Information Technology Reference
In-Depth Information
•
=
P
x
, for all
x
in
X
x
•
=
P
y
⃔
=
P
x
x
y
•
x
=
P
y
, and
y
=
P
z
imply
x
=
P
z
,
=
P
is an equivalence in
X
, and gives the quotient set
X
/
=
P
,ofthe
the relation
classes of equally-
P
elements. The predicate
P
is semi-rigid in
X
if
X
/
=
P
consists
in a finite number of classes. Of course, all predicates on a finite
X
are semi-rigid.
Example 1.1.1
Let it be
X
={
x
1
,...,
x
5
}
, and
P
a predicate inducing the preorder
given by the matrix with entries
1
,
if
x
i
P
x
j
entr y
(
i
,
j
)
=
0
,
otherwise,
that is,
⊛
⊝
⊞
⊠
10101
01010
10101
01010
10101
[
P
]=
The quotient set
X
/
=
P
has the two classes
{
x
1
,
x
3
,
x
5
}
and
{
x
2
,
x
4
}
.
If, in the same
X
,itis
Q
with primary use defined by
x
i
Q
x
j
⃔
i
j
,
it results
X
/
=
Q
with the 5 classes
{
x
1
}
,
{
x
2
}
,
{
x
3
}
,
{
x
4
}
,
{
x
5
}
.
Example 1.1.2
In
X
=[
0
,
10
]
, consider the predicate
P
=
around five
, with
P
defined by
•
If
x
,
y
∈[
0
,
5
]
,
x
P
y
⃔
x
y
•
If
x
,
y
∈
(
5
,
10
]
,
x
P
y
⃔
y
x
•
If
x
∈[
0
,
5
]
,
y
∈
(
5
,
10
]
,
x
and
y
are not
P
-comparable.
Obviously,
P
is a preorder, and
(
x
y
)
&
(
y
x
)
:
x
=
y
x
=
P
y
⃔
⃔
x
=
y
,
(
y
x
)
&
(
x
y
)
:
y
=
x
that is,
=
P
is the equality. Hence,
X
/
=
P
={{
x
};
x
∈[
0
,
10
]}
, and
P
is not semi-
rigid.
1.1.1 L-Degree
Let us suppose
P
in
X
given by its primary use
P
, and let
(
L
,
)
be a poset. An
μ
P
:
ₒ
P
y
,
L-degree, or L-measure, for
P
in
X
is a function
X
L
, such that if
x
Search WWH ::
Custom Search