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•
=
The family of
S
max, only contains this t-conorm.
The family of
T
pr od
•
contains all t-conorms of the form
pr od
˕
(
)
=
˕
−
1
pr od
∗
(˕(
)))
=
˕
−
1
a
,
a
(
a
), ˕(
b
(˕(
a
)
+
˕(
b
)
−
˕(
a
)
·
˕(
b
))
The family of
W
∗
contains all t-conorms of the form
•
W
∗
(
)
=
˕
−
1
W
∗
(˕(
))
=
˕
−
1
a
,
a
(
a
), ˕(
b
(
min
(
1
, ˕(
a
)
+
˕(
b
)))
Remark 2.2.27
The order-automorphism
˕
plays the role of a functional parameter.
x
r
, it follows, for example,
By taking,
˕(
x
)
=
r
max
r
min
W
˕
(
a
r
b
r
a
r
b
r
W
˕
(
a
,
b
)
=
((
0
,
+
−
1
),
a
,
b
)
=
((
1
,
+
)
giving a family of t-norms (t-conorms) depending on the numerical parameter
r
>
0.
x
r
,
Notice that with
˕(
x
)
=
√
a
r
r
b
r
Prod
˕
(
a
,
b
)
=
·
=
a
·
b
=
Prod
(
a
,
b
),
but
√
a
r
Prod
˕
(
)
=
˕
−
1
r
b
r
a
r
b
r
a
,
b
(˕(
a
)
+
˕(
b
)
−
˕(
a
)
·
˕(
b
))
=
+
−
·
.
2.2.6 Strong Negations
As it was said before, an strong negation is a function
N
:[
0
,
1
]ₒ[
0
,
1
]
such that
•
N
(
0
)
=
1
•
If
a
b
, then
N
(
b
)
N
(
a
)
,or
N
2
•
N
(
N
(
a
))
=
a
, for all
a
∈[
0
,
1
]
=
id.
Notice that
N
2
N
−
1
, that shows
N
is a continuous
=
id is equivalent to
N
=
function: It is
N
(
1
)
=
N
(
N
(
0
))
=
0
,
and if
a
<
b
it should be
N
(
b
)<
N
(
a
)
since
N
(
b
)
=
N
(
a
)
would imply
N
(
N
(
b
))
=
N
(
N
(
a
))
,or
a
=
b
. Hence, N is strictly
decreasing.
Since
N
is continuous, the equation
N
(
x
)
=
x
has solutions, but there is only
one. Suppose
N
(
x
1
)
=
x
1
and
N
(
x
2
)
=
x
2
. Either
x
1
x
2
,or
x
2
<
x
1
.Inthe
first case, it follows
N
(
x
2
)
N
(
x
1
)
,or
x
2
x
1
, and
x
1
=
x
2
. In the second case,
N
(
x
1
)<
N
(
x
2
)
,or
x
1
<
x
2
, that is absurd. Then, each strong negation has a single
fixed point
N
(
n
)
=
n
, in the open interval
(
0
,
1
)
, since
N
(
0
)
=
1
,
N
(
1
)
=
0show
that 0 and 1 are not fixed points.
Remark 2.2.28
In the classical case (a Boolean algebra
L
, or a power set
P
(
X
)
), the
transformation
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