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h
−
1
(μ
α
)), g
−
1
(
3
)λα
=
(λ
h
(λg(
v
α
))
,λ >
0
.
g
−
1
h
−
1
(
4
)α
λ
=
(λg(μ
α
)),
(λ
h
(
v
α
))
,λ >
0
.
, then the operational laws (1)-(4) can be trans-
formed into the corresponding operational laws (1)-(4) of Definition 1.3, which are
the ones based on Algebraic t-conorm and t-norm.
If
g(
Especially, if
g(
t
)
=−
log
(
t
)
log
2
−
t
, then
t
)
=
μ
α
1
+
μ
α
2
1
v
α
1
v
α
2
(
)α
1
⊕
α
2
=
+
μ
α
1
μ
α
2
,
.
5
1
+
(
1
−
v
α
1
)(
1
−
v
α
2
)
μ
α
1
μ
α
2
v
α
1
+
v
α
2
(
6
)α
1
⊗
α
2
=
−
μ
α
2
)
,
.
1
+
(
1
−
μ
α
1
)(
1
1
+
v
α
1
v
α
2
(
+
μ
α
)
λ
−
(
−
μ
α
)
λ
2
v
α
1
1
(
7
)λα
=
−
μ
α
)
λ
,
,λ>
0
.
+
μ
α
)
λ
+
(
α
))
λ
+
v
α
(
1
1
(
2
−
v
μ
α
v
α
)
λ
−
(
v
α
)
λ
−
μ
α
)
λ
+
μ
α
,
(
2
1
+
1
−
)α
λ
=
(
8
,λ>
0
,
(
(
+
v
α
)
λ
+
(
−
v
α
)
λ
2
1
1
where the operational laws (5) and (6) are the ones defined by Wang and Liu (2011)
based on Einstein t-conorm and t-norm.
If
g(
log
γ
+
(
1
−
γ)
t
t
,
t
)
=
γ
∈
(
0
,
+∞
)
, then
μ
α
1
+
μ
α
2
−
μ
α
1
μ
α
2
−
(
1
−
γ)μ
α
1
μ
α
2
(
9
)α
1
⊕
α
2
=
,
1
−
(
1
−
γ)μ
α
1
μ
α
2
v
1
v
α
α
2
.
γ
+
(
1
−
γ)(
v
+
v
−
v
1
v
)
α
α
α
α
1
2
2
μ
α
1
μ
α
2
(
10
)α
1
⊗
α
2
=
−
γ )(μ
α
1
+
μ
α
2
−
μ
α
1
μ
α
2
)
,
γ
+
(
1
v
α
1
+
v
α
2
−
v
α
1
v
α
2
−
(
1
−
γ)
v
α
1
v
α
2
.
1
−
(
1
−
γ)
v
α
1
v
α
2
)μ
α
)
λ
−
(
−
μ
α
)
λ
(
1
+
(γ
−
1
1
(
11
)λα
=
−
μ
α
)
λ
,
)μ
α
)
λ
+
(γ
−
(
1
+
(γ
−
1
1
)(
1
v
α
γ
,λ >
0
.
v
α
))
λ
+
(γ
−
v
α
(
1
+
(γ
−
1
)(
1
−
1
)
γμ
α
(
12
)α
λ
=
)μ
α
,
−
μ
α
))
λ
+
(γ
−
(
1
+
(γ
−
1
)(
1
1
v
α
)
λ
−
(
v
α
)
λ
(
+
(γ
−
)
−
1
1
1
,λ >
,
0
v
α
)
λ
+
(γ
−
v
α
)
λ
(
1
+
(γ
−
1
)
1
)(
1
−
which are the ones defined based on Hammer t-conorm and t-norm.
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