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[
μ
]↗
min
[
μ
]=[
μ
]
is reflexive, symmetrical and min-transitive, since
. Hence, all the
ʱ
-cuts are classical equivalence relations, each one defining a partition of
X
.The
different values of
ʱ
are 0.2, 0.6, 0.8 and 1 (
levels of crispness
of
μ
), and it is easy
to see that the corresponding partitions
ˀ
0
.
2
,
ˀ
0
.
6
,
ˀ
0
.
8
, and
ˀ
1
, can be located as the
partition tree:
1
3
4
6
2
5
1
2
5
1
3
4
6
0.8
2
5
1
3
4
6
0.6
1
3
4
6
2
5
0.2
0
This tree is called the
fuzzy quotient
of
X
by
μ
.
Example 4.5.3
The fuzzy relation
μ
:{
1
,
2
,...,
6
}ₒ[
0
,
1
]
given by
⊛
⊞
10
.
80
.
20
.
60
.
60
.
4
⊝
⊠
01000
60
00100
.
50
00010
.
[
μ
]=
4
000010
000001
.
60
.
is reflexive, antisymmetric and min-transitive. Hence, is a fuzzy ordering whose
ʱ
-cuts should be crisp partial orderings. Namely,
⊛
⊝
⊞
⊠
100000
010000
001000
000100
000010
000001
5
6
1.
[
μ
(
1
)
]=
, with partial order
2
3
4
1
that only connects the pairs
(
1
,
1
), (
2
,
2
),...(
6
,
6
)
, and is called the disconnected
partial order.
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