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Anyway,
a lot of laws typical of classical sets are not always valid
in all standard
algebras of fuzzy sets. For example,
c
(
P
(
),
∩
,
∪
,
)
X
is a Boolean algebra and no
X
one
(
[
0
,
1
]
,
T
,
S
,
N
)
is a Boolean algebra. In particular,
(
P
(
X
),
∩
,
∪
)
is a lattice
and the only standard algebra that is a lattice is that with
T
=
min and
S
=
max.
Let us study in which standard algebras some laws of crisp sets do hold.
2.2.8.1 Distributive Laws
With classical sets it always do hold the two distributive laws
1.
A
∩
(
B
∪
C
)
=
(
A
∩
B
)
∪
(
A
∩
C
)
2.
A
∪
(
B
∩
C
)
=
(
A
∪
B
)
∩
(
A
∪
C
)
and the question is for which triplets
(
T
,
S
,
N
)
do hold the corresponding laws with
fuzzy sets
1.
μ
·
(˃
+
ʻ)
=
μ
·
˃
+
μ
·
ʻ
,
2.
μ
+
(˃
·
ʻ)
=
(μ
+
˃)
·
(μ
+
ʻ)
.
This questions correspond to solve the functional equations in the unknowns
T
and
S
:
T
(
a
,
S
(
b
,
c
))
=
S
(
T
(
a
,
b
),
T
(
a
,
c
))
(2.3)
(
,
(
,
))
=
(
(
,
),
(
,
))
S
a
T
b
c
T
S
a
b
S
a
c
(2.4)
for all
a
,
b
,
c
in
[
0
,
1
]
.
Lemma 2.2.39
Equation
(
2.3
)
does hold if and only if S
=
max
.
Proof
With
b
=
c
=
1, is
T
(
a
,
S
(
1
,
1
))
=
S
(
T
(
a
,
1
),
T
(
a
,
1
))
or
a
=
S
(
a
,
a
)
. That
is
S
max.
Provided
S
=
=
max, the equation is
T
(
a
,
max
(
b
,
c
))
=
max
(
T
(
a
,
b
),
T
(
a
,
c
))
.If
either
b
c
or
c
b
, it is immediate to check its validity for all t-norm T.
Lemma 2.2.40
Equation
(
2.4
)
does hold if and only if T
=
min
.
Proof
With
b
=
c
=
0, is
S
(
a
,
0
)
=
a
=
T
(
a
,
a
)
. That is
T
=
min.
Provided
T
=
min, the equation is
S
(
a
,
min
(
b
,
c
))
=
min
(
S
(
a
,
b
),
S
(
a
,
c
))
.If
either
b
c
or
c
b
, it is immediate to check its validity for all t-conorm T.
Hence,
•
In all standard algebras with
(
T
,
max
)
, it holds
μ
·
(˃
+
ʻ)
=
μ
·
˃
+
μ
·
ʻ,
•
In all standard algebras with
(
min
,
S
)
, it holds
μ
+
(˃
·
ʻ)
=
(μ
+
˃)
·
(μ
+
ʻ)
•
The two distributive laws (
2.3
) and (
2.4
) do jointly hold if and only if
T
=
min
and
S
=
max.
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