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In-Depth Information
•
T
is commutative,
S
is commutative
•
T
is associative,
S
is associative
•
N
is involutive,
that is:
•
T
(
a
,
b
)
=
T
(
b
,
a
),
S
(
a
,
b
)
=
S
(
b
,
a
)
, for all
a
,
b
in
[
0
,
1
]
•
T
(
a
,
T
(
b
,
c
))
=
T
(
T
(
a
,
b
),
c
),
S
(
a
,
S
(
b
,
c
))
=
S
(
S
(
a
,
b
),
c
)
, for all
a
,
b
,
c
in
[
0
,
1
]
N
−
1
.
•
N
(
N
(
a
))
=
a
, for all
a
in
[
0
,
1
]
,or
N
ⓦ
N
=
id, or
N
=
Functions
T
and
S
are called t-norms and t-conorms, respectively. Functions N
are strong negations. Hence,
(
[
0
,
1
]
,
T
,
)
is an ordered semigroup with neutral 1,
(
[
,
]
,
,
)
and absorbent 0, and
0
1
S
is also an ordered semigroup but with neutral 0
(
)
=
and absorbent 1. Since
N
0, it seems that this two kind of ordered semigroups
should show some character of duality. This duality goes in the way:
1
•
If
T
is a t-norm,
T
N
=
N
ⓦ
S
ⓦ
(
N
×
N
)
is a t-conorm
•
is a t-norm
that are easy to prove. Of course, from Sect.
2.1.4
,
If
S
is a t-conorm,
S
N
=
N
ⓦ
S
ⓦ
(
N
×
N
)
•
If
T
is a t-norm,
T
min, and min is a t-norm
•
If
S
is a t-conorm, max
S
, and
max
is a t-conorm
Hence, for all t-norm
T
and all t-conorm
S
:
T
min
max
S
,
S
.
1
in particular,
T
Even more, the function
⊧
⊨
min
b
,
if
a
=
1
(
a
,
b
),
if
a
=
1or
b
=
1
Z
(
a
,
b
)
=
a
,
if
b
=
1
=
⊩
0
,
otherwise
,
0
,
otherwise
,
is obviously a t-norm such that
Z
T
for all t-norm T. Consequently,
⊧
⊨
b
,
if
a
=
0
Z
∗
(
a
,
b
)
=
1
−
Z
(
1
−
a
,
1
−
b
)
=
a
,
if
b
=
0
⊩
1
,
otherwise
,
max
(
a
,
b
),
if
a
=
0or
b
=
0
=
1
,
otherwise
,
Z
∗
is a t-conorm such that
S
for all t-conorm
S
. Hence, for all t-norm
T
and all
t-conorm
S
,
Z
∗
.
Z
T
min
max
S
1
Notice that
T
S
mean
T
(
a
,
b
)
S
(
a
,
b
)
,forall
(
a
,
b
)
∈[
0
,
1
]×[
0
,
1
]
.
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