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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 ) ( 0 . 8 ) ( 0 . 6 ) ( 0 . 4 ) ( 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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