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μ
P
(
)
μ
P
(
)
then
x
y
- in the order of the poset. The idea behind this definition is
μ
P
(
)
∈
that
L
evaluates up to which extent
x
is
P
, up to which extent
x
verifies the
property named by P. It can be written,
x
Degree up to which
x
is
P
=
μ
P
(
x
)
∈
L
,
and said that the primary use of
P
is gradable in
(
L
,
)
.If
x
=
P
y
, then
μ
P
(
x
)
=
μ
P
(
y
)
, as is easily proven. Hence,
μ
P
is constant in the equivalence's
classes modulo
=
P
.
Once
μ
P
is known, it can be defined the relation
μ
P
in
X
by
x
μ
P
y
⃔
μ
P
(
x
)
μ
P
(
y
),
with which it is
P
ↂ
μ
P
. When
P
=
μ
P
,
μ
P
perfectly reflects
the primary use
of
P
in
X
.
Relation
μ
P
is a preorder since it is obviously reflexive and transitive. The pair
(
P
,
μ
P
)
reflects
ause
of
P
in
X
, and once
P
and
(
L
,
)
are fixed, there can
exist again several uses of
P
in
X
depending on the
L
-degree
μ
P
.
L
-degrees are also
called
L
-
x
5.
When
L
, endowed with the linear order of the unit interval, the L-sets
are known as fuzzy sets. In this case, the triplet
=[
0
,
1
]
(
X
,
P
, μ
P
)
is a numerical quantity,
where of course, given
P
and
X
, neither the relation
P
, nor the measure
μ
P
are in
general unique.
When
L
={
+
;
}=C
a
ib
0
a
1&0
b
1
is endowed with the partial
order
a
1
+
ib
1
a
2
+
ib
2
⃔
a
1
a
2
&
b
1
b
2
,
L-sets are 'complex fuzzy sets'. Notice that
(
C
,
)
is isomorphic with the set of the
sub-intervals
[
a
,
b
]ↆ[
0
,
1
]
, once partially ordered by
[
a
1
,
b
1
] [
a
2
,
b
2
]⃔
a
1
a
2
&
b
1
b
2
.
When it is only known that the degree up to which “
x
is
P
” belongs to some sub-
interval
[
a
,
b
]
, it can be taken
μ
P
(
x
)
=
a
+
ib
. In science and technology, complex
quantities are not at all rare.
Remark 1.1.3
Provided
L
=[
0
,
1
]
, the relation
μ
P
is linear or total, since given
x
,
y
in
X
, it is either
μ
P
(
x
)
μ
P
(
y
)
,or
μ
P
(
x
)
≥
μ
P
(
y
)
; that is, either
x
μ
P
y
,
or
y
μ
P
x
.
Remark 1.1.4
The relation
P
is not always linear, there can exist elements
x
,
y
in
X
such that it is neither
x
P
y
, nor
y
P
x
, that is, elements which are not comparable
under
P
. Of course, in the cases in which
P
is not linear and
L
=[
0
,
1
]
, it cannot
be
P
=
μ
P
, the degree cannot perfectly reflect the primary use of
P
, and the
relation
μ
P
enlarges the primary use
P
with all the links between elements in
X
P
−
μ
P
, provided this difference-set is non empty.
contained in
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