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or of the functional equations
a
=
S
(
T
(
a
,
b
),
T
(
a
,
N
(
b
))),
a
=
T
(
S
(
a
,
b
),
S
(
a
,
N
(
b
)))
Lemma 2.2.46
The equation a
=
S
(
T
(
a
,
b
),
T
(
a
,
N
(
b
)))
holds if and only if T
=
W
˕
,
pr od
˕
,
S
=
N
=
N
.
˕
W
˕
,
Proof
If
T
=
pr od
˕
,
S
=
N
=
N
,itis
S
(
T
(
a
,
b
),
T
(
a
,
N
(
b
)))
=
˕
W
˕
(
)))
=
˕
−
1
W
∗
(˕(
pr od
˕
(
a
,
b
),
pr od
˕
(
a
,
N
˕
(
b
(
pr od
˕
(
a
,
b
)), ˕(
pr od
˕
(
a
,
N
˕
)))))
=
˕
−
1
W
∗
(˕(
))))
=
˕
−
1
(
b
(
a
)
·
˕(
b
), ˕(
a
)
·
˕(
N
˕
(
b
(
min
(
1
, ˕(
a
)
·
˕(
b
)
+
˕(
a
)
·
))))
=
˕
−
1
))
=
˕
−
1
˕(
N
˕
(
b
(
min
(
1
, ˕(
a
)
·
˕(
b
)
+
˕(
a
)
·
(
1
−
˕(
b
(
min
(
1
, ˕(
a
)
·
)))
=
˕
−
1
˕(
a
.
The proof of the reciprocal will be avoided since it is technically complex. Let us
say only that
a
b
)
+
˕(
a
)
−
˕(
a
)
·
˕(
b
(
min
(
1
, ˕(
a
))
=
=
S
(
T
(
a
,
b
),
T
(
a
,
N
(
b
)))
gives, with
a
=
1
,
1
=
S
(
b
,
N
(
b
))
, that
W
∗
and
N
˕
=
implies
S
N
.
=
(
(
,
),
(
,
(
)))
=
It can be also proven that
a
T
S
a
b
S
a
N
b
if and only if
T
W
˕
,
pr od
˕
,
S
=
N
=
N
˕
. Notice only that
a
=
0gives
T
(
b
,
N
(
b
))
=
0, or
T
=
W
˕
and
N
.
Notice that the verification of von Neumann's law require, in the case of fuzzy
sets, non-dual theories, like those given by the triplets
=
N
˕
W
˕
,
(
pr od
˕
,
N
˕
)
, and
pr od
˕
,
(
W
˕
,
N
˕
)
.
2.2.8.8 Which Standard Algebra Is Closer to a Boolean Algebra?
The results in last section can be summarized in the following Table
2.1
.
Hence, the algebras with the triplets
are the ones that preserve more
structural Boolean properties. Indeed, these algebras preserve all the basic Boolean
laws except those of non-contradiction and excluded-middle. They are distributive
pseudo-complemented lattices, that is, De Morgan algebras that, in addition and like
all algebras of fuzzy sets, verify the law of Kleene,
(
min
,
max
,
N
)
T
(
a
,
N
(
a
))
S
(
b
,
N
(
b
)),
for all
a
,
b
in
[
0
,
1
]
. The algebras given by the triplets
(
min
,
max
,
N
)
are De Morgan-
Kleene algebras.
2.2.8.9 Last Comments
It can be considered, in addition to the structural Boolean laws, the cases that can be
derived from them, for example,
B
c
c
A
c
B
c
(
A
∩
)
=
B
∪
(
∩
),
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