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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
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
L
,
such that
μ Q | P (
,
) =
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
P (
x 1 ), μ Q (
y 1 ))
J
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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