International Journal of Mathematical Archive-9(3), 2018,
97-106Available online throug
ISSN 2229 – 5046International Journal of Mathematical Archive- 9(3), March – 2018 97 CONFERENCE PAPER
National Conference dated 26th Feb. 2018, on “New Trends in Mathematical Modelling” (NTMM - 2018),
A STUDY ON FINE INTUITIONISTIC FUZZY TOPOLOGICAL RING SPACES
R. NANDHINI
1AND DR. D. AMSAVENI
21Department of Mathematics (Research Scholar),
Sri Sarada College for Women, Salem - 636 016, Tamilnadu, India.
2Department of Mathematics (Assistant Professor),
Sri Sarada College for Women, Salem - 636 016, Tamilnadu, India. E-mail: [email protected]1, [email protected]2.
ABSTRACT
I
n this paper the notions of fine intuitionistic fuzzy topological spaces are introduced and the concepts of fine intuitionistic fuzzy ring structure spaces are defined also its characterizations are investigated. Finally, the fine intuitionistic fuzzy normal subrings are introduced and studied few of its properties.Keywords: Fine intuitionistic fuzzy set (FIfS), Fine intuitionistic fuzzy topological space (FIfTS), Fine intuitionistic fuzzy ring structure space(FIfTRSS).
2010 Subject Classification Primary: 54A40, 03E72, 54F45.
1. INTRODUCTION
Lofti A. Zadeh[10] introduced the fuzzy set theory in 1965. The notion of a fuzzy topology was introduced by Chang [3] in 1968. The concept of intuitionistic fuzzy set was first publishedby Krassimir T. Attanassov[1] as a generalization of the notion of fuzzy sets. Coker.D [4] developed the idea of an intuitionistic fuzzy topological spaces in 1997.Meena.K and Thomas.V[5]introduced, an intuitionisticL-fuzzy Subrings. Fine topological space was introduced by Power P. L. and RajakK [8]. Fine fuzzy topological spaces are introduced and studied the concept of fine fuzzy sp-closed sets by Nandhini. R [6]. In the present paper, we introduced a fine intuitionistic fuzzy set and fineintuitionistic fuzzy topological spaces and discussed some of its properties. The concepts of fine intuitionistic fuzzy ring structure spaces are defined and its characterizations are also examined. Finally, the fine intuitionistic fuzzy normal subrings are studied with few of its properties.
2. PRELIMINARIES
Definition 2.1 [3]: Let X be a non-empty set and
I
be the unit interval. A fuzzy set of X is an element of the setI
x
of all functions from X toI
.Definition 2.2[4]: Let X be a nonempty fixed set. An intuitionistic fuzzy set A is an object having the form
{
x
x
x
x
X
}
A
=
,
µ
A(
),
γ
A(
)
:
∈
where the functionsµ
A(
x
)
:
X
→
I
andγ
A(
x
)
:
X
→
I
denote the degree of membership and the degree of nonmembership of each element x∈Xto the set A, respectively and1
0
≤
µ
A+
γ
A≤
for each x∈X .Definition 2.3 [4]: An intuitionistic fuzzy topology (IFT) on a nonempty set X is a family
τ
of IFSs inX
satisfying the following conditions:(a)
.
~
1
,
~
0
∈
τ
(c)
G
i∈
τ
for any arbitrary family{
G
i:
i
∈
J
}
⊆
τ
.
The pair
(
X
,
τ
)
is called an intuitionistic fuzzy topological space (abbreviated as IFTS) and any IFS inτ
is known as an intuitionistic fuzzy open set (IFOS) in X.Definition 2.4 [4]: Let A and B be intuitionistic fuzzy set of X. Then A is said to be quasi coincident with B is denoted byAqB if there exist an x
∈
X such thatγ
B(
x
)
<
µ
A(
x
)
orγ
A(
x
)
<
µ
B(
x
).
Definition 2.5[8]: Let (X, τ) be a topological space and define τ(𝐴𝛼) = 𝜏𝛼 (say) = {𝐺𝛼(≠X) : 𝐺𝛼∩𝐴𝛼 ≠ 𝜙, for 𝐴𝛼∈𝜏 and 𝐴𝛼
≠ 𝜙, 𝑋, for some α ∈J, where J is the index set }.Define 𝜏𝑓 = {𝜙, 𝑋,
J
{T
}
∈
α α }. The collection 𝜏𝑓 of subsets of Xis
called the fine collection of subsets of X and (X, τ, 𝜏𝑓) is said to be fine space X generated by the topology τ on X.
