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⎧
⎨
⎫
⎬
l
l
l
0
,
◦
n
α
j
,τ
,
◦
n
β
j
,τ
1
≤
j
≤
1
≤
j
≤
k
1
k
1
k
1
,
α
,τ
, β
,τ
∗
n
μ
k
1
,
l
j
,τ
,
∗
n
ν
k
i
,
l
j
,τ
=
|
τ
∈
T
.
1
≤
j
≤
1
≤
j
≤
⎩
⎭
.
.
k
m
m
k
m
,
α
,τ
, β
,τ
∗
n
μ
k
m
,
l
j
,τ
,
∗
n
ν
k
i
,
l
j
,τ
1
≤
j
≤
1
≤
j
≤
Let
•
and
•
be two other dual operations, that can coincide with
(
◦
,
◦
)
or
(
∗
,
∗
)
,
or to be different. The following new aggregation operation can be defined:
(
◦
,
•
,
∗
)
−
T
-aggregation
ρ
(
◦
,
•
,
∗
)
(
A
∗
(T ))
l
1
l
1
l
1
,
•
•
τ
∈
T
α
,τ
,
τ
∈
T
β
,τ
...
k
1
k
1
k
1
,
◦
◦
∗
∗
τ
∈
T
α
,τ
,
τ
∈
T
β
,τ
τ
∈
T
μ
k
1
,
l
1
,τ
,
τ
∈
T
ν
k
1
,
l
1
,τ
...
=
.
.
.
...
k
m
,
◦
k
m
◦
m
,τ
∗
∗
τ
∈
T
α
,τ
,
τ
∈
T
β
τ
∈
T
μ
k
m
,
l
1
,τ
,
τ
∈
T
ν
k
m
,
l
1
,τ
...
l
n
l
n
...
l
n
,
α
,τ
, β
,τ
...
∗
τ
∈
T
μ
k
1
,
l
n
,τ
,
τ
∈
T
ν
k
1
,
l
n
,τ
∗
.
.
.
...
∗
τ
∈
T
μ
k
m
,
l
n
,τ
,
τ
∈
T
ν
k
m
,
l
n
,τ
∗
The above operations are extensions of the operations from IFIMs, EIFIMs and
TIFIMs and have similar properties.
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