In this paper, we shall introduce the concept of rough antifuzzy subring and prove some theorems in this context. We have, if
µ
is an antifuzzy subring, then
µ
is a rough antifuzzy subring. Also we give some properties of homomorphism and antihomomorphism on rough antifuzzy subring.
AMS Mathematics Subject Classification : 03E72, 08A72, 06E20.
1. Introduction
The fuzzy set introduced by L.A. Zadeh in 1965 and the rough set introduced by Z. Pawlak in 1982 are generalisations of the classical set theory. Both these set theories are new mathematical tool to deal the uncertain, vague and imprecise data. In Zadeh’s fuzzy set theory, the degree of membership of elements of a set plays the key role, whereas in Pawlak’s rough set theory, equivalence classes of a set are the building blocks for the upper and lower approximations of the set, in which a subset of universe is approximated by the pair of ordinary sets, called upper and lower approximations. Combining the theory of rough set with abstract algebra is one of the trends in the theory of rough set. Some authors studied the concept of rough algebraic structures. On the other hand, some authors substituted an algebraic structure for the universal set and studied the roughness in algebraic structure. Biswas and Nanda introduced the notion of rough subgroups. The concept of rough ideal in a semigroup was introduced by Kuroki. And then B. Davvaz studied relationship between rough sets and ring theory and considered ring as a universal set and introduced the notion of rough ideals of a rings in
[4]
. A further study of this work is done by Osman Kazanci and B Davaaz in
[8]
. Extensive researches has also been carried out to compare the theory of rough sets with other theories of uncertainty such as fuzzy sets and conditional events. There have been many papers studying the connections and differences of fuzzy set theory and rough set theory. Dubois and Prade were one of the first who combined fuzzy sets and rough sets in a fruitful way by defining rough fuzzy sets and fuzzy rough sets.
This paper deals with a relationship between rough sets, fuzzy sets and ring theory. In section 2, we review some basic definitions. Section 3 deals with some properties of rough antifuzzy subring. In section 4,we give some homomorphic properties of rough antifuzzy subring. Section 5 deals with antihomomorphic properties of rough antifuzzy subring.
2. Preliminaries
Definition 2.1.
Let
θ
be an equivalence relation on
R
, then
θ
is called a full congruence relation if
(
a
,
b
) ∈
θ
implies(
a
+
x
,
b
+
x
), (
ax
,
bx
), (
xa
,
xb
) ∈
θ
for all
x
∈
R
.
A full congruence relation
θ
on
R
is called complete if [
ab
]
_{θ}
= [
a
]
_{θ}
[
b
]
_{θ}
.
Definition 2.2.
Let
θ
be a full congruence relation on
R
and
A
a subset of
R
.
Then the sets
are called, respectively , the
θ
 lower and
θ
 upper approximations of the set
A. θ
(
A
) = (
θ
_{−}
(
A
) ,
θ
^{−}
(
A
)) is called a rough set with respect to
θ
if
θ
_{−}
(
A
)≠
θ
^{−}
(
A
)
Definition 2.3
(
[9]
). Let
X
and
Y
be two nonempty sets,
f
:
X
→
Y
,
µ
and
σ
be fuzzy subsets of
X
and
Y
respectively. Then
f
(
µ
), the image of
µ
under
f
is a fuzzy subset of
Y
defined by
f
^{−1}
(
σ
), the preimage of
σ
under
f
is a fuzzy subset of
X
defined by
Definition 2.4
(
[9]
). For a function
f
:
R
_{1}
→
R
_{2}
, a fuzzy subset
µ
of a ring
R
_{1}
is called finvariant if
f
(
x
) =
f
(
y
) implies
µ
(
x
) =
µ
(
y
),
x
,
y
∈
R
_{1}
.
We say that a fuzzy subset
µ
of a ring
R
_{1}
has the sup property if for any subset
T
of
R
_{1}
, there exists
t
_{0}
∈
T
such that
µ
(
t
_{0}
) = sup
_{t∈T}
µ
(
t
).
Definition 2.5.
A fuzzy subset
µ
of a ring
R
is called upper rough finvariant if
θ
^{−}
(
µ
) is finvariant and a lower rough finvariant if
θ
_{−}
(
µ
) is a finvariant.