Definition 2.6 [6]: Let (X,T) be a fuzzy topological space and let
T
(
λ
α)
=
T
α=
{
µ
α(
≠
1
X)
:
µ
α∧
λ
α≠
0
X,
forλ
α∈
T
andλ
α≠
0
X,
1
X in someα
∈
J
,
where Jis the index set}
be the collection of fine fuzzy sets of X.Then the collection
{0
,
1
,
{T
}
}
J X
X
∈=
α αf
T
is said to be the fine fuzzy collection of subsets of X and)
T
,
T
,
X
(
f is called the Fine fuzzy topological space (abbreviated as) FfTS.Definition 2.7[5]: An Intuitionistic L-fuzzy subset A = {x, µA(x), νA(x)/x ∈ R} of R is said to be an Intuitionistic L-fuzzy subring of R (ILFSR) if for all x, y ∈ R
(i) µA(x − y) ≥ µA(x) ∧ µA(y) (ii) µA(xy) ≥ µA(x) ∧ µA(y) (iii) νA(x − y) ≤ νA(x) ∨νA(y)(iv) νA(xy) ≤ νA(x) ∨νA(y).
Definition 2.8 [9]: Let
( , , .)
R
+
be a ring. An intuitionistic fuzzy subring A of R is said to be an intuitionistic fuzzy normal subring of R if it satisfies the following axioms(i)
µ
A(
xy
)
=
µ
A(
yx
)
(ii)
ν
A(
xy
)
=
ν
A(
yx
)
∀
x y
,
∈
R
.
3. FINE INTUITIONISTIC FUZZY TOPOLOGICAL SPACE
Definition 3.1: Let X be a nonempty set. Let a fine intuitionistic fuzzy setbe defined as
X
x
x
A
x
A
x
A
=
,
(
(
)),
(
(
))
:
∈
α
α
τ
γ
µ
τ
for allx
∈
X
~
~ ~
( 1 ) :
quasi
, where ( )
is degree of membership
(
( ))
function with
( )
, and
1 , 0 , for some
, is an index set
AA
X
A
x
I
A
A
A
A
x
x
J J
A
α
α α α α
µ
µ α
λ
λ
δ
λ
τ
τ µ
δ
τ
α
≠
∈
=
=
∈
≠
∈
~
~ ~
( 0 ) : not quasi , where is degree of non membership
( ( ))
function with and 0 1 for some , is an index set
X
α α α α
A α
A α
νA νA θA ν (x) IA
τγ τ γA x
θ (x) τA γ , ,
α
J J = ∈
= =
∈ = ∈
Where
0
~≤
+
≤
1
~.
A
A γ
µ
τ
τ
Definition 3.2: A fine intuitionistic fuzzy topology (FIfT) on a nonempty set X is a family
τ
f of FIfSs in X satisfyingthe following axioms
1.
0
X~,
1
X~∈
τ
f.2. If for any
µ
1,
µ
2∈
τ
fthenµ
1
µ
2∈
τ
f.3.
µ
i∈
I
τ
f for any arbitrary family{
G
i:
i
∈
J
}
⊆
τ
f.
The triplet
(
X
,
τ
,
τ
f)
is called a Fine intuitionistic fuzzy topological space FIfTS (for short) and any FIfS inτ
f isExample 3.1: Let X = {a, b, c} and
=
2
.
0
3
.
0
4
.
0
,
8
.
0
7
.
0
4
.
0
,
a
b
c
a
b
c
x
A
,
=
2
.
0
2
.
0
4
.
0
,
8
.
0
8
.
0
6
.
0
,
a
b
c
a
b
c
x
B
,
=
1
.
0
1
.
0
3
.
0
,
85
.
0
9
.
0
7
.
0
,
a
b
c
a
b
c
x
C
Then the family
τ
=
{
0
~,
1
~,
A
,
B
,
C
}
is an intuitionistic fuzzy topological space in X.Let us define a fineintuitionistic fuzzy set as follows
≤
≤
=
=
79
.
0
,
69
.
0
,
39
.
0
1
.
0
,
1
.
0
,
1
.
0
))
(
(
Ax
a
b
c
Aa
b
c
A
τ
µ
αµ
ατ
µ~ ~
(
( ))
0 , 1 ,
,
,
,
,
0.1 0.1 0.1
0.39 0.29 0.19
A A A
a
b
c
a
b
c
x
α α
γ
τ
=
τ γ
=
≤
γ
≤
,
≤
≤
=
=
69
.
0
,
79
.
0
,
59
.
0
1
.
0
,
1
.
0
,
1
.
0
))
(
(
Bx
a
b
c
Ba
b
c
B
τ
µ
αµ
ατ
µ~ ~
(
( ))
0 , 1 ,
,
,
,
,
0.1 0.1 0.1
0.39 0.19 0.19
B B B
a
b
c
a
b
c
x
α α
γ
τ
=
τ γ
=
≤
γ
≤
,
≤
≤
=
=
84
.
0
,
89
.
0
,
69
.
0
1
.
0
,
1
.
0
,
1
.
