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We notice that an additive generator of a continuous Archimedean t-norm is a strictly
decreasing function
g
:
[
,
]→
,
]
such that
g(
)
=
0. If we assign specific forms
to the function
g
, then some well-known t-cornorms and t-norms can be obtained
(Xia et al. 2012b):
0
1
0
1
1
e
−
t
,
and Algebraic t-conorm and t-norm (Beliakov et al. 2007) are obtained as follows:
,
g
−
1
e
−
t
,
h
−
1
(1) Let
g(
t
)
=−
log
t
, then
h
(
t
)
=−
log
(
1
−
t
)
(
t
)
=
(
t
)
=
1
−
s
A
(
˙
x
,
y
)
=
x
+
y
−
xy
,τ
A
(
x
,
y
)
=
xy
(1.190)
log
2
−
t
, then
(2) Let
g(
t
)
=
log
2
−
(
1
−
t
)
2
2
g
−
1
h
−
1
h
(
t
)
=
,
(
t
)
=
1
,
(
t
)
=
1
−
(1.191)
1
−
t
e
t
+
e
t
+
1
and we can get Einstein t-conorm and t-norm (Beliakov et al. 2007):
x
+
y
xy
˙
s
E
(
x
,
y
)
=
xy
,τ
E
(
x
,
y
)
=
(1.192)
1
+
1
+
(
1
−
x
)(
1
−
y
)
log
γ
+
(
1
−
γ)
t
t
,
(3) Let
g(
)
=
γ
∈
(
,
+∞
)
t
0
, then we have
log
γ
+
(
−
γ)
−
t
)
γ
γ
1
1
, g
−
1
h
−
1
h
(
t
)
=
(
t
)
=
1
,
(
t
)
=
1
−
e
t
e
t
1
−
t
+
γ
−
+
γ
−
1
(1.193)
and Hamacher t-conorm and t-norm (Beliakov et al. 2007) are obtained as follows:
x
+
y
−
xy
−
(
1
−
γ)
xy
˙
s
H
(
x
,
y
)
=
,γ
∈
(
0
,
+∞
)
(1.194)
1
−
(
1
−
γ)
xy
xy
τ
H
(
x
,
y
)
=
)
,γ
∈
(
0
,
+∞
)
(1.195)
γ
+
(
1
−
γ)(
x
+
y
−
xy
1, then Hamacher t-conorm and t-norm reduce to Algebraic
t-conorm and t-norm, respectively; if
Especially, if
γ
=
2, then Hamacher t-conorm and t-norm
reduce to Einstein t-conorm and t-norm, respectively.
(4) Let
g(
γ
=
log
γ
−
1
γ
1
,
t
)
=
γ
∈
(
1
,
+∞
)
, then
t
−
log
e
γ
γ
γ
−
1
+
g
−
1
g(
t
)
=
,
(
t
)
=
,
(
1
−
γ)
t
−
1
e
γ
γ
h
−
1
(
t
)
=
1
−
e
t
+
γ
−
1
(1.196)
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