Let
µ
be a fuzzy subset of
R
and
θ
(
µ
) = (
θ
_{−}
(
µ
) ,
θ
^{−}
(
µ
)) a rough fuzzy set. If
θ
_{−}
(
µ
) and
θ
^{−}
(
µ
) are finvariant, then (
θ
_{−}
(
µ
) ,
θ
^{−}
(
µ
)) is called rough finvariant.
3. Rough AntiFuzzy Subring
As it is well known in the fuzzy set theory established by Zadeh, a fuzzy subset
µ
of a set
R
is defined as a map from
R
to the unit interval [0, 1].
Definition 3.1
(
[8]
). Let
θ
be an equivalence relation on
R
and
µ
a fuzzy subset of
R
. Then we define the fuzzy sets
θ
_{−}
(
µ
) ,
θ
^{−}
(
µ
) as follows:
The fuzzy sets
θ
_{−}
(
µ
) and
θ
^{−}
(
µ
) are called , respectively the
θ
 lower and
θ
upper approximations of the fuzzy set
µ
.
θ
(
µ
) = (
θ
_{−}
(
µ
) ,
θ
^{−}
(
µ
)) is called a rough fuzzy set with respect to
θ
if
θ
_{−}
(
µ
) ≠
θ
^{−}
(
µ
).
Definition 3.2.
A fuzzy subset
µ
of a ring
R
is called a fuzzy subring of
R
if
for all
x
,
y
∈
R
.
Definition 3.3.
A fuzzy subset
µ
of a ring
R
is called an antifuzzy subring of
R
if
for all
x
,
y
∈
R
.
Definition 3.4.
A fuzzy subset
µ
of a ring
R
is called an upper rough fuzzy subring of
R
if
θ
^{−}
(
µ
) is a fuzzy subring of
R
and a lower rough fuzzy subring of
R
if
θ
_{−}
(
µ
) is a fuzzy subring of
R
.
Let
µ
be a fuzzy subset of
R
and
θ
(
µ
) = (
θ
_{−}
(
µ
) ,
θ
^{−}
(
µ
)) a rough fuzzy set. If
θ
_{−}
(
µ
) and
θ
^{−}
(
µ
) are fuzzy subrings of
R
, then (
θ
_{−}
(
µ
) ,
θ
^{−}
(
µ
)) is called a rough fuzzy subring.
Definition 3.5.
A fuzzy subset
µ
of a ring
R
is called an upper rough antifuzzy subring of
R
if
θ
^{−}
(
µ
) is an antifuzzy subring of
R
and a lower rough antifuzzy subring of
R
if
θ
_{−}
(
µ
) is an antifuzzy subring of
R
.
Let
µ
be a fuzzy subset of
R
and
θ
(
µ
) = (
θ
_{−}
(
µ
) ,
θ
^{−}
(
µ
)) a rough fuzzy set. If
θ
_{−}
(
µ
) and
θ
^{−}
(
µ
) are antifuzzy subrings of
R
, then (
θ
_{−}
(
µ
) ,
θ
^{−}
(
µ
)) is called a rough antifuzzy subring.
Theorem 3.6.
Let θ be a complete congruence relation on R. If µ is an antifuzzy subring of R, then θ
^{−}
(
µ
)
is an antifuzzy subring of R
.
Proof
. For
x
,
y
∈
R
,
Hence
θ
^{−}
(
µ
)(
x − y
) ≤
θ
^{−}
(
µ
)(
x
) ∨
θ
^{−}
(
µ
)(
y
). Also we have,
Hence
θ
^{−}
(
µ
)(
xy
) ≤
θ
^{−}
(
µ
)(
x
) ∨
θ
^{−}
(
µ
)(
y
). Therefore,
θ
^{−}
(
µ
) is an antifuzzy subring of
R
.
Theorem 3.7.
Let θ be a complete congruence relation on R. If µ is an antifuzzy subring of R, then θ_{−}(µ) is an antifuzzy subring of R
.
