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μ
·
˃
=
ⓦ
(μ
×
˃)
Example 5.3.3
With an intersection
T
,(
T
a continuous t-norm)
define
E
T
(μ, ˃)
=
Sup
(μ
·
˃).
Obviously,
E
T
(μ, ˃)
=
E
T
(˃, μ)
, since T is a commutative operation. To have,
E
T
(μ, μ)
=
1, or
Sup T
ⓦ
(μ
×
μ)
=
1
,
since
T
ⓦ
(μ
×
μ)
≤
μ
, it should be
Sup
μ
=
1, that is, since
X
is a finite set,
μ
should
be a normalized fuzzy set, a fuzzy set for which it exists
x
0
∈
X
such that
μ(
x
0
)
=
1.
X
From now on let's consider the set
N(
X
)
={
μ
∈[
0
,
1
]
;
Sup
μ
=
1
}
.The
mapping
E
T
:
N(
X
)
×
N(
X
)
ₒ[
0
,
1
]
,
is reflexive and symmetric, that is,
E
T
is a similarity in
N(
X
)
. Obviously,
E
T
≤
E
min
, for all continuous t-norm T. Since,
T
(
E
T
(μ, ˃),
E
T
(˃, ʻ))
=
T
(
Sup T
(μ, ˃),
Sup T
(˃, ʻ))
=
Sup T
(
T
(μ, ˃),
T
(˃, ʻ))
≤
Sup T
(μ, ʻ)
=
E
T
(μ, ʻ),
it results that, in addition,
E
T
is a
T
∗
-indistinguishability for all continuous t-norm
T
∗
such that
T
∗
≤
T
. In particular,
E
min
is not only a min-indistinguishability but a
T-indistinguishability for all continuous t-norm T.
For example with
X
={
1
,
2
,
3
,
4
}
, and the two fuzzy sets
μ
=
1
|
1
+
0
.
4
|
2
+
0
.
8
|
3
+
0
.
7
|
4,
˃
=
0
.
6
|
1
+
0
.
5
|
2
+
0
.
8
|
3
+
1
|
4
it follows
μ
·
˃
=
0
.
6
|
1
+
T
(
0
.
4
,
0
.
5
)
|
2
+
T
(
0
.
8
,
0
.
8
)
|
3
+
0
.
7
|
4
,
and
•
E
min
(μ, ˃)
=
0
.
8, since
min
(
0
.
8
,
0
.
8
)
=
0
.
8,
min
(
0
.
4
,
0
.
5
)
=
0
.
4
•
E
prod
(μ, ˃)
=
0
.
64, since
prod
(
0
.
8
,
0
.
8
)
=
0
.
64,
prod
(
0
.
4
,
0
.
5
)
=
0
.
2
•
E
W
(μ, ˃)
=
0
.
6, since
W
(
0
.
8
,
0
.
8
)
=
0
.
6,
W
(
0
.
4
,
0
.
5
)
=
0
Remark 5.3.4
Provided
T
=
min
or
T
=
prod
˕
,if
E
T
(μ, ˃)>
0, and
E
T
(˃, ʱ)>
0,
it is
0
<
T
(
E
T
(μ, ˃),
E
T
(˃, ʱ))
≤
E
T
(μ, ʱ),
and
E
T
(μ, ʱ)>
0. In these cases,
E
T
is said to be strictly transitive.
Nevertheless, in the case in which a fuzzy relation
E
:
X
×
X
ₒ[
0
,
1
]
is
W
˕
-
transitive, from
E
(
x
,
y
)>
0 and
E
(
y
,
z
)>
0
,
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