We define and characterize a somewhat fuzzy
γ
irresolute continuous mapping and a somewhat fuzzy irresolute
γ
open mapping on a fuzzy topological space.
AMS Mathematics Subject Classification : 54A40.
1. Introduction
The concept of fuzzy
γ
continuous mappings on a fuzzy topological space was introduced and studied by I. M. Hanafy in
[2]
. Also, the concept of fuzzy
γ
irresolute continuous mappings on a fuzzy topological space were introduced and studied by Y. B. Im et al. in
[8]
and fuzzy irresolute
γ
open mappings on a fuzzy topological space was introduced and studied by Y. B. Im in
[3]
.
Recently, somewhat fuzzy
γ
continuous mappings on a fuzzy topological space were introduced and studied by G. Thangaraj and V. Seenivasan in
[9]
.
In this paper, we define and characterize a somewhat fuzzy
γ
irresolute continuous mapping and a somewhat fuzzy irresolute
γ
open mapping which are stronger than a somewhat fuzzy
γ
continuous mapping and a somewhat fuzzy
γ
open mapping respectively. Besides, some interesting properties of those mappings are also given.
2. Preliminaries
A fuzzy set
μ
on a fuzzy topological space
X
is called
fuzzy γ
open if
μ
≤ ClInt
μ
∨ IntCl
μ
and
μ
is called
fuzzy γ

closed
if
μ^{c}
is a fuzzy
γ
open set on
X
.
A mapping
f
:
X
→
Y
is called
fuzzy γ

continuous
if
f
^{−1}
(
ν
) is a fuzzy
γ
open set on
X
for any fuzzy open set
ν
on
Y
and a mapping
f
:
X
→
Y
is called
fuzzy γ
open if
f
(
μ
) is a fuzzy
γ
open set on
Y
for any fuzzy open set
μ
on
X
. It is clear that every fuzzy continuous mapping is a fuzzy
γ
continuous mapping. And every fuzzy open mapping is a fuzzy
γ
open mapping from the above definitions. But the converses are not true in general
[2]
.
A mapping
f
:
X
→
Y
is called
fuzzy γirresolute continuous
if
f
^{−1}
(
ν
) is a fuzzy
γ
open set on
X
for any fuzzy
γ
open set
ν
on
Y
and a mapping
f
:
X
→
Y
is called
fuzzy irresolute γopen
if
f
(
μ
) is a fuzzy
γ
open set on
Y
for any fuzzy
γ
open set
μ
on
X
. It is clear that every fuzzy
γ
irresolute continuous mapping is a fuzzy
γ
continuous mapping. And every fuzzy irresolute
γ
open mapping is a fuzzy open mapping from the above definitions. But the converses are not true in general
[8]
and
[3]
.
A mapping
f
:
X
→
Y
is
called somewhat fuzzy γ

continuous
if there exists a fuzzy
γ
open set
μ
≠ 0
_{X}
on
X
such that
μ
≤
f
^{−1}
(
ν
)≠ 0
_{X}
for any fuzzy open set
ν
on
Y
. It is clear that every fuzzy
γ
continuous mapping is a somewhat fuzzy
γ
continuous mapping. But the converse is not true in general.
A mapping
f
:
X
→
Y
is called
somewhat fuzzy γ

open
if there exists a fuzzy
γ
open set
ν
≠ 0
_{Y}
on
Y
such that
ν
≤
f
(
μ
)≠0
_{Y}
for any fuzzy open set
μ
on
X
. Every fuzzy open mapping is a somewhat fuzzy
γ
open mapping but the converse is not true in general
[9]
.
3. Somewhat fuzzy γirresolute continuous mappings
In this section, we introduce a somewhat fuzzy
γ
irresolute continuous mapping and a somewhat fuzzy irresolute
γ
open mapping which are stronger than a somewhat fuzzy
γ
continuous mapping and a somewhat fuzzy
γ
open mapping respectively. And we characterize a somewhat fuzzy
γ
irresolute continuous mapping and a somewhat fuzzy irresolute
γ
open mapping.
Definition 3.1.
A mapping
f
:
X
→
Y
is called somewhat fuzzy
γ
irresolute continuous if there exists a fuzzy
γ
open set
μ
≠ 0
_{X}
on
X
such that
μ
≤
f
^{−1}
(
ν
) for any fuzzy
γ
open set
ν
≠ 0
_{Y}
on
Y
.
It is clear that every fuzzy
γ
irresolute continuous mapping is a somewhat fuzzy
γ
irresolute continuous mapping. And every somewhat fuzzy
γ
irresolute continuous mapping is a fuzzy
γ
continuous mapping from the above definitions. But the converses are not true in general as the following examples show.
Example 3.2.
Let
μ
_{1}
,
μ
_{2}
and
μ
_{3}
be fuzzy sets on
X
= {
a
,
b
,
c
} and let
ν
_{1}
,
ν
_{2}
and
ν
_{3}
be fuzzy sets on
Y
= {
x
,
y
,
z
} with