0
))
(
(
Cx
a
b
c
Ca
b
c
C α α
µ
µ
τ
τ
µ~ ~
(
( ))
0 , 1 ,
,
,
,
,
0.1 0.1 0.1
0.29 0.09 0.09
C C C
a
b
c
a
b
c
x
α α
γ
τ
=
τ γ
=
≤
γ
≤
Where
0
~≤
τ
µA+
τ
γA≤
1
~Then
~ ~
, , , , , 0 ,1 ,~ ~ , , , ,
0.1 0.1 0.1 0.39 0.69 0.79 0.1 0.1 0.1 0.39 0.29 0.19
0 , 1 , , , , , , 0 ,1 ,~ ~ , ,
0.1 0.1 0.1 0.59 0.79 0.69 0.1 0.1 0.1 0.39
A A
f X X B B
a b c a b c a b c a b c
a b c a b c a b c a
α α
α α
µ γ
τ µ γ
≤ ≤ ≤ ≤ = ∨ ≤ ≤ ≤ ≤
,0.19 0.19, ,, , , , , , 0 ,1 ,~ ~ , , , ,
0.1 0.1 0.1 C 0.69 0.89 0.84 0.1 0.1 0.1 C 0.29 0.09 0.09
b c
a b c a b c a b c a b c
α α µ γ ≤ ≤ ≤ ≤
is a fine intuitionistic fuzzy topology defined on X
Then
(
X
,
τ
,
τ
f)
is called a fine intuitionistic fuzzytopological space.Definition 3.3: Let
(
,
,
)
1
1 f
X
τ
τ
and(
,
,
)
2
2 f
X
τ
τ
be two fine intuitionistic fuzzy topological space on X. Then 1f
τ
is contained in 2 f
τ
if 2 fτ
δ
∈
for each1
f
τ
δ
∈
. In this case we say that1
f
τ
δ
∈
is coarser than 2f
τ
.Definition 3.4: Let
(
X
,
τ
1,
τ
f1)
be a fine intuitionistic fuzzy topological space andδ
=
x
,
τ
(
λ
δ),
τ
(
γ
δ)
be afineintuitionistic fuzzy set in X and fineintuitionistic fuzzy interior and fine intuitionistic fuzzy closure of
δ
is defined byFIfcl(
δ
) =
{
λ
:
λ
is
an
FIfCS
in
X
and
δ
⊆
λ
}
FIfint(
δ
) =
{
µ
:
µ
is
an
FIfOS
in
X
and
µ
⊆
δ
}
.Remark 3.1:
a.
δ
is an FIfCS in X iffFIfcl(δ
).Proposition 3.1: Let
{
τ
fi:
i
∈
J
}
be a family of fine intuitionistic fuzzy topological space on X. Then
τ
fiisafineintuitionistic fuzzy topological space on X. Furthermore,
i f
τ
is the coarsest fine intuitionistic fuzzy topological space on X containing alli f
τ
Proof: It is simple by definition 3. 3.
Proposition 3.2: Let
(
X
,
τ
,
τ
f)
be an FIfTS, then for any FIfSθ
in(
X
,
τ
,
τ
f)
in we havea) FIfcl(
θ
)=FIfInt
(
θ
)
).b) FIfInt(
θ
) =FIfcl
(
θ
)
Proof: Let
=
x
,
(
(
x
)),
(
(
x
))
:
x
∈
X
α α
τ
γ
θ
θ
µ
τ
θ
. Then we getFIfInt(
θ
) =x
,
(
(
x
)),
(
(
x
))
i
i α
α
τ
γ
θ
θ
µ
τ
∧
∨
and hence,FIfInt
(
θ
)
=
,
(
),
(
)
α
α θ
θ
τ
µ
γ
τ
∨
∧
x
.Now
,
(
),
(
)
α
α θ
θ
τ
µ
γ
τ
θ
=
x
and,
Gi θ Gi θ
µ µ γ γ
τ
≤
τ τ
≥
τ
,FIfcl
(
θ
)
=
FIfInt
(
θ
)
. Similarly we prove b.Proposition 3.3: Let
(
X
,
τ
,
τ
f)
be an FIfTSA and B be FIfSs in X. Then the following properties are hold.a)
FIfint(A)
⊆
A
and
A
⊆
FIfcl(A).
b)
A
⊆
B
⇒
FIfInt
(
A
)
⊆
FIf
int
(
B
)
andA
⊆
B
⇒
FIfcl
(
A
)
⊆
FIfcl
(
B
).
c)
FIfInt
(
FIfInt
(
A
))
=
FI
fint(A)
andFIfcl
(
FIfcl
(
A
))
=
FIfcl
(
A
).
d)
FIfInt
(
A
B
)
=
FIf
int(
A
)
FIfInt
(
B
)
andFIfcl
(
A
B
)
=
FIfcl
(
A
)
FIfcl
(
B
).
e)
~ ~
)
1
1
(
X XFIfInt
=
,~ ~
)
0
0
(
X XFIfInt
=
.Proof:
a)
FIfint(A)
=
{
B
:
B
is
an
FIfOS
in
X
∋
B
⊆
A
}
SinceFint(A)
⊆
A
⊆
Fcl
(
A)
and by definition{
C
C
is
an
FIfCS
in
X
A
C
}
FIfcl(A)
=
:
∋
⊆
.Hence,
FIfint(A)
⊆
A
and
A
⊆
FIfcl(A).
