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since,
μ(
x
,
y
)
≥
ʱ, μ(
y
,
z
)
≥
ʱ
⃒
T
(ʱ, ʱ)
T
(μ(
x
,
y
)μ(
y
,
z
))
μ(
x
,
z
).
Hence,
only if
μ
is min-transitive
, it is sure that all
ʱ
-cuts are classical preorders,
and
μ
results decomposed in the family of preorders
{
μ
ʱ
;
ʱ
∈
(
0
,
1
)
}
.
Example 4.5.1
With
X
={
1
,
2
,
3
,
4
}
, the matrix
⊛
⊝
⊞
⊠
,
10
.
610
.
6
.
.
.
0
310
30
3
[
μ
]=
.
.
10
610
6
0
.
40
.
80
.
41
is obviously reflexive but not symmetric, and verifies
[
μ
]↗
min
[
μ
]=[
μ
]
. Hence,
μ
is a min-preorder. Its different
ʱ
-cuts are
⊛
⊞
⊛
⊞
⊛
⊞
1010
0100
1010
0001
1010
0100
1010
0101
1111
0100
1111
0101
⊝
⊠
,
[
μ
(
0
.
8
)
]=
⊝
⊠
,
[
μ
(
0
.
6
)
]=
⊝
⊠
,
[
μ
(
1
)
]=
⊛
⊝
⊞
⊠
,
[
μ
(
0
.
3
)
]=
⊛
⊝
⊞
⊠
,
1111
0100
1111
1111
1111
1111
1111
1111
[
μ
(
0
.
4
)
]=
that give the classical
(ʱ)
preorders, that follows:
•
(
1
)
:
1
(
1
)
1
,
2
(
1
)
2
,
3
(
1
)
3
,
4
(
1
)
4
,
1
(
1
)
3
,
3
(
1
)
1.
•
(
0
.
8
)
:
1
(
0
.
8
)
1
,
2
(
0
.
8
)
2
,
3
(
0
.
8
)
3
,
4
(
0
.
8
)
4
,
1
(
0
.
8
)
3
,
3
(
0
.
8
)
(
0
.
8
)
2.
•
(
0
.
6
)
:
1
,
4
1
(
0
.
6
)
1
,...,
4
(
0
.
6
)
4
,
1
(
0
.
6
)
3
,
3
(
0
.
6
)
1
,
4
(
0
.
6
)
2
,
1
(
0
.
6
)
2
,
1
(
0
.
6
)
4
,
3
(
0
.
6
)
2
,
3
(
0
.
6
)
4.
•
(
0
.
4
)
:
1
(
0
.
4
)
1
,...,
4
(
0
.
4
)
4
,
1
(
0
.
4
)
3
,...,
4
(
0
.
4
)
1
,
4
(
0
.
4
)
3.
•
(
0
.
3
)
:
1
(
0
.
3
)
1
,...,
4
(
0
.
3
)
3
,
2
(
0
.
3
)
1
, ...,
4
(
0
.
3
)
2,
.
.
.
.
(
1
)
ↂ
(
0
.
8
)
ↂ
(
0
.
6
)
ↂ
(
0
.
4
)
ↂ
(
0
.
3
)
and, since, 0
3
0
4
0
6
0
8
1, verify
.
Example 4.5.2
The fuzzy relation
μ
:
X
×
X
ₒ[
0
,
1
]
, with
X
={
1
,
2
,
3
,
4
,
5
,
6
}
,
given by
⊛
⊝
⊞
⊠
10
.
210
.
60
.
20
.
6
0
.
210
.
20
.
20
.
80
.
2
10
.
210
.
60
.
20
.
6
[
μ
]=
0
.
60
.
20
.
610
.
20
.
8
.
.
.
.
.
0
20
80
20
210
8
.
.
.
.
.
0
60
20
60
80
81
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