μ1(a) = 0.1,μ1(b) = 0.1,μ1(c) = 0.1,

μ2(a) = 0.2,μ2(b) = 0.2,μ2(c) = 0.2,

μ3(a) = 0.5,μ3(b) = 0.5,μ3(c) = 0.5 and

ν1(x) = 0.3,ν1(y) = 0.2,ν1(z) = 0.3,

ν2(x) = 0.5,ν2(y) = 0.5,ν2(z) = 0.5,

ν3(x) = 0.5,ν3(y) = 0.2,ν3(z) = 0.5.
Let
be fuzzy topologies on
X
and let
τ
^{∗}
= {0
_{Y}
,
ν
_{1}
,
ν
_{2}
, 1
_{Y}
} be fuzzy topologies on
Y
. Consider the mapping
f
: (
X
,
τ
) → (
Y
,
τ
^{∗}
) defined by
f
(
a
) =
y
,
f
(
b
) =
y
and
f
(
c
) =
y
. Then we have
μ
_{1}
≤
f
^{−1}
(
ν
_{1}
) =
μ
_{2}
,
f
^{−1}
(
ν
_{2}
) =
μ
_{3}
and
μ
_{1}
≤
f
^{−1}
(
ν
_{3}
) =
μ
_{2}
. Since
μ
_{1}
is a fuzzy
γ
open set on (
X
,
τ
),
f
is somewhat fuzzy
γ
irresolute continuous. But
f
^{−1}
(
ν
_{1}
) =
μ
_{2}
and
f
^{−1}
(
ν
_{3}
) =
μ
_{2}
are not fuzzy
γ
open sets on (
X
,
τ
). Hence
f
is not a fuzzy
γ
irresolute continuous mapping.
Example 3.3.
Let
μ
_{1}
,
μ
_{2}
and
μ
_{3}
be fuzzy sets on
X
= {
a
,
b
,
c
} and let
ν
_{1}
,
ν
_{2}
and
ν
_{3}
be fuzzy sets on
Y
= {
x
,
y
,
z
} with

μ1(a) = 0.2,μ1(b) = 0.2,μ1(c) = 0.2,

μ2(a) = 0.5,μ2(b) = 0.5,μ2(c) = 0.5, and

ν1(x) = 0.3,ν1(y) = 0.2,ν1(z) = 0.3,

ν2(x) = 0.5,ν2(y) = 0.5,ν2(z) = 0.5.
Let
be fuzzy topologies on
X
and let
τ
^{∗}
= {0
_{Y}
,
ν
_{2}
, 1
_{Y}
} be fuzzy topologies on
Y
. Consider the mapping
f
: (
X
,
τ
) → (
Y
,
τ
^{∗}
) defined by
f
(
a
) =
y
,
f
(
b
) =
y
and
f
(
c
) =
y
. Since
f
^{−1}
(
ν
_{2}
) =
μ
_{2}
is fuzzy
γ
open sets on (
X
,
τ
),
f
is fuzzy
γ
continuous. But the inverse images 0
_{X}
≤
f
^{−1}
(
ν
_{1}
) =
μ
_{1}
of a fuzzy
γ
open set
ν
_{1}
on (
Y
,
τ
^{∗}
) is not fuzzy
γ
open on (
X
,
τ
). Hence
f
is not a fuzzy somewhat
γ
irresolute continuous mapping.
Definition 3.4
(
[9]
). A fuzzy set
μ
on a fuzzy topological space
X
is called fuzzy
γ
dense if there exists no fuzzy
γ
closed set
ν
such that
μ
<
ν
< 1.
Theorem 3.5.
Let f
:
X
→
Y be a mapping. Then the following are equivalent:

(1) f is somewhat fuzzy γirresolute continuous.