FIfInt
(
A
B
)
⊆
FIfInt
(
B
)
Similarly for b) and c).
d)
FIfInt
(
A
B
)
⊆
FIfint
(
A
)
andFIfInt
(
A
B
)
⊆
FIfint
(
B
)
we get
FIfInt
(
A
B
)
⊆
FIfint
(
A
)
FIfint
(
B
)
---(1)A
FIfint(A)
⊆
andFIfint(B)
⊆
B
B
A
B
FIfint
A
FIfint
(
)
(
)
⊆
andFIfint
(
A
)
FIfint
(
B
)
∈
τ
f,Hence,
FIfint
(
A
)
FIfint
(
B
)
⊆
FIfint(
A
B
)
---(2) From (1) and (2) we getFIfInt
(
A
B
)
=
FIfint
(
A
)
FIfint
(
B
)
. Similarly forFIfcl
(
A
B
)
=
FIfcl
(
A
)
FIfcl
(
B
)
.Proof of e) is easy.
4. FINEINTUITIONISTIC FUZZY TOPOLOGICAL RING
Definition 4.1: Let R be a ring. A fineintuitionistic fuzzy set
A
=
x
,
τ
λA,
τ
νA in R is called afine intuitionisticfuzzy topological ringon R if it satisfies the following conditions
i.
(
x
y
)
(
x
)
(
y
)
A A
A µ µ
µ
τ
τ
τ
+
≥
∧
.ii.
(
xy
)
(
x
)
(
y
)
A A
A µ µ
µ
τ
τ
iii.
(
x
y
)
(
x
)
(
y
).
A AA γ γ
γ
τ
τ
τ
+
≤
∨
iv.
xy
x
y
x
y
R
A A
A
(
)
≤
γ(
)
∨
γ(
)
∀
,
∈
γ
τ
τ
τ
.Definition 4.2: Let
(
R
,
+
,
•
)
be a ring. A familyσ
of a fineintuitionistic fuzzy topological ring on R if it satisfies the following axioms.1.
0
X~,
1
X~∈
σ
f .2. If for any
G
1,
G
2∈
τ
f thenµ
1
µ
2∈
σ
f .3.
µ
i∈
σ
f for any arbitrary family{
G
i:
i
∈
J
}
⊆
σ
f.
The triplet
(
R
,
σ
,
σ
f)
is called a fine intuitionistic fuzzy topological ring (FIfTR)or fine intuitionistic fuzzy ringstructure space (FIfRSS)for short and any FIfTR in
σ
fis known as a fine intuitionistic fuzzy open ring (FIfOR) in)
,
,
(
R
σ
σ
f and its complement is a fine intuitionistic fuzzy closed ring(FIfCR).Example 4.1: Let
R
=
{
0
,
1
}
be a set of integers of modulo 2 with two binary operations as followsThen
(
R
,
+
,
•
)
is a ring.Define a fine intuitionistic fuzzy rings B and C as follows
29
.
0
)
1
(
,
49
.
0
)
0
(
65
.
0
)
1
(
,
35
.
0
)
0
(
=
=
=
=
B B
B
B µ
and
γ γµ
τ
τ
τ
τ
10
.
0
)
1
(
,
39
.
0
)
0
(
89
.
0
)
1
(
,
40
.
0
)
0
(
=
=
=
=
C C
C
C µ
and
γ γµ
τ
τ
τ
τ
Then
{
0
,
1
,
,
}
~ ~ X
B
C
X f
=
σ
is a fine intuitionistic fuzzy topological ring structure on R. Thus(
R
,
σ
,
σ
f)
is called a fine intuitionistic fuzzy ring structure space (FIfRSS).Notation: Let
(
R
,
σ
,
σ
f)
be anyfine intuitionistic fuzzy ring structure space. Then FIfO(R) denotes the family of allfine intuitionistic fuzzy open ring in
(
R
,
σ
,
σ
f)
. Then FIfC(R) denotes the family of all fine intuitionistic fuzzyclosed ring in
(
R
,
σ
,
σ
f)
.Definition 4.3: Let
(
R
,
σ
,
σ
f)
be a fine intuitionistic fuzzy ringstructure space andA A
x
A
=
,
τ
µ,
τ
γ be a fineintuitionistic fuzzy ring in
R
. A fine intuitionistic fuzzyring interior and fine intuitionistic fuzzy ring closure of A is defined byFIfRcl(
A
) ={
B
x
,
,
:
B
is
a
FIfC
(
R
)
in
X
and
B
A
}
BB
⊆
=
τ
µτ
γ
FIfRint(
A
) ={
B
x
,
,
:
B
is
a
FIfO
(
R
)
in
X
and
A
B
}
BB
⊆
=
τ
µτ
γ
.Corollary 4.1: Let
(
R
,
σ
,
σ
f)
be a fine intuitionistic fuzzy ring structure space. Then the following statements arehold for anyfine intuitionistic fuzzy ring structure in R.
i. FIfRint(
A
)=A iff A is afine intuitionistic fuzzy topological open ring. ii. FIfRcl(A
)=A iff A is a fine intuitionistic fuzzy topological closed ring. iii.FIfRint
(
A
)
⊆
A
⊆
FIfRcl
(
A
)
.iv.