Proof
. For
x,y
∈
R
,
Hence
θ
_{−}
(
µ
)(
x
−
y
) ≤
θ
_{−}
(
µ
)(
x
) ∨
θ
_{−}
(
µ
)(
y
). Also we have,
Hence
θ
_{−}
(
µ
)(
xy
) ≤
θ
_{−}
(
µ
)(
x
) ∨
θ
_{−}
(
µ
)(
y
). Therefore,
θ
_{−}
(
µ
) is an antifuzzy subring of
R
.
Corollary 3.8.
Let µ be an antifuzzy subring of R, then µ is a rough antifuzzy subring of R
.
Proof
. This follows from Theorems 3.6 and 3.7.
Remark 3.1.
From here onwards
θ
,
θ
_{1}
and
θ
_{2}
denote full congruence relations on the rings
R
,
R
_{1}
and
R
_{2}
respectively.
Definition 3.9.
Let
µ
be a fuzzy subset of
R
. Then the sets
µ_{t}
= {
x
∈
R

µ
(
x
) ≤
t
},
= {
x
∈
R

µ
(
x
) <
t
}, where
t
∈ [0, 1] are called respectively,
t
lower level subset and tstrong lower level subset of
µ
.
Theorem 3.10.
Let µ be a fuzzy subset of R and t
∈[0, 1],
then
Proof
. (1) We have
(2) Also we have
Theorem 3.11.
Let µ be a fuzzy subset of R. Then µ is an antifuzzy subring if and only if µ_{t} and
are, if they are nonempty, subrings of R for every t
∈ [0, 1].
Proof.
Suppose
µ
is an antifuzzy subring of
R
. Let
x, y
∈
µ_{t}
. Then
µ
(
x
) ≤
t
and
µ
(
y
) ≤
t
.
Since
µ
is an antifuzzy subring, we have
µ
(
x
−
y
) ≤
µ
(
x
) ∨
µ
(
y
) ≤
t
. Therefore
x
−
y
∈
µ_{t}
. Also since
µ
(
xy
) ≤
µ
(
x
) ∨
µ
(
y
),
µ
(
xy
) ≤
t
. Therefore
xy
∈
µ_{t}
. Hence
µ_{t}
is a subring of
R
. Similarly we can show that
is also a subring of
R
.
Conversely, let
µ_{t}
be a subring of
R
. Let
x,y
∈
R
. Assume
µ
(
x
) ≤
µ
(
y
) and
µ
(
y
) =
t
. Then
x,y
∈
µ_{t}
. Since
µ_{t}
is a subring,
x
−
y
∈
µ_{t}
. Hence
µ
(
x
−
y
) ≤
t
=
µ
(
y
) =
µ
(
x
) ∨
µ
(
y
). Thus
µ
(
x
−
y
) ≤
µ
(
x
) ∨
µ
(
y
). Again since
xy
∈
µ_{t}
,
µ
(
xy
) ≤
t
=
µ
(
y
) =
µ
(
x
) ∨
µ
(
y
). Hence
µ
(
xy
) ≤
µ
(
x
) ∨
µ
(
y
). Therefore
µ
is an antifuzzy subring of
R
.
4. Homomorphism on Rough AntiFuzzy Subring
Definition 4.1.
Let
R
and
R
^{′}
be any two rings. Then the function
f
:
R
→
R
^{′}
is said to be a homomorphism if for all
x,y
∈
R
Theorem 4.2
(
[8]
).
Let f be a homomorphism of a ring R
_{1}
onto a ring R
_{2}
and let A be a subset of R
_{1}
.
Then

(1)θ1= {(a, b)(f(a),f(b)) ∈θ2}is a full congruence relation on R1.

(2)f(θ1−(A)) =θ2(f(A))

(3)f(θ1−(A) ⊆θ2−(f(A)).If f is one to one, then f(θ1−(A)) =θ2−(f(A))
Theorem 4.3
(
[3]
).
Let f be a homomorphism from ring R
_{1}
onto a ring R
_{2}
and let µ be a fuzzy subset of R
_{1}
.
Then
(1)
f
(
θ
_{1}
^{}
(
µ
)) =
θ
_{2}
^{}
(
f
(
µ
))
(2)
f
(
θ
_{1−}
(
µ
)) ⊆
θ
_{2−}
(
f
(
µ
)).