(2) If ν is a fuzzy γclosed set of Y such that f−1(ν)≠ 1X,then there exists a fuzzy γclosed set μ≠ 1Xof X such that f−1(ν) ≤μ.

(3) If μ is a fuzzy γdense set on X, then f(μ)is a fuzzy γdense set on Y.
Proof
. (1) implies (2): Let
ν
be a fuzzy
γ
closed set on
Y
such that
f
^{−1}
(
ν
)≠ 1
_{X}
. Then
ν
^{c}
is a fuzzy
γ
open set on
Y
and
f
^{−1}
(
ν
^{c}
) = (
f
^{−1}
(
ν
))
^{c}
≠ 0
_{X}
. Since
f
is somewhat fuzzy
γ
irresolute continuous, there exists a fuzzy
γ
open set
λ
≠ 0
_{X}
on
X
such that
λ
≤
f
^{−1}
(
ν
^{c}
). Let
μ
=
λ
^{c}
. Then
μ
≠ 1
_{X}
is fuzzy
γ
closed such that
f
^{−1}
(
ν
) = 1 −
f
^{−1}
(
ν
^{c}
) ≤ 1 −
λ
=
λ
^{c}
=
μ
.
(2) implies (3): Let
μ
be a fuzzy
γ
dense set on
X
and suppose
f
(
μ
) is not fuzzy
γ
dense on
Y
. Then there exists a fuzzy
γ
closed set
ν
on
Y
such that
f
(
μ
) <
ν
< 1. Since
ν
< 1 and
f
^{−1}
(
ν
) ≠ 1
_{X}
, there exists a fuzzy
γ
closed set
δ
≠ 1
_{X}
such that
μ
≤
f
^{−1}
(
f
(
μ
)) <
f
^{−1}
(
ν
) ≤
δ
. This contradicts to the assumption that
μ
is a fuzzy
γ
dense set on
X
. Hence
f
(
μ
) is a fuzzy
γ
dense set on
Y
.
(3) implies (1): Let
ν
≠ 0
_{Y}
be a fuzzy
γ
open set on
Y
and
f
^{−1}
(
ν
) ≠ 0
_{X}
. Suppose there exists no fuzzy
γ
open
μ
≠ 0
_{X}
on
X
such that
μ
≤
f
^{−1}
(
ν
). Then (
f
^{−1}
(
ν
))
^{c}
is a fuzzy set on
X
such that there is no fuzzy
γ
closed set
δ
on
X
with (
f
^{−1}
(
ν
))
^{c}
<
δ
< 1. In fact, if there exists a fuzzy
γ
open set
δ
^{c}
such that
δ
^{c}
≤
f
^{−1}
(
ν
), then it is a contradiction. So (
f
^{−1}
(
ν
))
^{c}
is a fuzzy
γ
dense set on
X
. Then
f
((
f
^{−1}
(
ν
))
^{c}
) is a fuzzy
γ
dense set on
Y
. But
f
((
f
^{−1}
(
ν
))
^{c}
) =
f
(
f
^{−1}
(
ν
^{c}
))≠
ν
^{c}
< 1. This is a contradiction to the fact that
f
((
f
^{−1}
(
ν
))
^{c}
) is fuzzy
γ
dense on
Y
. Hence there exists a
γ
open set
μ
≠ 0
_{X}
on
X
such that
μ
≤
f
^{−1}
(
ν
). Consequently,
f
is somewhat fuzzy
γ
irresolute continuous.
A fuzzy topological space
X
is
product related
to a fuzzy topological space
Y
if for fuzzy sets
μ
on
X
and
ν
on
Y
whenever
(in which case(
γ
^{c}
× 1) ∨ (1 ×
δ
^{c}
) ≥ (
μ
×
ν
)) where
γ
is a fuzzy open set on
X
and
δ
is a fuzzy open set on
Y
, there exists a fuzzy open set
γ
_{1}
on
X
and a fuzzy open set
δ
_{1}
on
Y
such that
and
[1]
.
Theorem 3.6.
Let X
_{1}
be product related to X
_{2}
and Y
_{1}
be product related to Y
_{2}
.
Then the product f
_{1}
×
f
_{2}
:
X
_{1}
×
X
_{2}
→
Y
_{1}
×
Y
_{2}
of somewhat fuzzy γ