~
~ X
X
1
FIfRint
(
)
=
1
and~
~ X
X
0
FIfRint
(
)
=
0
.v.
~
~ X
X
1
FIfRcl
(
)
=
1
and~
~ X
X
0
FIfRcl
(
)
=
0
.vi.
FIfRcl
(
A
)
=
FIfRint
(
A
)
FIfRint
(
A
)
=
FIfRcl
(
A
)
.vii.
(
)
(
i)
1 i i
1 i
A
FIfRcl
A
FIfRcl
∞
= ∞
=
⊆
.+ 0 1
0 0 1
1 1 0
•
0 10 0 0
viii.
(
)
(
i)
n1 i i
n
1 i
A
FIfRcl
A
FIfRcl
=
=
=
.ix.
(
)
(
i)
1 i i
1 i
A
FIfRcl
A
FIfRcl
∞
= ∞
=
⊆
x.
(
)
(
i).
1 i i
1 i
A
FIfRint
A
FIfRint
∞
= ∞
=
⊆
Proof: The Proof is simple
Definition 4.4: Let
(
R
,
σ
,
σ
f)
be a fine intuitionistic fuzzy ring structure space and ifA
=
x
,
τ
µA,
τ
γA is anyfineintuitionistic fuzzyring in
R
. Then FIfRint(A
) is called a fine intuitionistic fuzzy ring exterior of A is denoted by FIfRExt(A).Proposition 4.1: Let
(
R
,
σ
,
σ
f)
be a fineintuitionistic fuzzy ring structure space. Then the following statements arehold for any two fine intuitionistic fuzzy rings A and B.
i. FIfRExt(
A
)⊆
A
.ii.
FIfRExt
(
A
)
=
FIfRcl
(
A
)
.iii.
FIfRExt
(
FIfRExt
(
A
))
=
FIfint
(
FIfRcl
(
A
))
. iv. If A⊆
B thenFIfRExt
(
A
)
⊇
FIfRExt
(
B
)
.v.
~
~ X
X
1
FIfRExt
(
)
=
0
,~
~ X
X
0
FIfRExt
(
)
=
1
. vi.
FIfRExt
(
A
B
)
=
FIfRExt
(
A
)
FIfRExt
(
B
).
Proof It is easy by using above definition.
Definition 4.5: Let
(
R
,
σ
,
σ
f)
be any fine intuitionistic fuzzyring structure space. LetA A
x
A
=
,
τ
µ,
τ
γ be a fineintuitionistic fuzzy ring in R. Then A is said to be a fine intuitionistic fuzzy
G
δ ring in(
R
,
σ
,
σ
f)
if ii
A
A
=
∝=1 ,
where
A A
x
A
=
,
τ
µ,
τ
γ is a fine intuitionistic fuzzy open ring in(
R
,
σ
,
σ
f)
.The complement of a fine intuitionisticfuzzy
G
δ ring in(
R
,
σ
,
σ
f)
is a fine intuitionisticfuzzyF
σring in(
R
,
σ
,
σ
f)
.Definition 4.6: Let
(
R
,
σ
,
σ
f)
be any fine intuitionistic fuzzy ring structure space. LetA A
x
A
=
,
τ
µ,
τ
γ be a fineintuitionistic fuzzy ring in R. Then A is said to be a fine intuitionistic fuzzy dense ring in
(
R
,
σ
,
σ
f)
if there exists nofine intuitionistic fuzzy closed ring B in
(
R
,
σ
,
σ
f)
such thatA
⊂
B
⊂
1
X~and fine intuitionistic fuzzy nowheredense ring in
(
R
,
σ
,
σ
f)
if there exists no fine intuitionistic fuzzy open ring B in(
R
,
σ
,
σ
f)
such that)
(
A
FIfRcl
B
⊂
(i.e)FIfRint(FI
fRcl(A))
=
0
X~.Definition 4.7: Let
(
R
,
σ
,
σ
f)
be any fine intuitionistic fuzzy structure ring space. LetA
=
x
,
τ
µA,
τ
γA be a fineintuitionistic fuzzy ring in R. Then A is said to be a fine intuitionistic fuzzy first category ring in
(
R
,
σ
,
σ
f)
ifi
i
A
A
=
∝=1 where Ai’s are fine intuitionistic fuzzy nowhere dense rings in
(
R
,
σ
,
σ
f)
and its complement is a fineintuitionistic fuzzy residual ring in
(
R
,
σ
,
σ
f)
.Proposition 4.2: Let
(
R
,
σ
,
σ
f)
be a fine intuitionistic fuzzyring structure space. If A is a fine intuitionistic fuzzyδ
G
ring and the fine intuitionistic fuzzy ring exterior ofA
is a fine intuitionistic fuzzy dense ring in(
R
,
σ
,
σ
f)
thenProof: Since A is fine intuitionistic fuzzy
G
δ ring in(
R
,
σ
,
σ
f)
, ii
A
A
=
∝=1 where Ai’s are fine intuitionistic
fuzzy open rings. Since the fine intuitionistic fuzzy ring exterior of
A
is afine intuitionistic fuzzy dense ringin)
,
,
(
R
σ
σ
f ,FIfRcl(FIf
RExt(
A
))
=
1
X~.SinceFIfRExt(
A
)
⊆
A
⊆
FIfRcl(A),
FIfRExt(
A
)
⊆
FIfRcl(A)
from this we have
FIfRcl(FIf
RExt(
A
))
⊆
FIfRcl(A),
1
X~⊆
FIfRcl
(
A
).
