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(
J T (
,
),
J T (
,
))
J T (
,
)
,
,
[
,
]
T
a
b
b
c
a
c
for all a
b
c in
0
1
To avoid some difficult technicalities, we will exemplify this general result in the
particular case J W (
a
,
b
) =
min
(
1
,
1
a
+
b
)
:
W
(
J W (
a
,
b
),
J W (
b
,
c
)) =
W
(
min
(
1
,
1
a
+
b
),
min
(
1
,
1
b
+
c
))
=
(
,
(
, b a ) +
(
,
b + c ))
max
0
min
0
min
1
1
min
(
1
,
1
b
+
c
) (
1
,
1
a
+
c
),
if a
>
b
=
max
(
0
,
b
a
+
min
(
1
,
1
b
+
c
))
min
(
1
,
1
a
+
c
),
if a
b
J W ( a , c ).
Then, with the T-transitivity of J T ,itis
T
(
E T (
a
,
b
),
E T (
b
,
c
)) =
T
(
min
(
J T (
a
,
b
),
J T (
b
,
a
)),
min
(
J T (
b
,
c
),
J T (
c
,
b
)))
T
(
J T (
a
,
b
),
J
(
b
,
c
)) =
E T (
a
,
c
),
hence, all relations E T are T-transitive. Then,
- All relations J T are T-Preorders, and
- All relations E T are T-Indistinguishabilities
If
{
Ri
;
i
I
}
is a collection of T-Preorders , Inf i I R i (
x
,
y
) =
R
(
x
,
y
)
, is also a
T-Preorder .
Obviously, R
(
x
,
x
) =
Inf i I R i (
x
,
x
) =
1, and
(
(
,
),
(
,
)) =
(
R i (
,
),
R i (
,
))
T
R
a
b
R
b
c
T
Inf
i I
a
b
Inf
i I
b
c
Inf
i
T
(
R i (
a
,
b
),
R i (
b
,
c
))
I
R i (
,
) =
(
,
),
Inf
i
a
c
R
a
c
I
since T is a continuous t-norm.
If
{
E i ;
i
I
}
is a collection of T-indistinguishabilities , Inf i I E i (
x
,
y
) =
E
(
x
,
y
)
is
also a T-indistinguishability .
Obviously,
E
(
a
,
a
) =
Inf
i
E i (
a
,
a
) =
1
,
I
and
E
(
a
,
b
) =
Inf
i
E i (
a
,
b
) =
Inf
i
E i (
b
,
a
) =
E
(
b
,
a
).
I
I
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