irresolute continuous mappings f
_{1}
:
X
_{1}
→
Y
_{1}
and f
_{2}
:
X
_{2}
→
Y
_{2}
is also somewhat fuzzy γ

irresolute continuous
.
Proof
. Let
λ
= ∨
_{i,j}
(
μ
_{i}
×
ν
_{j}
) be a fuzzy
γ
open set on
Y
_{1}
×
Y
_{2}
where
are fuzzy
γ
open sets on
Y
_{1}
and
Y
_{2}
respectively. Then
Since
f
_{1}
is somewhat fuzzy
γ
irresolute continuous, there exists a fuzzy
γ
open set
such that
And, since
f
_{2}
is somewhat fuzzy
γ
irresolute continuous, there exists a fuzzy
γ
open set
such that
Now
and
is a fuzzy
γ
open set on
X
_{1}
×
X
_{2}
. Hence
is a fuzzy
γ
open set on
X
_{1}
×
X
_{2}
such that
Therefore,
f
_{1}
×
f
_{2}
is somewhat fuzzy
γ
irresolute continuous.
Theorem 3.7.
Let f
:
X
→
Y be a mapping
.
If the graph g
:
X
→
X
×
Y of f is a somewhat fuzzy γ

irresolute continuous mapping
,
then f is also somewhat fuzzy γ

irresolute continuous
.
Proof
. Let
ν
be a fuzzy
γ
open set on
Y
. Then
f
^{−1}
(
ν
) = 1∧
f
^{−1}
(
ν
) =
g
^{−1}
(1×
ν
). Since
g
is somewhat fuzzy
γ
irresolute continuous and 1×
ν
is a fuzzy
γ
open set on
X
×
Y
, there exists a fuzzy
γ
open set
μ
≠ 0
_{X}
on
X
such that
μ
≤
g
^{−1}
(1×
ν
) =
f
^{−1}
(
ν
) ≠ 0
_{X}
. Therefore,
f
is somewhat fuzzy
γ
irresolute continuous.
Definition 3.8.
A mapping
f
:
X
→
Y
is called somewhat fuzzy irresolute
γ
open if there exists a fuzzy
γ
open set
ν
≠ 0
_{Y}
on
Y
such that
ν
≤
f
(
μ
) for any fuzzy
γ
open set
μ
≠ 0
_{X}
on
X
.
It is clear that every fuzzy irresolute
γ
open mapping is a somewhat fuzzy irresolute
γ
open mapping. And every somewhat fuzzy irresolute
γ
open mapping is a fuzzy
γ
open mapping. Also, every fuzzy
γ
open mapping is a somewhat fuzzy
γ
open mapping from the above definitions. But the converses are not true in general as the following examples show.
Example 3.9.
Let
μ
_{1}
and
μ
_{2}
be fuzzy sets on
X
= {
a
,
b
,
c
} and let
ν
_{1}
and
ν
_{2}
be fuzzy sets on
Y
= {
x
,
y
,
z
} with

μ1(a) = 0.1,μ1(b) = 0.1,μ1(c) = 0.1,

μ2(a) = 0.2,μ2(b) = 0.2,μ2(c) = 0.2 and

ν1(x) = 0.0,ν1(y) = 0.1,ν1(z) = 0.0,

ν2(x) = 0.0,ν2(y) = 0.2,ν2(z) = 0.0,

ν3(x) = 0.0,ν3(y) = 0.8,ν3(z) = 0.0,

ν4(x) = 0.0,ν4(y) = 0.9,ν4(z) = 0.0.
Let
be fuzzy topologies on
X
and let
be fuzzy topologies on
Y
. Consider the mapping
f
: (
X
,
τ
) → (
Y
,
τ
^{∗}
) defined by
f
(
a
) =
y
,
f
(
b
) =
y
and
f
(
c
) =
y
. Since
and
f
is somewhat fuzzy irresolute
γ
open. But
f
(
μ
_{2}
) =
ν
_{2}
is not a fuzzy
γ
open set on (
Y
,
τ
^{∗}
). Hence
f
is not a fuzzy irresolute
γ
open mapping.
Example 3.10.
Let
μ
_{1}
,
μ
_{2}
and
μ
_{3}
be fuzzy sets on
X
= {
a
,
b
,
c
} and let
ν
_{1}
and
ν
_{2}
be fuzzy sets on
Y
= {
x
,
y
,
z
} with

