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In-Depth Information
ʣ
)
Representation Theorems with Respect to
(
ʣ
)isagraded consequence relation.
Proof.
(GC1) is proved as in Theorem 3.1. (GC2) is obtained using (i) and Lemma 3.8.
(GC3) inf
ʲ
∈
Z
gr
(
X
Theorem 3.3.
For any
{
T
i
}
I
,
|≈
in the sense of (
i
∈
|≈
ʲ
∗
∪
|≈
ʱ
)
gr
(
X
Z
)
Z
r
[
i
∈
I
{
x
∈
X
T
i
(
x
)
r
[
i
∈
I
{
x
∈
X
∪
Z
T
i
(
x
)
=inf
ʲ
∈
→
I
T
i
(
ʲ
)
}
]
∗
→
I
T
i
(
ʱ
)
}
]
=
r
[
ʲ
∈
Z
{
i
∈
I
(
x
∈
X
T
i
(
x
)
r
[
i
∈
I
{
x
∈
X
∪
Z
T
i
(
x
)
→
I
T
i
(
ʲ
))
}
]
∗
→
I
T
i
(
ʱ
)
}
] (Lemma 3.5)
=
r
[
i
∈
I
{
ʲ
∈
Z
(
x
∈
X
T
i
(
x
)
r
[
i
∈
I
{
x
∈
X
∪
Z
T
i
(
x
)
→
I
T
i
(
ʲ
))
}
]
∗
→
I
T
i
(
ʱ
)
}
] (Lemma 3.6)
r
[
i
∈
I
{
ʲ
∈
Z
(
x
∈
X
T
i
(
x
)
r
[
i
∈
I
{
x
∈
X
∪
Z
T
i
(
x
)
≤
→
I
T
i
(
ʲ
))
}
]
∗
→
I
T
i
(
ʱ
)
}
](Corollary 3.1)
=
r
[
i
∈
I
(
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
r
[
i
∈
I
{
x
∈
X
∪
Z
T
i
(
x
)
ʲ
))
}
]
∗
→
I
T
i
(
ʱ
)
}
] (by(vii))...(1)
Also we obtain,
r
[
i
∈
I
[
{
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
}∗
I
{
x
∈
X
∪
Z
T
i
(
x
)
ʲ
)
→
I
T
i
(
ʱ
)
}
]]
r
[
i
∈
I
{
x
∈
X
T
i
(
x
)
≤
→
I
T
i
(
ʱ
)
}
] (by(vi) andLemma3.8).
...(2)
Now
i
∈
I
(
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
}∗
I
i
∈
I
{
x
∈
X
∪
Z
T
i
(
x
)
ʲ
))
→
I
T
i
(
ʱ
)
}
≤
I
i
∈
I
[
{
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
}∗
I
{
x
∈
X
∪
Z
T
i
(
x
)
ʲ
)
→
I
T
i
(
ʱ
)
}
] (Lemma 3.2)
≤
I
i
∈
I
[
{
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
}∗
I
{
x
∈
X
∪
Z
T
i
(
x
)
ʲ
)
→
I
T
i
(
ʱ
)
}
]. (Lemma 3.9)
i.e.,
r
[
i
∈
I
(
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
r
[
i
∈
I
{
x
∈
X
∪
Z
T
i
(
x
)
ʲ
))
}
]
∗
→
I
T
i
(
ʱ
)
}
]
r
[
i
∈
I
[
{
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
}∗
I
{
x
∈
X
∪
Z
T
i
(
x
)
≤
ʲ
)
→
I
T
i
(
ʱ
)
}
]]. (Lemma3.7)...(3)
i
∈
I
(
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
}ↆ
e
i
∈
I
(
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
ʲ
))
ʲ
))
}
. (Lemma 3.10)
So,
r
[
i
∈
I
(
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
r
[
i
∈
I
(
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
ʲ
))
}
]
≤
ʲ
))
}
].
