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ϕ(β, α)
=
(
{
,
−
+
,
−
+
}
,
{
,
{
−
,
−
}}
)
=
(
,
)
min
1
1
1
0
1
1
0
max
0
min
1
0
1
0
0
1
With the intuitionistic fuzzy Lukasiewicz implication, the traditional triangle
product and the square product (Kohout and Bandler 1980), below we further intro-
duce an intuitionistic fuzzy triangle product and an intuitionistic fuzzy square product
respectively:
Definition 2.39
(Wang et al. 2012) Let
α
={
α
1
,α
2
,...,α
l
}
,γ
={
γ
1
,γ
2
,...,γ
m
}
and
β
={
β
1
,β
2
,...,β
n
}
be three sets of IFVs,
Z
1
∈
F
(α
×
γ)
and
Z
2
∈
F
(γ
×
β)
two
intuitionistic fuzzy relations, then an intuitionistic fuzzy triangle product
Z
1
Z
2
∈
F
(α
×
β)
of
Z
1
and
Z
2
can be defined as:
1
m
m
m
1
m
(
Z
1
Z
2
)(α
i
,β
j
)
=
1
μ
Z
1
(α
i
,γ
k
)
→
Z
2
(γ
k
,β
j
)
,
v
Z
1
(α
i
,γ
k
)
→
Z
2
(γ
k
,β
j
)
,
k
=
k
=
1
for any
(α
i
,β
j
)
∈
(α, β),
i
=
1
,
2
,...,
l
;
j
=
1
,
2
,...,
n
(2.203)
where “
→
” represents the intuitionistic fuzzy Lukasiewicz implication.
Similarly, Wang et al. (2012) defined an intuitionistic fuzzy square product
Z
1
Z
2
∈
F
(α
×
β)
of
Z
1
and
Z
2
as:
m
μ
min
(
Z
1
(α
i
,γ
k
)
→
Z
2
(γ
k
,β
j
),
Z
2
(γ
k
,β
j
)
→
Z
1
(α
i
,γ
k
))
,
v
min
(
Z
1
(α
i
,γ
k
)
→
Z
2
(γ
k
,β
j
),
Z
2
(γ
k
,β
j
)
→
Z
1
(α
i
,γ
k
))
(
Z
1
Z
2
)(α
i
,β
j
)
=
min
1
≤
k
≤
for any
(α
i
,β
j
)
∈
(α, β),
i
=
1
,
2
,...,
l
;
j
=
1
,
2
,...,
n
(2.204)
for short, and the same with others.
As a result, Eqs. (
2.203
) and (
2.204
) can be respectively simplified as:
For convenience, we denote
z
ik
as
Z
(α
i
,γ
k
)
⎛
⎞
m
m
1
m
1
m
⎝
⎠
(
Z
1
Z
2
)(α
i
,β
j
)
=
1
μ
z
ik
→
z
kj
,
v
z
ik
→
z
kj
(2.205)
j
=
j
=
1
m
μ
min
(
z
ik
→
z
kj
,
z
kj
→
z
ik
)
,
v
min
(
z
ik
→
z
kj
,
z
kj
→
z
ik
)
(2.206)
(
Z
1
Z
2
)(α
i
,β
j
)
=
min
1
≤
k
≤
Indeed, the intuitionistic fuzzy triangle product and the intuitionistic fuzzy square
product are very closely-related with each other. That is, the former is the basis of
the latter, due to that
(
Z
1
Z
2
)(α
i
,β
j
)
is directly derived from
(
Z
1
Z
2
)(α
i
,β
j
)
and
(
Z
2
Z
1
)(α
i
,β
j
)
.
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