Anti Fuzzy Meet Subsemilattice
1
G.Mehboobnisha,
2B.Chellappa
1Research Scholar (Part Time-Mathematics), Alagappa University, Karaikudi-630003,
TamilNadu, India.
2
Principal, Nachiappa Swamigal Arts and Science College, Karaikudi-630003, Tamilnadu, India.
Abstract: In this paper, we made an attempt to define and study some properties of Anti Fuzzy meet subsemilattice and we introduce some definitions and theorems on the union and intersection of Anti Fuzzy meet subsemilattice.
Key words: Fuzzy meet semilattice, Fuzzy meet subsemilattice,Anti fuzzy meet subsemilattice,Anti ,Fuzzy level meet subsemilattice.
Introduction: The notion of Fuzzy sets was introduced by Zadeh,L.A.[38] in 1965.He has initiated fuzzy set theory as a modification of ordinary set theory. In this paper we define Anti fuzzy subsemilattice,Anti Fuzzy level meet subsemilattice and some related theorems.
Definition 1: Let A be a Fuzzy meet semilattice. A
fuzzy subset
T
:
A
[
0
,
1
]
of a fuzzy meet semilattice A is called a anti Fuzzy meetsubsemilattice of A if
,
,
y
A
x
T
(
x
y
)
max{
T
(
x
),
T
(
y
)}
Example 1: Let A={0,a,b,c,1}. Let
T
:
A
[
0
,
1
]
be a Fuzzy subset in A defined by,
6
.
0
)
(
,
7
.
0
)
(
,
4
.
0
)
0
(
T
a
T
b
T
5
.
0
)
(
c
T
,T
(
1
)
0
.
9
Thus,
T
is an anti fuzzy meet subsemilattice.Definition 2: Let
T
be any anti fuzzy meetsubsemilattice of a fuzzy meet semilattice A and let
].
1
,
0
[
t
ThenT
t
{
x
A
/
T
(
x
)
t
}
iscalled anti fuzzy level meet subsemilattice of
T
.Example 2: From Example(1), Let t=0.6.Then
T
t={a,1}. Then
T
t is an anti fuzzy level meetsubsemilattice of
T
.Remark:
T
t
T
S whenever t>sDefinition 3: Let
T
1 andT
2 be any two anti fuzzymeet subsemilattices of a fuzzy meet semi lattice A.
Then
T
1 is said to be contained inT
2 ifA
x
x
T
x
T
1(
)
2(
),
and is denoted by. 2
1
T
T
Definition 4: Let
T
1 andT
2 be any two anti fuzzymeet subsemilattices of a fuzzy meet semi lattice A.
If
T
1(
x
)
T
2(
x
),
x
A
, thenT
1 andT
2are said to be equal and it is written as
T
1=T
2.Definition 5: The complement of a anti fuzzy meet
subsemilattice
T
of a fuzzy meet semilattice Asymbolized by
~
T
(
x
)
1
T
(
x
),
x
A
.Definition 6: The intersection of two anti fuzzy meet
subsemilattices
T
1 andT
2 of a fuzzy meetsemilattice A is defined as
[
T
1
T
2](
x
)
min{
T
1(
x
),
T
2(
x
)},
x
A
Definition 7: The union of two anti fuzzy meet
subsemilattices
T
1 andT
2 of a fuzzy meetsemilattice A is defined as
[
T
1
T
2](
x
)
max{
T
1(
x
),
T
2(
x
)},
x
A
.
Lemma 1: Let
T
be an anti fuzzy meetsubsemilattice of a fuzzy meet semilattice A and let
t,s
Im
T
.ThenT
t=T
s if t=s.Proof:
If t=s ,then
T
t
T
s .Conversely,
Let
T
t
T
s.Since t
Im
T
,
x
A such thatT
(
x
)
t
,
t
T
sHence t=
T
(
x
)
s
--- (1)Similarly,
it can be proved that s
t--- (2)Then from (1) and (2), t=s.
