v
r1(y)]
=0⟹
u
r1( )
x
v
r1(y)
……….
2
And
x
[ ( ), ( )],
u r u r
y
[ ( ), ( )]
v r v r
;⟹
d u r u r
( ( ), ( ))
d v r v r
( ( ), ( ))
⟹
(
u r
)( )
x
(
v r
)( )
y
……….
3
Finallyfrom
1 , 2 , 3
and by definition 3.1 it follows thatu
v
(ii) According to the second condition of definition3.6 and case(i) it is obvious.
Lemma3.17Let
u v
,
be two discrete fuzzy number. Thenu
v
andu
v
if and only ifu
v
Proof
If
u
v
andu
v
then by the definition
3.1
it follows that(i) ∀ 𝑥 ∈
u
─
r ≤ 1, 𝑦 ∈
v
─
r ≤ 1,if
d x u r
( , ( ))
d y v r
( , ( ))
then
u
r1( )
x
=
v
r1(y)
(ii)
d u r u r
( ( ), ( ))
d v r v r
( ( ), ( ))
then(
u r
)( )
x
(
v r
)( )
y
(iii) ∀ 𝑥 ∈
u
r ≤ 1, 𝑦 ∈
v
r ≤ 1,ifd x u r
( , ( ))
d y v r
( , ( ))
then
u
r1( )
x
=
v
r1(y)
This implies
1
( )
1(y)
r r
u
x
v
,
x
X
,
y Y
.
Conversely
u
v
by the definition
3.3
it is obvious.Lemma3.18 Let
( ,*)
G
be a fuzzy group with ambiguity rankG
ar1
(0, 0, 0, 0, 0).
Then
u v
,
G
;e
u
e
vProof
According to the lemma 3.16 and by definition3.5, we have
e
u
e
vAnd
u u
v v
u e
e
u
u
v e
e
v
v
Then
[ ]
[ ]
[ ]
[ ]
[ ]
[v]
[ ]
[ ]
[v]
[v]
r u u r r
r v v r r
av u
av e
av e
av u
av u
av
av e
av e
av
av
………..(4)
The system (4) is a system at the crisp space, thus
av e
[ ]
u
av e
[ ]
v ………(5)Therefore from the case (5) and the definition (3.4) we have
e
u
e
vFinally, by lemma3.17 ,
e
u
e
vTheorem3.19Algebraic group is a special case of the fuzzy group with ambiguity rank
G
ar1
(0, 0, 0, 0, 0).
Proof
Let
( ,*)
G
be a fuzzy group with ambiguity rankG
ar1
(0, 0, 0, 0, 0).
Then to show that
(i)
u v w G
, ,
,(
u v
)
w
u
(
v w
)
………(6) and(ii) there exists
e
G
;u
G
andu e
e u
u
………(7)According to the (4) and (5) conditions on fuzzy group,using lemma3.16 and lemma3.17we get the case (6).
4. REFERENCES
[1] S.Abdullah.M.Aslam,T.A. Khan and M.Naeem, A new type of fuzzy normal subgroups and fuzzy cosets,Journal of Intelligent and Fuzzy Systems 25(2013),37-47.
[2] S.Abdullah.M.Aslam and K.Hila, A new generalization of fuzzy bi-ideals in semigroups its Applications in fuzzy finite state machines,
Multiple-Valued Logic and Soft Computing 24 (2015),599-623.
[3] S.Abdullah,N-dimensional
( ; )
-Fuzzy Ideals of hemirings, International journal of Machine Learning and Cybernetics5 (2014),635-645.[4]S.Abdullah.M.Aslam and H.Hedayathi,Interval valued
( ; )
- intuitionstic fuzzy ideals in hemirings, Journal of Intelligent and fuzzySystems 26 (2014),2873-2888 .
[5] H.Aktaÿs and Cÿaćgman, Atype of fuzzy ring ,Arch Math Logic 46 (2007), 165-177. [6] J.J. Buckley ,” Fuzzy complex numbers” Fuzzy Sets and Systems,33(1989), 333-345,1989.
[7]S.S.L.Chang , and L.A.Zadeh,” On fuzzy mapping and control ”, IEEE Trans Man Cybernet., 2(1972),30-34. [8] Diamond.D and Kloeden, “Metric space of fuzzy sets”, Fuzzy Sets and Systems,100(1999), 63-71. [9] K.A.Dib,N.Gaalhum and A.AM.Hassan,Fuzzy ring and fuzzy ideals,Fuzzy Math 4(2)(1996),245-261. [10] Dubois and Prade .H “Operations on fuzzy numbers”,Internet J Systems,9(1978),613-626. [11] D.Dubois and H.Prade,Fuzzy Sets Systems: Theory and Applications,1998.
[12] Goetschel R. andW.Voxman, “Elementry calculus”,Fuzzy Sets and Systems,18(1986),31-43.
[13] K.Hila and S.Abdullah,A study on intuitionistic fuzzy sets in
- semihypergroups, Journal of Intelligent and Fuzzy Systems26(2014),1695-1710.
[14]G.J.Klir and B.Yuan,Fuzzy sets and Fuzzy Logic:theory and applications,Prentice-Hall PTR, Upper Saddlie River,1995. [15]W.Liu,Fuzzy invariant subgroups and fuzzy ideals,Fuzzy Sets and Systems (1982),133-139.
[16] D.S.Malik and J.N. Mordeson,Fuzzy commutative Algebra, World Scientific Publishing, Singapore,1998. [17] D.S.Malik and J.N.Mordeson, Fuzzy relations on rings and groups. Fuzzy Setsand Systems 43(1991),117-123. [18] Y.C.Ren,Fuzzy Ideals and quotient rings,Fuzzy math 4(1985).19-26
[19] A.Rosenfield,Fuzzygroups,J Math Anal Appl35 (1971), 512-517
[20] VoxmanW ,”canonical representations of dicrete fuzzy numbers”, Fuzzy Sets and Systems 11(2001), 457-466.
[21] Wang Guixiang, Li Youming and Wen Chenglin, “On fuzzy n- cell numbers and n- dimension fuzzy vectors”, Fuzzy Sets and Systems, 158(2007), 71-84.
[22] WangGuixiang,Wu Cong and Chunhui Zhao, “Representation and Operations of Discrete Fuzzy Numbers”, Southeast Asian Bulletin of Mathematics, 28(2005),1003-1010.
[23] Wang Guixiang, and Wu Congxin, “ Fuzzy n-cell numbers and the differential of fuzzy n-cell number valuemappings”,Fuzzy Sets and Systems,130(2002), 367-381.