μ1(a) = 0.4,μ1(b) = 0.1,μ1(c) = 0.4,

μ2(a) = 0.5,μ2(b) = 0.5,μ2(c) = 0.5,

μ3(a) = 0.1,μ3(b) = 0.0,μ3(c) = 0.1 and

ν1(x) = 0.0,ν1(y) = 0.1,ν1(z) = 0.0,

ν2(x) = 0.0,ν2(y) = 0.5,ν2(z) = 0.0.
Let
τ
= {0
_{X}
,
μ
_{1}
,
μ
_{2}
, 1
_{X}
} be fuzzy topologies on
X
and let
τ
^{∗}
= {0
_{Y}
,
ν
_{2}
, 1
_{Y}
} be fuzzy topologies on
Y
. Consider the mapping
f
: (
X
,
τ
) → (
Y
,
τ
^{∗}
) defined by
f
(
a
) =
y
,
f
(
b
) =
y
and
f
(
c
) =
y
. Since
f
(
μ
_{1}
) =
ν
_{1}
and
f
(
μ
_{2}
) =
ν
_{2}
are fuzzy
γ
open sets on (
Y
,
τ
^{∗}
),
f
is fuzzy
γ
open. But
μ
_{3}
≠ 0
_{X}
is a fuzzy
γ
open set on (
X
,
τ
) and
f
(
μ
_{3}
) = 0
_{Y}
. Hence
f
is not a fuzzy somewhat irresolute
γ
open mapping.
Example 3.11.
Let
μ
_{1}
,
μ
_{2}
and
μ
_{3}
be fuzzy sets on
X
= {
a
,
b
,
c
} with

μ1(a) = 0.1,μ1(b) = 0.1,μ1(c) = 0.1,

μ2(a) = 0.2,μ2(b) = 0.2,μ2(c) = 0.2 and

μ3(a) = 0.3,μ3(b) = 0.3,μ3(c) = 0.3.
Let
and
τ
^{∗}
= {0
_{X}
;
μ
_{1}
,
μ
_{3}
, 1
_{X}
}be fuzzy topologies on
X
. Consider the identity mapping
i_{X}
: (
X
,
τ
) → (
X
,
τ
^{∗}
). We have
Since
μ
_{3}
is a fuzzy
γ
open set on (
X
,
τ
),
i_{X}
is somewhat fuzzy
γ
open. But
is not a fuzzy
γ
open set on (
X
;
τ
^{∗}
). Hence
i_{X}
is not a fuzzy
γ
open mapping.
Theorem 3.12.
Let f
:
X
→
Y be a bijection
.
Then the following are equivalent:
(1) f is somewhat fuzzy irresolute γ

open
.
(2) If μ is a fuzzy γ

closed set on X such that f
(
μ
) ≠ 1
_{Y}
,
then there exists a fuzzy γ

closed set ν
≠ 1
_{Y} on Y such that f
(
μ
) <
ν
.
Proof
. (1) implies (2): Let
μ
be a fuzzy
γ
closed set on
X
such that
f
(
μ
) ≠ 1
_{Y}
. Since
f
is bijective and
μ
^{c}
is a fuzzy
γ
open set on
X
,
f
(
μ
^{c}
) = (
f
(
μ
))
^{c}
≠ 0
_{Y}
. And, since
f
is somewhat fuzzy irresolute
γ
open, there exists a
γ
open set
δ
≠ 0
_{Y}
on
Y
such that
δ
<
f
(
μ
^{c}
) = (
f
(
μ
))
^{c}
. Consequently,
f
(
μ
) <
δ
^{c}
=
ν
≠ 1
_{Y}
and
ν
is a fuzzy
γ
closed set on
Y
.
(2) implies (1): Let
μ
be a fuzzy
γ
open set on
X
such that
f
(
μ
) ≠ 0
_{Y}
. Then
μ
^{c}
is a fuzzy
γ
closed set on
X
and
f
(
μ
^{c}
) ≠ 1
_{Y}
. Hence there exists a fuzzy
γ
closed set
ν
≠ 1
_{Y}
on
Y
such that
f
(
μ
^{c}
) <
ν
. Since
f
is bijective,
f
(
μ
^{c}
) = (
f
(
μ
))
^{c}
<
ν
. Hence
ν
^{c}
<
f
(
μ
) and
ν
^{c}
≠ 0
_{X}
is a fuzzy
γ
open set on
Y
. Therefore,
f
is somewhat fuzzy irresolute
γ
open.
Theorem 3.13.
Let f
:
X
→
Y be a surjection
.
Then the following are equivalent:
(1) f is somewhat fuzzy irresolute γ

