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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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