Therefore,FIfRcl(A)
=
1
X~.
(i.e.) i X~
1
i
A
1
FIfRcl
A
FIfRcl
(
)
=
(
∝=)
=
.ButFIfRcl
A
FIfRcl(A
i)
1 i i
1
i
∝= ∝
=
)
⊆
(
.Hence,
(
i)
1 i ~
X
FIfRcl
A
1
⊆
∝= i X~1
i
FIfRcl(A
)
=
1
⇒
∝= . This implies thatFIfRcl(A
i)
=
1
X~Foreach
A
i∈
τ
f. ThusFIfRcl(FIf
Rint(
A
))
=
1
X~.Consider,
FIfRint(FI
fRcl(
A
i))
=
FIfRint
(FIfRint(A
i)
)
=
FIfRcl
(
FIfRint(A
i))
=
0
X~.Therefore,
A
i is the fine intuitionistic fuzzy nowhere dense ring in(
R
,
σ
,
σ
f)
. Now,i 1 i i 1
i
A
A
A
=
∝==
∝= . Hence, i1
i
A
A
=
∝= whereA
i ’s are the fineintuitionistic fuzzynowhere dense rings in
(
R
,
σ
,
σ
f)
. Therefore,A
is a fine intuitionistic fuzzy first category ring in(
R
,
σ
,
σ
f)
.Proposition 4.3: If A is a fine intuitionistic fuzzy first category ring in a fine intuitionistic fuzzy ring structure space
)
,
,
(
R
σ
σ
f such thatB
⊆
A
where B is non-zero fine intuitionistic fuzzyG
δ ring and the fine intuitionistic fuzzyring exterior of
B
is a fine intuitionistic fuzzy dense ring in(
R
,
σ
,
σ
f)
then A is a fine intuitionistic fuzzy nowheredense ring in
(
R
,
σ
,
σ
f)
.Proof: Let A be a fine intuitionistic fuzzy first category ring in
(
R
,
σ
,
σ
f)
. Then, ii
A
A
=
∝=1
where
A
i’s are the fine intuitionistic fuzzy nowhere dense rings in(
R
,
σ
,
σ
f)
. Now,)
(
A
iFIfRcl
is a fine intuitionistic fuzzy open ring in(
R
,
σ
,
σ
f)
. LetB
i 1FIfRcl
(
A
i)
∝ =
=
.Then, B is non-zero fine intuitionistic fuzzy
G
δ ring in(
R
,
σ
,
σ
f)
.Consider
B
FIfRcl
A
FIfRcl
A
A
iA
i i i
i
i
=
⊆
=
=
∝=
∝=
∝=1 1
1
(
)
(
)
Hence,
B
⊆
A
. Then,A
⊆
B
.Let
FIfRint(
FIfRcl
(
A
)
⊆
FIfRint(
FIfRcl
(
(
B
)
)
)
)
(
(
FIfR
int
B
FIfRint
=
)
)
(
B
int
FIfR
FIfRcl(
=
)
)
(
B
Ext
FIfR
FIfRcl(
=
We know that
FIfR
Ext
(
B
)
is a fine intuitionistic fuzzy dense ring in(
R
,
σ
,
σ
f)
,FIfRcl
(
FIfR
Ext
(
B
))
=
1
X~.
Therefore
FIfint(
FIfR
cl
(
A
))
⊆
0
X~. ThenFIfint(
FIfR
cl
(
A
))
=
0
X~. Hence, A is a fine intuitionistic fuzzynowhere dense ring in
(
R
,
σ
,
σ
f)
.Definition 4.8: Let
(
R
,
σ
,
σ
f)
be any fine intuitionistic fuzzy ring structure space. Let A be a fine intuitionisticif
FIfR
cl(FIfint
(
A
))
=
A
.
The complement of a fine intuitionistic fuzzy regular closed ring in(
R
,
σ
,
σ
f)
is a fineintuitionistic fuzzy regular open ring in
(
R
,
σ
,
σ
f)
.Remark 4.1: Every fine intuitionistic fuzzy regular closed ring is a fine intuitionistic fuzzy closed ring.