If f is one to one, then f
(
θ
_{1−}
(
µ
)) =
θ
_{2−}
(
f
(
µ
))
Remark 4.1.
Let
f
be a homomorphism from ring
R
_{1}
to a ring
R
_{2}
and
µ
be a fuzzy subset of
R
_{1}
. Let
y
∈
⇐⇒
f
(
µ
)(
y
) <
t
⇐⇒
sup _{f}
_{(}
_{x}
_{)=}
_{y}
µ
(
x
) <
t
⇐⇒
µ
(
x
) <
t
∀
x
such that
f
(
x
) =
y
⇐⇒
x
∈
for
f
(
x
) =
y
⇐⇒
y
∈
f
(
). Then
f
(
) =
Remark 4.2.
Let
f
be a homomorphism (antihomomorphism) from ring
R
_{1}
onto a ring
R
_{2}
, and let
σ
be a fuzzy subset of
R
_{2}
. Then
f
^{−1}
(
σ
) is a fuzzy subset of
R
_{1}
. Hence by theorem 4.3, we get
f
(
θ
_{1}
^{−}
(
f
^{−1}
(
σ
)) =
θ
_{2}
^{−}
(
f
(
f
^{−1}
(
σ
))). Further if
f
is one to one and onto,
θ
_{1}
^{−}
(
f
^{−1}
(
^{σ}
)) =
f
^{−1}
(
θ
_{2}
^{−}
(
σ
)).
Theorem 4.4.
Let f be an isomorphism from ring R
_{1}
onto ring R
_{2}
and let µ be a fuzzy subset of R
_{1}
.
Then
(1)
θ
_{1}
^{−}
(
µ
)
is an antifuzzy subring of R
_{1}
if and only if θ
_{2}
^{−}
(
f
(
µ
))
is an antifuzzy subring of R
_{2}
.
(2)
θ
_{1}
^{−}
(
µ
)
is an antifuzzy subring of R
_{1}
if and only if θ
_{2−}
(
f
(
µ
))
is an antifuzzy subring of R
_{2}
.
Proof
. (1) By theorem 3.11,
θ
_{1}
^{−}
(
µ
) is an antifuzzy subring of
R
_{1}
if and only if
is, if it is nonempty, a subring of
R
_{1}
for every
t
∈ [0, 1]. Again by theorem 3.10, we have
=
θ
_{1−}
(
). We know that
θ
_{1−}
(
) is a subring of
R
_{1}
if and only if
f
(
θ
_{1−}
(
) is a subring of
R
_{2}
. Now by theorem 4.2,
f
(
θ
_{1−}
(
) =
θ
_{2−}
f
(
). By remark 4.1,
f
(
) =
. From this and theorem 3.10, we have,
θ
_{2−}
f
(
) =
θ
_{2−}
(
f
(
) =
. By theorem 3.11, we obtain
is a subring of
R
_{2}
for every
t
∈ [0, 1] if and only if
θ
_{2}
^{−}
(
f
(
µ
))is an antifuzzy subring of
R
_{2}
.
The proof of (2) is similar to the proof of (1).
Theorem 4.5.
Let f be a homomorphism from a ring R
_{1}
onto a ring R
_{2}
and let σ be an upper rough antifuzzy subring of R
_{2}
.
Then f
^{−1}
(
σ
)
is an upper rough antifuzzy subring of R
_{1}
.
Proof
. Let
σ
be an upper rough antifuzzy subring of
R
_{2}
. Then
θ
_{2}
^{−}
(
σ
) is an antifuzzy subring of
R
_{2}
. For
x,y
∈
R
_{1}
,
Therefore,
f
^{−1}
(
θ
_{2}
^{−}
(
σ
))(
x
−
y
) ≤
f
^{−1}
(
θ
_{2}
^{−}
(
σ
))(
x
) ∨
f
^{−1}
(
θ
_{2}
^{−}
(
σ
))(
y
). Also
Therefore
f
^{−1}
(
θ
_{2}
^{−}
(
σ
))(
xy
) ≤
f
^{−1}
(
θ
_{2}
^{−}
(
σ
))(
x
) ∨
f
^{−1}
(
θ
_{2}
^{−}
(
σ
))(
y
). Thus
f
^{−1}
(
θ
_{2}
^{−}
(
σ
)) is an antifuzzy subring of
R
_{1}
. By Remark 4.2,
f
^{−1}
(
θ
_{2}
^{−}
(
σ
)) =
θ
_{1}
^{−}
(
f
^{−1}
(
σ
)) is an antifuzzy subring of
R
_{1}
. Therefore,
f
^{−1}
(
σ
) is an upper rough antifuzzy subring of
R
_{1}
. Hence the theorem is proved.