open
.
(2) If ν is a fuzzy γ

dense set on Y
,
then f
^{−1}
(
ν
)
is a fuzzy γ

dense set on X
.
Proof
. (1) implies (2): Let
ν
be a fuzzy
γ
dense set on
Y
. Suppose
f
^{−1}
(
ν
) is not fuzzy
γ
dense on
X
. Then there exists a fuzzy
γ
closed set
μ
on
X
such that
f
^{−1}
(
ν
) <
μ
< 1. Since
f
is somewhat fuzzy irresolute
γ
open and
μ
^{c}
is a fuzzy
γ
open set on
X
, there exists a fuzzy
γ
open set
δ
≠ 0
_{Y}
on
Y
such that
δ
≤
f
(Int
μ
^{c}
) ≤
f
(
μ
^{c}
). Since
f
is surjective,
δ
≤
f
(
μ
^{c}
) <
f
(
f
^{−1}
(
ν
^{c}
)) =
ν
^{c}
. Thus there exists a
γ
closed set
δ
^{c}
on
Y
such that
ν
<
δ
^{c}
< 1. This is a contradiction. Hence
f
^{−1}
(
ν
) is fuzzy
γ
dense on
X
.
(2) implies (1): Let
μ
be a fuzzy open set on
X
and
f
(
μ
) ≠ 0
_{Y}
. Suppose there exists no fuzzy
γ
open
ν
≠ 0
_{Y}
on
Y
such that
ν
≤
f
(
μ
). Then (
f
(
μ
))
^{c}
is a fuzzy set on
Y
such that there exists no fuzzy
γ
closed set
δ
on
Y
with (
f
(
μ
))
^{c}
<
δ
< 1. This means that (
f
(
μ
))
^{c}
is fuzzy
γ
dense on
Y
. Thus
f
^{−1}
((f(
μ
))
^{c}
) is fuzzy
γ
dense on
X
. But
f
^{−1}
((
f
(
μ
))
^{c}
) = (
f
^{−1}
(
f
(
μ
)))
^{c}
≤
μ
^{c}
< 1. This is a contradiction to the fact that
f
^{−1}
((
f
(
ν
))
^{c}
is fuzzy
γ
dense on
X
. Hence there exists a
γ
open set
ν
≠ 0
_{Y}
on
Y
such that
ν
≤
f
(
μ
). Therefore,
f
is somewhat fuzzy irresolute
γ
open.
BIO
Young Bin Im received his B.S. and Ph.D. at Dongguk University under the direction of Professor K. D. Park. Since 2009 he has been a professor at Dongguk University. His research interests are fuzzy topological space and fuzzy matrix.
Faculty of General Education, Dongguk University, Seoul 100715, Korea.
email : philpen@dongguk.edu
Joo Sung Lee received his B.S. from Dongguk University and Ph.D. at University of Florida under the direction of Professor B. Brechner. Since 1995 he has been a professor at Dongguk University. His research interests are topological dynamics and fuzzy theory.
Department of Mathematics, Dongguk University, Seoul 100715, Korea.
email : jsl@dongguk.edu
Yung Duk Cho received his B.S. and Ph.D. at Dongguk University under the direction of Professor J. C. Lee. Since 2008 he has been a professor at Dongguk University. His research interests are fuzzy category and fuzzy algebra.
Faculty of General Education, Dongguk University, Seoul 100715, Korea.
email : joyd@dongguk.edu
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