Definition 4.9: Let
(
R
,
σ
,
σ
f)
be any fine intuitionistic fuzzy ring structure space. Then(
R
,
σ
,
σ
f)
is called a fineintuitionistic fuzzy ring
G T
δ 1/ 2space if every nonzero fine intuitionistic fuzzyG
δ ring in(
R
,
σ
,
σ
f)
is a fineintuitionistic fuzzy open ring in
(
R
,
σ
,
σ
f)
.Proposition 4.4: If the fine intuitionistic fuzzy ring structure space
(
R
,
σ
,
σ
f)
is a fine intuitionistic fuzzy ring2 / 1
T
G
δ space and if A is a fine intuitionistic fuzzy first category ring in(
R
,
σ
,
σ
f)
, then A is not a fine intuitionisticfuzzy dense ring in
(
R
,
σ
,
σ
f)
.Proof: Suppose that A is a fine intuitionistic fuzzy first categoryring in
(
R
,
σ
,
σ
f)
such that A is a fine intuitionisticfuzzy dense ring in
(
R
,
σ
,
σ
f)
,that is,FIfRcl
(
A
)
=
1
X~.Then,FIfRcl FIfRExt B( ( )) 1 ,= X~ A=i∝=1Ai whereA
i’s arefineintuitionistic fuzzy nowhere dense rings in
(
R
,
σ
,
σ
f)
.Now,
FIfRcl
(
A
i)
is a fine intuitionistic fuzzy open ring in(
R
,
σ
,
σ
f)
.LetB
11FIfRcl
(
A
i)
∝ =
=
.Then, B isnon-zero, fine intuitionistic fuzzy
G
δ ring in(
R
,
σ
,
σ
f)
.Consider
B
FIfRcl
A
B
FIfRcl
A
A
iA
i i i
i
i
=
=
⊆
=
=
∝ ∝= ∝==
1(
)
1(
)
1 . HenceB
⊆
A
. Then~
0
)
(
)
int(
)
int(
B
FIfR
A
FIfRcl
A
XFIfR
⊆
⊆
=
. That is,FIfR
int(
B
)
=
0
X~.Since(
R
,
σ
,
σ
f)
is a fineintuitionistic fuzzy ring
G
δT
1/2 space,FIfR
int(
B
)
=
B
, which implies thatB
=
0
X~. This is a contradiction. Therefore, A is not a fine intuitionistic fuzzy dense ring in(
R
,
σ
,
σ
f)
.5. FINEINTUITIONISTIC FUZZY NORMALSUBRING
Definition 5.1: A fine intuitionistic fuzzy subset
δ
of a ring( , ,.)
R
+
is said to be a fineintuitionistic fuzzy subringofa ring
( , ,.)
R
+
is denoted by (FIfSR) if it satisfies the following conditions a.τ
µδ(
x
−
y
)
≥
Min
{
τ
µδ( ),
x
τ
µδ( )}
y
b.
τ
µδ(
xy
)
≥
Min
{
τ
µδ( ),
x
τ
µδ( )}
y
c.
(
x
y
)
Max
{
( ),
x
( )}
y
δ δ δ
γ γ γ
τ
−
≤
τ
τ
d.
(
xy
)
Max
{
( ),
x
( )}
y
x y
,
R
.
δ δ δ
γ γ γ
τ
≤
τ
τ
∀
∈
Definition 5.2: Let
( , , .)
R
+
be a ring. A fine intuitionistic fuzzy subsetδ
of R is said to be a fine intuitionistic fuzzy normal subring of R is denoted by FIfNSR if it satisfiesi.
τ
µδ(
xy
)
=
τ
µδ(
yx
)
ii.
τ
γδ(
xy
)
=
τ
γδ(
yx
)
∀
x y
,
∈
R
.
Let
δ
andθ
be two fine intuitionistic fuzzy subsets of the fine intuitionistic fuzzy rings with an identity R1 and R2 andδ θ
×
is a fine intuitionistic fuzzy subring of R1×
R2.thefollowingholdsi.
Ifτ
µδ( )
x
≤
τ
µθ( )
e
1 andτ
γδ( )
x
≥
τ
γθ( )
e
1 thenδ
is a fine intuitionistic fuzzy subring of R1.ii.
Ifτ
µθ( )
x
≤
τ
µδ( )
e
andτ
γθ( )
x
≥
τ
γδ( )
e
thenθ
is a fine intuitionistic fuzzy subring of R2.iii.
Eitherδ
is a fine intuitionistic fuzzy subring of R1(or) fine intuitionistic fuzzy subring of R2.Definition 5.4: Let A and B be two fine intuitionistic fuzzy subring of rings of rings R1 and R2respectively. The product of A and B denoted by
A B
×
is defined as1 2
{ ( , ),
( , ),
(
,
) :
for all
and
}
A B A B
x
A B
× =
x y
τ
µ×x y
τ
γ ×x y
∈
R
y
∈
R
where( , )
Min{
( ),
( )}
A B
x y
Ax
By
µ µ µ
τ
×=
τ
τ
and( , )
Max{
( ),
( )}
A B
x y
Ax
By
γ γ γ
τ
×=
τ
τ
.
Theorem 5.1: Let
( , ,.)
R
+
be a ring. If Aand B are two fine intuitionistic fuzzy normal subrings of R. Then theirintersection A
B is a fine intuitionistic fuzzy normal subring of R.The intersection A
Bof any two fine intuitionistic fuzzy normal subrings of a ring( , ,.)