Theorem 4.6.
Let f be an isomorphism from a ring R
_{1}
onto a ring R
_{2}
and let σ be a lower rough antifuzzy subring of R
_{2}
.
Then f
^{−1}
(
σ
)
is a lower rough antifuzzy subring of R
_{1}
.
Proof
. The proof is similar to that of theorem 4.5.
Corollary 4.7.
Isomorphic preimage of a rough antifuzzy subring is a rough antifuzzy subring
.
Proof
. This follows from Theorems 4.5 and 4.6.
Theorem 4.8.
Let f be a homomorphism from a ring R
_{1}
onto a ring R
_{2}
and let µ be an upper rough finvariant antifuzzy subring of R
_{1}
with sup property
.
Then f
(
µ
)
is an upper rough antifuzzy subring of R
_{2}
.
Proof
. Let
µ
be an upper rough antifuzzy subring of
R
_{1}
. Then
θ
_{1}
^{−}
(
µ
) is an antifuzzy subring of
R
_{1}
. Let
x
_{0}
∈
f
^{−1}
[
f
(
x
)] and
y
_{0}
∈
f
^{−1}
[
f
(
y
)] be such that
For
f
(
x
),
f
(
y
) ∈
R
_{2}
,
Therefore,
f
(
θ
_{1}
^{−}
(
µ
))(
f
(
x
) −
f
(
y
)) ≤
f
(
θ
_{1}
^{−}
(
µ
))
f
(
x
) ∨
f
(
θ
_{1}
^{−}
(
µ
))
f
(
y
). Also
Hence,
f
(
θ
_{1}
^{−}
(
µ
))(
f
(
x
)
f
(
y
)) ≤
f
(
θ
_{1}
^{−}
(
µ
))
f
(
x
) ∨
f
(
θ
_{1}
^{−}
(
µ
))
f
(
y
).
Therefore,
f
(
θ
_{1}
^{−}
(
µ
)) is an antifuzzy subring of
R
_{2}
. By theorem 4.3,
f
(
θ
_{1}
^{−}
(
µ
))=
θ
_{2}
^{−}
(
f
(
µ
)) is an antifuzzy subring of
R
_{2}
. Hence
f
(
µ
) is an upper rough antifuzzy subring of
R
_{2}
. This proves the theorem.
Theorem 4.9.
Let f be an isomorphism from a ring R
_{1}
onto a ring R
_{2}
and let µ be a lower rough finvariant antifuzzy subring of R
_{1}
with sup property. Then f
(
µ
)
is a lower rough antifuzzy subring of R
_{2}
.
Proof
. Let
µ
be a lower rough antifuzzy subring of
R
_{1}
. Then
θ
_{1−}
(
µ
) is an antifuzzy subring of
R
_{1}
. For
f
(
x
),
f
(
y
) ∈
R
_{2}
,
Therefore,
f
(
θ
_{1−}
(
µ
))(
f
(
x
) −
f
(
y
)) ≤
f
(
θ
_{1−}
(
µ
))
f
(
x
) ∨
f
(
θ
_{1−}
(
µ
))
f
(
y
). Also
Hence,
f
(
θ
_{1−}
(
µ
))(
f
(
x
)
f
(
y
)) ≤
f
(
θ
_{1−}
(
µ
))
f
(
x
) ∨
f
(
θ
_{1−}
(
µ
))
f
(
y
).
Therefore,
f
(
θ
_{1−}
(
µ
)) is an antifuzzy subring of
R
_{2}
. By theorem 4.3,
f
(
θ
_{1−}
(
µ
))=
θ
_{2−}
(
f
(
µ
)) is an antifuzzy subring of
R
_{2}
. Hence
f
(
µ
) is a lower rough antifuzzy subring of
R
_{2}
. This proves the theorem.
