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0and1(like
T
pr od
and
W
), or have some idempotents different from 0 and 1. In
any case, since it is always
T
(
,
)
=
(
,
)
=
0
0
0 and
T
1
1
1, 0 and 1 are idempotent
elements for all t-norms.
Remark 2.2.25
Analogous considerations can be made for t-conorms. There are
discontinuous t-conorms like
Z
∗
, and continuous ones like
T
pr od
and
W
∗
. The only
for which all elements in
[
0
,
1
]
are idempotent is
S
=
max. Since
S
(
0
,
0
)
=
0
and
S
1, 0 and 1 are always idempotent, and there are t-conorms that only
have these two idempotents (like
T
pr od
and
W
∗
), as well as those that have some
idempotents different from 0,1. There are t-conorms without one-divisors, like max
and
T
pr od
, and t-conorms with one-divisors like
W
∗
, for example,
W
∗
(
(
1
,
1
)
=
0
.
5
,
0
.
5
)
=
min
(
1
,
1
)
=
1.
Remark 2.2.26
There is not a characterization theorem for all t-norms (t-conorms),
but it is a characterization of the continuous t-norms
(t-conorms) that will be pre-
sented by means of the following, and easy to prove, results:
•
If
˕
:[
0
,
1
]ₒ[
0
,
1
]
verifies, (1) If
x
y
, then
˕(
x
)
˕(
y
)
,(2)
˕
is bijective,
(3)
˕(
0
)
=
0
, ˕(
1
)
=
1(
˕
is an order-automorphism of the ordered interval
), and
T
is a t-norm, then
T
˕
=
˕
−
1
(
[
0
,
1
]
,
)
ⓦ
T
ⓦ
(˕
×
˕)
is also a t-norm.
Given
T
,theset
{
T
˕
;
˕
an order-automorphism
}
is called the
family
of
T
.
•
T
is a continuous t-norm if and only if all t-norms
T
˕
are continuous.
If
S
is a t-conorm, then
S
˕
=
˕
−
1
•
ⓦ
S
ⓦ
(˕
×
˕)
is also a t-conorm, and
S
{
S
˕
;
˕
is continuous if and only if all t-conorms
S
˕
are continuous, the set
an
}
order-automorphism
is called the
family
of
S
.
In particular,
˕
−
1
•
The family of
T
=
min, is reduced to the only t-norm min, since
(
min
(˕(
a
),
(˕
−
1
)), ˕
−
1
˕(
b
))
=
min
(˕(
a
(˕(
b
)))
=
min
(
a
,
b
)
•
The family of
T
pr od
contains all continuous t-norms of the form prod
˕
(
a
,
b
)
=
˕
−
1
(˕(
a
).(˕(
b
))
.
)
=
˕
−
1
•
The family of
W
contains all t-norms of the form
W
˕
(
a
,
b
(
W
(˕(
a
),
))
=
˕
−
1
˕(
b
(
max
(
0
, ˕(
a
)
+
˕(
b
)
−
1
))
, and all of them are continuous t-norms.
Notice that no t-norm in the family
{
pr od
˕
}
has zero-divisors, since
pr od
˕
(
a
,
b
)
=
0
⃔
˕(
a
)
·
˕(
b
)
=
0
⃔
˕(
a
)
=
0or
˕(
b
)
=
0
⃔
a
=
0, or
b
=
0. Instead
all t-norms
W
have zero-divisors, since
W
˕
(
a
,
b
)
=
0
⃔
max
(
0
, ˕(
a
)
+
˕(
b
)
−
˕
1
)
⃔
˕(
a
)
+
˕(
b
)
1. Of course, neither t-norms
pr od
, nor
W
,havemore
˕
˕
idempotents than 0 and 1:
)
=
˕
−
1
•
a
=
W
˕
(
a
,
a
(
max
(
0
,
2
˕(
a
)
−
1
))
⃔
˕(
a
)
=
max
((
0
,
2
˕(
a
)
−
1
)
⃔
˕(
a
)
=
0or
˕(
a
)
=
1or
a
=
0or
a
=
1.
)
=
˕
−
1
•
a
=
pr od
˕
(
a
,
a
(˕(
a
).(˕(
a
))
⃔
˕(
a
)
=
˕(
a
).(˕(
a
)
⃔
˕(
a
)
=
0or
˕(
a
)
=
1or
a
=
0or
a
=
1.
Analogously,
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