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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
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 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
) =
(
0
, +∞ )
(1.194)
1
(
1
γ)
xy
xy
τ H (
x
,
y
) =
) (
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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