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⎛
⎝
⎞
⎠
1
p
+
q
1
α
j
n
1
n
(
n
−
1
)
v
p
α
i
v
q
+
1
−
−
i
,
j
=
1
i
=
j
⎛
⎝
⎞
⎠
1
p
+
q
n
1
q
1
n
(
n
−
1
)
p
≤
1
−
1
−
−
(
1
−
μ
α
i
)
(
1
−
μ
α
j
)
i
,
j
=
1
i
=
j
⎛
⎝
⎞
⎠
1
p
+
q
n
1
q
1
n
(
n
−
1
)
p
(1.130)
+
1
−
−
(
1
−
μ
α
i
)
(
1
−
μ
α
j
)
=
1
i
,
j
=
1
i
=
j
which completes the proof of Theorem 1.13.
Then, in what follows, we introduce some desirable properties of the IFGBM (Xia
et al. 2012a):
(1) (Idempotency) If all
α
i
(
i
=
1
,
2
,...,
n
)
are equal, i.e.,
α
i
=
α
=
(μ
α
,
v
α
,π
α
)
,
for all
i
, then
IFGBM
p
,
q
(α
1
,α
2
,...,α
n
)
IFGB
p
,
q
=
(α,α,...,α)
⎛
⎞
⎛
⎞
1
1
n
⊗
n
⊗
⎝
⎠
=
⎝
⎠
1
n
(
n
−
1
)
1
n
(
n
−
1
)
=
(
p
α
⊕
q
α)
((
p
+
q
)α)
p
+
q
p
+
q
i
,
j
=
1
i
,
1
i
=
j
j
=
i
=
j
1
n
)
n
(
n
−
1
)
(
n
−
1
=
q
((
p
+
q
)α)
=
α
(1.131)
p
+
(2) (Commutativity)
IFGBM
p
,
q
IFGBM
p
,
q
(α
1
,α
2
,...,α
n
)
=
( α
1
, α
2
,..., α
n
)
(1.132)
( α
1
, α
2
,..., α
n
)
(α
1
,α
2
,...,α
n
)
where
is any permutation of
.
Proof
Since
( α
1
, α
2
,..., α
n
)
is any permutation of
(α
1
,α
2
,...,α
n
)
, then
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