Theorem 1: Two anti fuzzy meet subsemilattices
T
and
T
of a fuzzy meet semilattice A such that thecard Im
T
are equal iff ImT
=ImT
and
S
F
=F
S whereF
S={T
t/
T
t is an anti fuzzymeet subsemilattices of A for all t
Im
T
} andF
S
{
T
t/
T
t is an anti fuzzy level meet subsemilattice of A for all t
ImT
}
.Proof: Let
T
andT
be two anti fuzzy meet subsemilattices of a fuzzy meet semilattice A suchthat the card Im
T
Assume that
T
andT
are equal(ie)
T
(
x
)
T
(
x
)
---(1)
Similarly ,
it can be proved that
Im
T
Im
T
--- (3)(2)and(3)
Im
T
Im
T
(
4
)
)
1
(
),
0
(
)
0
(
,
).
4
(
)
1
(
,
Im
,
)
(
Im
,
)
(
)
0
(
,
,
by
T
T
t
T
x
and
by
T
t
t
x
T
T
t
t
x
T
T
t
T
x
and
F
LetT
t t S t
t tT
T
Similarly, it can be proved that
T
t
T
t)
7
(
)
6
(
&
)
5
(
)
6
(
,
)
5
(
,
,
S S S S S S S t S t t tF
F
F
F
Similarly
F
F
F
T
but
F
T
T
T
Hence
Equation (4) and (7) completes the proof.
Conversely, assume that Im
T
=ImT
and
S
S
F
F
To prove:
T
andT
are equal.Suppose
T
(
x
)
T
(
x
)
forx
A
Then either Im
T
ImT
orF
S
F
S.This is a contradiction
Hence
T
(
x
)
T
(
x
),
x
A
.
Therefore
T
and
,
T
are equal.Theorem 2
If B is a fuzzy meet semilattice of A, B≠A ,Then the
anti fuzzy meet subsemilattice
T
of A and isdefined by
B
A
x
if
t
B
x
if
s
x
T
~
,
,
)
(
where s,t
[ 0,1], s>t is an anti fuzzy meet subsemilattice of A.Proof:
Let x,y
ATo prove:
T
is an anti fuzzy meet subsemi lattice ofA.
(ie)To Prove:
.
T
(
x
y
)
max{
T
(
x
),
T
(
y
)}
It is proved by considering exhaustive three cases.
Case(i)
Let x,y
B.T
(
x
)
s
,
T
(
y
)
s
.
As x,y
B
,x
y
B
,since B is a fuzzy meet semilattice.)}
(
),
(
max{
)
(
x
y
T
x
T
y
T
= max{s.s}
=s
Case(ii):
Let x
B,y
A~B.T
(
x
)
s
,
T
(
y
)
t
As x
B,y
A~B,x
y
A
Now
T
(
x
)
s
t
T
(
y
)
(ie)
T
(
x
)
T
(
y
)
)}
(
),
(
max{
)
(
)
(
,
y
T
x
T
y
x
T
T
B
for
y
x
B
y
x
=max{s,s}
=s
)}
(
),
(
max{
)
(
)
(
~
,
~
y
T
x
T
y
x
T
T
B
A
for
y
x
B
A
y
x
=max{t,t}
=t
Hence
T
is an anti fuzzy meet subsemi lattice.Case(iii)
Let x, y
A~B,T
(
x
)
t
,
T
(
y
)
t
As x,y
A~B,x
y
A
~
B
or BIf x
y
A
~B, then)}
(
),
(
max{
)
(
)
(
T
x
y
T
x
T
y
T
=max{t,t}
=t
If
x
y
B
,then)}
(
),
(
max{
)
(
x
y
T
x
T
y
T
=max{s,s}
=s
Hence
T
is an anti fuzzy meet subsemilattice of Ain all the three cases.
Proposition 1:
A non empty fuzzy meet subset C of A is a fuzzy
meet semilattice of A iff
C is an anti fuzzy meetsubsemilattice of A.
Proof:
C
is nothing but characteristic function of the antifuzzy meet subsemi lattice of C.
(ie)
C
A
x
C
x
x
C
~
0
1
)
(
Where s,t
[
0
,
1
]
Then by previous theorem,the proof is complete.
