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In-Depth Information
1.4.3 Constrained Predicates
Let
P
and
Q
predicates in
X
and
Y
, respectively, with uses
(
P
, μ
P
), (
Q
, μ
Q
)
.
Each relation
∅ =
R
(
P
,
Q
)
:
(
x
is
P
,
y
is
Q
)
∈
R
(
P
,
Q
),
allows to define the
constrained predicate Q
|
P
=
'
Q
if
P
', in
X
×
Y
, given by
(
x
,
y
)
∈
Q
|
P
⃔
(
x
is
P
,
y
is
Q
)
∈
R
(
P
,
Q
).
An example is given by the interpretation
then
y
is
Q
⃔
'
x
is
P
implies
y
is
Q
.
(
x
,
y
)
∈
Q
|
P
⃔
'If
x
is
P
,
Provided
Q
|
P
induces a preorder
Q
|
P
in
X
×
Y
, and that there is an L-degree
μ
Q
|
P
:
X
×
Y
ₒ
L
,
(
x
1
,
y
1
)
Q
|
P
(
x
2
,
y
2
)
⃒
μ
Q
|
P
(
x
1
,
y
1
)
μ
Q
|
P
(
x
2
,
y
2
),
it could be studied how to express
μ
Q
|
P
by means of
μ
P
and
μ
Q
.
Notice that there are several possibilities for obtaining
Q
|
P
from both
P
and
Q
, i.e.,
Q
|
P
=
P
×
Q
,
Q
|
P
=
−
1
×
Q
,
etc.
P
μ
Q
|
P
is said to be decomposable, or functionally expressible, if there
is an operation
J
The degree
:
L
×
L
ₒ
L
,
such that
μ
Q
|
P
(
,
)
=
(μ
P
(
), μ
Q
(
)),
x
y
J
x
y
for all
(
x
,
y
)
∈
X
×
Y
, and it again remains to be tested that
μ
Q
|
P
is actually a
L-degree for
Q
|
P
. For example,
•
If
Q
|
P
=
Q
×
P
, and
J
is non-decreasing in both variables, it is
(
x
1
,
y
1
)
Q
|
P
(
x
2
,
y
2
)
⃔
x
1
P
x
2
,
y
1
Q
y
2
⃒
μ
P
(
x
1
)
μ
P
(
x
2
)
, and
μ
Q
(
y
1
)
μ
Q
(
y
2
)
⃒
J
(μ
P
(
x
1
), μ
Q
(
y
1
))
J
(μ
P
(
x
2
), μ
Q
(
y
2
)),
or
μ
Q
|
P
(
x
1
,
y
1
)
μ
Q
|
P
(
x
2
,
y
2
).
Q
|
P
=
−
1
×
−
Q
•
If
, and
J
is decreasing in both variables, it also follows the
P
same conclusion,
Q
|
P
=
−
P
×
Q
, and
J
is decreasing in its first variable, and non-decreasing
in the second, it also follows the same conclusion.
Etc.
•
If
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