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(μ, ˃))
˃
˃
(μ, ˃)
⃒
(
max
max
(μ, ˃))
(μ
, ˃
),
imply
(
max
min
or
(μ
, ˃
))
(
min
max
(μ, ˃).
(2.2)
Now, from (
2.2
) and (
2.1
), follows the result.
Theorem 2.2.10
(Kleene's Law)
In all general algebra of fuzzy sets it holds the law
μ
·
μ
˃
+
˃
,
X
.
for all
μ, ˃
in
[
0
,
1
]
(μ
·
μ
)(
)
(˃
+
˃
)(
Proof
We have to prove that, for any
x
∈
X
,itis
x
x
)
.But
it only can be either
μ(
x
)
˃(
x
)
,or
˃(
x
)
μ(
x
)
for each
x
∈
X
. In the firs
(μ
·
μ
)(
), μ
(
), ˃
(
case, it is
x
)
min
(μ(
x
x
))
μ(
x
)
˃(
x
)
max
(˃(
x
x
))
=
(˃, ˃
)(
)
(˃
+
˃
)(
μ
(
)
˃
(
(
max
x
x
)
. In the second case, it is
x
x
)
, and
(μ
·
μ
)(
), μ
(
))
μ
(
)
˃
(
), ˃
(
x
)
min
(μ(
x
x
x
x
)
max
(˃(
x
x
))
=
(˃, ˃
)(
)
(˃
+
˃
)(
(
max
x
x
)
. Notice that provided
μ
and
˃
were crisp sets, the
Kleene's law is reduced to
μ
0
μ
1
.
Remark 2.2.11
Concerning duality, Theorem
2.2.9
only states that the algebra given
, with
involutive, are dual algebras. But they are not the
only dual algebras. For example, with
,
)
(
,
by the triplets
min
max
·=
product,
(μ
·
˃)(
x
)
=
μ(
x
)
·
˃(
x
),
∀
x
∈
X
,
μ
(
it is easy to proof that taking
x
)
=
1
−
μ(
x
)
, and
(μ
+
˃)(
x
)
=
1
−
(
1
−
μ(
x
))(
1
−
˃(
x
))
=
μ(
x
)
+
˃(
x
)
−
μ(
x
)
·
˃(
x
),
X
,
·
,
+
,
)
it is
(
[
0
,
1
]
an algebra of fuzzy sets that since it is
μ
+
˃
=
(μ
·
˃
)
,
μ
(
is a dual algebra. Nevertheless, with
x
)
=
1
−
μ(
x
), (μ
·
˃)(
x
)
=
μ(
x
)
·
˃(
x
)
,
and
(μ
+
˃)(
x
)
=
max
(μ(
x
), ˃(
x
))
, we get an algebra that is not dual since
(μ
·
˃
)
(
x
)
=
μ(
x
)
+
˃(
x
)
−
μ(
x
)
·
˃(
x
)
does not coincides with max
(μ(
x
), ˃(
x
))
, as it is easy to see.
X
,
·
,
+
,
)
Remark 2.2.12
It is easy to prove that, for each algebra of fuzzy sets
(
[
0
,
1
]
,
the operation
μ
+
˃
=
(μ
·
˃
)
,
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