Thuswehave,
r
[
i
∈
I
(
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
r
[
i
∈
I
{
x
∈
X
∪
Z
T
i
(
x
)
ʲ
))
}
]
∗
→
I
T
i
(
ʱ
)
}
]
r
[
i
∈
I
(
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
r
[
i
∈
I
{
x
∈
X
∪
Z
T
i
(
x
)
≤
ʲ
))
}
]
∗
→
I
T
i
(
ʱ
)
}
]
r
[
i
∈
I
[
{
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
}∗
I
{
x
∈
X
∪
Z
T
i
(
x
)
≤
ʲ
)
→
I
T
i
(
ʱ
)
}
]] (by 3)
r
[
i
∈
I
{
x
∈
X
T
i
(
x
)
≤
→
I
T
i
(
ʱ
)
}
] (by (2)).
Hence (GC3) is proved.
Theorem 3.4.
For any graded consequence relation
|∼
there is a collection of interval-
ʣ
) coincides with
valued fuzzy sets such that
|≈
generated in the sense of (
|∼
.
Proof.
Given a graded consequence relation
|∼
,let
{
T
X
}
X
∈
P
(
F
)
be such that
T
X
(ʱ) =
|∼
ʱ) =
r
[
Y
∈
P
(
F
)
{
x
∈
X
T
Y
(
x
)
→
i
T
Y
(ʱ)
}
].
[
gr
(
X
|∼
ʱ)]
BIR
, and we want to show
gr
(
X
Arguing as Theorem 3.2 we have,
T
X
(
ʲ
)
≤
I
∧
Y
∈
P
(
F
)
[
∧
ʱ
∈
X
T
Y
(
ʱ
)
→
I
T
Y
(
ʲ
)]
ↆ
e
∩
Y
∈
P
(
F
)
[
∧
ʱ
∈
X
T
Y
(
ʱ
)
→
I
T
Y
(
ʲ
)] (Lemma 3.9)
r
(
T
X
(
ʲ
))
≤
r
(
∧
Y
∈
P
(
F
)
[
∧
ʱ
∈
X
T
Y
(
ʱ
)
→
I
T
Y
(
ʲ
)])
≤
r
(
∩
Y
∈
P
(
F
)
[
∧
ʱ
∈
X
T
Y
(
ʱ
)
→
I
T
Y
(
ʲ
)]) ...(1)
∩
Y
∈
P
(
F
)
[
∧
ʱ
∈
X
T
Y
(
ʱ
)
→
I
T
Y
(
ʲ
)]
ↆ
e
∩
X
ↆ
Y
∈
P
(
F
)
[
∧
ʱ
∈
X
T
Y
(
ʱ
)
→
I
T
Y
(
ʲ
)]
...(2)
Following the proof of Theorem 3.2, for
X
ↆ
Y
,wehave
∧
ʱ
∈
X
T
Y
(
ʱ
)
→
I
T
Y
(
ʲ
) =
T
Y
(
ʲ
).
From (2) and Lemma 3.10,
∩
Y
∈
P
(
F
)
[
∧
ʱ
∈
X
T
Y
(
ʱ
)
→
I
T
Y
(
ʲ
)]
ↆ
e
∧
X
ↆ
Y
T
Y
(
ʲ
) =
T
X
(
ʲ
).
Thus,
r
(
T
X
(
ʲ
)) =
r
(
∩
Y
∈
P
(
F
)
[
∧
ʱ
∈
X
T
Y
(
ʱ
)
→
I
T
Y
(
ʲ
)]) =
gr
(
X
|∼
ʱ
).
U
,∗
I
,→
I
,
[0
,
0]
,
[1
,
1]
,
r
, as mentioned in Remark 1, is
taken; only the differences are: (i)
U
is endowed with both the lattice order relations
≤
I
and
Here the same structure
ↆ
e
, and (ii) a function
r
:
U
→
[0
,
1], different from
l
, is considered here.
Let us present, below, a diagram to visualize the beauty and purpose of dealing with
two lattice structures over the same domain. We consider a linear scale
D
and intervals
over
D
. We consider
D
=
{
3
,
5
,
7
,
9
}
and
I
D
=
{
[
a
,
b
] :
a
≤
b
and
a
,
b
∈
D
}
.