Corollary 4.10.
Let f be an isomorphism from a ring R
_{1}
onto a ring R
_{2}
and let µ be a rough finvariant antifuzzy subring of R
_{1}
with sup property. Then f
(
µ
)
is a rough antifuzzy subring of R
_{2}
.
Proof
. This follows from theorems 4.8 and 4.9.
5. Antihomomorphism on Rough AntiFuzzy Subring
Definition 5.1.
Let
R
and
R′
be any two rings. Then the function
f
:
R
→
R′
is said to be an anti homomorphism if for all
x
,
y
∈
R
Theorem 5.2
(
[3]
).
Let f be an antihomomorphism from a ring R
_{1}
onto a ring R
_{2}
and let µ be a fuzzy subset of R
_{1}
.
Then
(1)
f
(
θ
_{1}
^{−}
(
µ
)) =
θ
_{2}
^{−}
(
f
(
µ
))
(2)
f
(
θ
_{1−}
(
µ
)) ⊆
θ
_{2−}
(
f
(
µ
)).
If f is one to one f
(
θ
_{1−}
(
µ
)) =
θ
_{2−}
(
f
(
µ
)).
The following theorems in antihomomorphisms can be proved in similar way as the corresponding theorems in homomorphism.
Theorem 5.3.
Anti homomorphic image of an upper rough finvariant antifuzzy subring with sup property is an upper rough antifuzzy subring.
Theorem 5.4.
Anti isomorphic image of a lower rough finvariant antifuzzy subring with sup property is a lower rough antifuzzy subring.
Corollary 5.5.
Anti isomorphic image of a rough finvariant antifuzzy subring with sup property is a rough antifuzzy subring.
Theorem 5.6.
Anti homomorphic preimage of an upper rough antifuzzy subring is an upper rough antifuzzy subring.
Theorem 5.7.
Anti isomorphic preimage of a lower rough antifuzzy subring is a lower rough antifuzzy subring.
Theorem 5.8.
Anti isomorphic preimage of a rough antifuzzy subring is a rough antifuzzy subring.
6. Conclusion
In this paper, we have shown that the theory of rough sets can be extended to rings. We discussed the concept of rough antifuzzy subring. Also, we discussed homomorphic and antihomomorphic properties of rough antifuzzy subrings. In a similar fashion the theory of rough sets can be extended to other topics in ring theory.
BIO
Paul Isaac received Ph.D. from Cochin University of Science and Technology,Kochi, India. His research interests include fuzzy algebra, KacMoody Lie Algebra.
Department of Mathematics, Bharata Mata College Thrikkakara, Kochi682 021, Kerala, India.
email: pibmct@gmail.com
C.A. Neelima is doing her Ph.D. Her research interests are fuzzy mathematics, functional analysis.
Department of Mathematics and Statistics, SNM College Maliankara, Ernakulam (Dt.)683 516, Kerala, India.
email: neelimaasokan@gmail.com
Biswas R.
,
Nanda S.
(1994)
Rough groups and rough subgroups
Bull. Polish Acad. Sci. Math.
42
251 
254
Neelima C.A.
,
Isaac Paul
(2014)
Rough semi prime ideals and rough biideals in rings
Int. J. Math. Sci. Appl.
4
29 
36
Neelima C.A
,
Isaac Paul
(2014)
Antihomomorphism on Rough Prime Fuzzy Ideals and Rough Primary Fuzzy Ideals
Ann.Fuzzy Math.Inform.
8
549 
559
Paul Isaac
,
Neelima C.A
(2014)
Antihomomorphism on Rough Prime Ideals and Rough Primary ideals
Advances in Theoretical and Applied Mathematics
9
1 
9
Kazanci Osman
,
Davvaz B.
(2008)
On the structure of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in commutative rings
Inform. Sci.
178
1343 
1354
DOI : 10.1016/j.ins.2007.10.005
Mordeson J.N.
,
Malik D.S.
(1998)
Fuzzy Commutative Algebra
World Scientific
ISBN 981023628X
Shah Tariq
,
Saeed Mohammad
(2012)
On Fuzzy ideals in rings and Antihomomorphism
Int. Math. Forum.
7
753 
759