R
+
is afine intuitionistic fuzzy normal subring of R.Proof: Let x, y
∈
R. Let{ ,
( ),
( ) :
}
A A
A
=
x
τ
µx
τ
γx
x
∈
R
andB
=
{ ,
x
τ
µB( ),
x
τ
γB( ) :
x
x
∈
R
}
be a fineintuitionistic fuzzy subring of a ring R. Let C = A
B and{ ,
( ),
( ) :
}
C C
C
=
x
τ
µx
τ
γx
x
∈
R
where min{
( ),
( )}
( )
A
x
Bx
Cx
µ µ µ
τ
τ
=
τ
and max{
( ),
( )}
( )
A
x
Bx
Cx
γ γ γ
τ
τ
=
τ
. Hence, C is a fine intuitionistic fuzzy subring of a ring R.Since A and B are two intuitionistic fuzzy subrings of a ring R.
(
)
C
xy
µ
τ
=
min{(
),
(
)
A
x y
Bx y
µ µ
τ
τ
} = min{(
),
(
)
A
yx
Byx
µ µ
τ
τ
} =(
)
C
yx
µ
τ
.(
)
C
xy
µ
τ
=
(
)
C
yx
µ
τ
∀
x y
,
∈
R
,also
(
)
C
xy
γ
τ
=
max{(
),
(
)
A
xy
Bxy
γ γ
τ
τ
} = max{(
),
(
)
A
yx
Byx
γ γ
τ
τ
} =(
)
C
yx
γ
τ
.(
)
C
xy
γ
τ
=
(
)
C
yx
γ
τ
∀
x y
,
∈
R
.Hence, intersection of any two fine intuitionistic fuzzy normal subring is a fine intuitionistic fuzzy normal subring of a ring R.
Theorem 5.2: Let A and B be fine intuitionistic fuzzy subsets of the rings with an identity R1 and R2 and A
×
B is a fine intuitionistic fuzzy normal subring of R1×
R2the following holdsi. If
( )
( ')
A
x
Be
µ µ
τ
≤
τ
and( )
( ')
A
x
Be
γ γ
τ
≥
τ
then A is a fine intuitionistic fuzzy normal subring of R1. ii. Ifτ
µB( )
x
≤
τ
µA( )
e
andτ
γB( )
x
≥
τ
γA( )
e
then B is a fine intuitionistic fuzzy normal subring of R2.iii. either A is afine intuitionistic fuzzy normal subring of R1 or B is afine intuitionistic fuzzy normal subring of R2.
Proof: Let A
×
B be a fine intuitionistic fuzzy normal subring of R1×
R2 andx, y in R1 ande
'
∈
R2. Then( , ')
x e
and( , ')
y e
are in R1×
R2and Clearly, A×
B is a fine intuitionistic fuzzy subring of R1.(
)
A
xy
µ
τ
=
min{(
),
( ' ')
A
xy
Be e
µ µ
τ
τ
} =((
), ( ' '))
A B
xy
e e
µ
τ
× =(( , '), ( , '))
A B
x e
y e
µ
τ
× =(( , '), ( , '))
A B
y e
x e
µ
τ
× = min{(
),
( ' ')
A
yx
Be e
µ µ
τ
τ
} =(
)
A
yx
µ
τ
. Thus,(
)
A
xy
µ
τ
=
(
)
A
yx
µ
τ
∀
x y
,
∈
R
.(
)
A
xy
γ
τ
=
max{(
),
( ' ')
A
xy
Be e
γ γ
τ
τ
} =((
), ( ' '))
A B
xy
e e
γ
τ
× =(( , '), ( , '))
A B
x e
y e
γ
τ
× =(( , '), ( , '))
A B
y e
x e
γ
τ
× =((
), ( ' '))
A B
yx
e e
γ
τ
× = max{(
),
( ' ')
A
yx
Be e
γ γ
τ
τ
} =(
)
A
yx
γ
τ
.Thus,
(
)
A
xy
γ
τ
=
(
)
A
yx
γ
τ
∀
x y
,
∈
R
. Hence, A is a fine intuitionistic fuzzy subring of R1.( )
B
x
µ
τ
≤
( )
A
e
µ
τ
,( )
B
x
γ
τ
≥
τ
γA( )
e
,∀ ∈
x
R
2 and letx y
,
∈
R
2and e∈
R1.Then( , )
e x
and( , )
e y
are in R1×
R2. Thus B is a fine intuitionistic fuzzy subring of R2.(
)
B
xy
µ
τ
=
min{(
),
( )
B
xy
Aee
µ µ
τ
τ
} = min{( ),
(
)
A
ee
Bxy
µ µ
τ
τ
}=(( ), (
))
A B
ee
xy
µ
τ
×=
(( , ), ( , ))
A B
e x
e y
µ
τ
× =(( , ), ( , ))
A B
e y
e x
µ
τ
× =(( ), (
))
A B
ee
yx
µ
τ
× =min{(( ),
(
))
A
ee
Byx
µ µ
τ
τ
} =(
)
B
yx
µ
τ
.(
xy
)
µ
