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For example,
•
T
=
min, all min-preorders are of the form,
1
,
if
˃(
x
)
≤
˃(
y
)
J
min
=
μ(
x
,
y
)
=
Inf
˃
∈
Inf
˃
∈
˃(
y
),
if
˃(
x
)>˃(
y
)
T
(μ)
T
(μ)
•
=
T
prod
˕
,all
prod
˕
-preorders are of the form,
1
,
if
˃(
x
)
≤
˃(
y
)
J
prod
˕
=
μ(
x
,
y
)
=
Inf
˃
∈
Inf
˃
∈
(
˕(
y
)
˕
−
1
˕(
x
)
),
if
˃(
x
)>˃(
y
)
T
(μ)
T
(μ)
•
T
=
W
˕
,all
W
˕
-preorders are of the form
J
W
(
μ(
x
,
y
)
=
Inf
˃
∈
x
,
y
)
=
Inf
˃
∈
min
(
1
,
1
−
˃(
x
)
+
˃(
y
))
T
(μ)
T
(μ)
Remark 5.2.1
As an immediate consequence of all that has been said, for any fam-
ily of fuzzy sets
X
, the T-preorder
Inf
J
T
F ↂ[
0
,
1
]
has the elements in
F
as
˃
∈
F
J
T
(
J
T
,
∀
˃
∈ F
T-states. If
μ(
x
,
y
)
=
Inf
x
,
y
)
, it follows
μ
≤
, or, equivalently
˃
∈
F
˃
∈
T
(μ),
∀
˃
∈ F
,or
F ↂ
T
(μ)
. For example, with
X
=[
0
,
1
]
and the two functions
˃
1
(
x
)
=
x
,
˃
2
(
x
)
=
1
−
x
,itis
J
˃
1
T
J
˃
2
T
μ(
x
,
y
)
=
min
(
(
x
,
y
),
(
x
,
y
))
a T-preorder. With
T
=
W
,
μ(
x
,
y
)
=
min
(
min
(
1
,
1
−
˃
1
(
x
)
+
˃
1
(
y
)),
min
(
1
,
1
−
˃
2
(
x
)
+
˃
2
(
y
)))
=
min
(
1
,
1
−
x
+
y
,
1
+
x
−
y
),
is a W-preorder.
5.3 The Characterization of T-Indistinguishabilities
Let us consider, the T-indistinguishabilities
E
T
(
J
T
(
J
T
(
x
,
y
)
=
min
(
x
,
y
),
y
,
x
))
=
min
(
J
T
(˃(
x
), ˃(
y
)),
J
T
(˃(
y
), ˃(
x
))).
X
×
X
, consider
T
If
μ
is a fuzzy relation in
[
0
,
1
]
(μ)
. Obviously
E
T
(
μ(
x
,
y
)
≤
Inf
˃
∈
x
,
y
)
, for all x, y in
X
.
T
(μ)
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