Theorem 3:
The intersection of two anti fuzzy meet subsemilattices of a fuzzy meet semilattice A is also an anti fuzzy meet subsemilattice of A.
Proof:
Let A be a fuzzy meet semilattice.
Let
T
1 andT
2be any two anti fuzzy meetsubsemilattices of A.
T.P:
T
1
T
2is an anti fuzzy meet subsemilatticeof A.
Let a,b
T
1
T
2Then a,b
T
1 and a,b
T
2.2
1
,
and
a
b
T
T
b
a
Hence
a
b
T
1
T
2.Therefore
T
1
T
2 is an anti fuzzy meetsubsemilattice of A.
Theorem 4: The union of anti fuzzy meet subsemilattices of a fuzzy meet semilattice A is also an anti fuzzy meet subsemilattice of A iff one contained in the other.
Proof:
Let A be a fuzzy meet semilattice.
Let
T
1 andT
2be any two anti fuzzy meetsubsemilattices of A such that one contained in the other.
1
T
T
2 orT
2
T
12
1
T
T
=T
1 orT
1
T
2=T
2Then
T
1
T
2 is an anti fuzzy meet subsemilatticeof A.
Conversely, suppose
T
1
T
2 is an anti fuzzymeet subsemilattice of A.
T.P:
T
1
T
2 orT
2
T
1Suppose
T
1 is not contained inT
2Then there exist an element a,b such that
)
2
(
,
)
1
(
,
1 2
2 1
T
b
and
T
b
T
a
and
T
a
Clearly, a,b
T
1
T
2Since
T
1
T
2 is an antifuzzy meet subsemilatticeof A,
a
b
T
1 orT
2Case(i)
Let
a
b
T
11
1
)
(
)
(
a
b
a
a
b
b
b
T
a
Which is a contradiction to b
T
1,by
(
2
)
Case(ii)
Let
a
b
T
2Since
b
T
2and
,
b
T
2Hence
(a
b
)
b
a
(
b
b
)
a
1
a
T
2Which is a contradiction to the assumption
a
T
2by
(
1
)
Hence the assumption that
T
1 is not contained in1
T
andT
2 is not contained inT
1 is false.Therefore either
T
1
T
2 orT
2
T
1.Theorem 5 :
Let T be any fuzzy meet subsemilattice of a fuzzy
meet semilattice A Such that Im T = {0, t}, where
t [0,1]. If T = Tµ T , where Tµ, and T
are fuzzy meet subsemilattices of A, then either Tµ
T, or T Tµ .
Proof
Suppose Tµ T or T Tµ then there exist some x,
y A such that
Tµ(x) > T (x) and T (y) > Tµ (y)
Then t = T(x) = [Tµ
T] (x).= max {Tµ, (x), T(x)}
= Tµ (x)
0, Since Tµ (x) > T(x)and t = T (y) = [Tµ
T] (y)= max [Tµ (y), T (y)}
= T (y)
0, Since T (y) > Tµ (y)Therefore T(x) = t = T (y)
Tµ (x) = t = T (y)
T (y) < T (x) and Tµ (x) < Tµ (y)Then Tµ (x y) ≤ max Tµ(x), Tµ (y)}
= Tµ (y) < t ………..(1)
and T (x y) ≤ max {T (x), T (y)}
=
T(x) < t………..…(2)hence t = T(x y) = [Tµ T] (x y)
= max {Tµ (x y), T (x y)}
=max{Tµ(y),T(x)}
< t by (1) and (2) which is a
contradiction.
Therefore, if T = Tµ T, then either Tµ T or T
Tµ .
Conclusion: Thus in this paper, we have defined Anti fuzzy meet subsemilattice and some related theorems.
References:
[1] Chellappa.B and Anand .B ,Fuzzy join semi L-ideal, Indian Journal of Mathematics and Mathematical sciences,Vol-7,Dec 2011, pp103-109.
[2] Chellappa.B and Anand.B, Fuzzy join subsemilattices, vol-7, No.2,Dec 2011, pp 111-119.
