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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) = + ˃) · + 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) = + ˃) · + 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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