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In-Depth Information
,
V
1
⇒
V
2
=
W
\
{
a
∈
W
|
a
b
}
\
c
∈
V
3
{
b
∈
W
b
∈
W
|
c
b
}
(
c
∈
V
1
\
V
2
↑
where
V
3
=
\
c)
.
Dually, an autoreferential frame F, for any complete Heyting algebra
W
H
=
(
Υ
,
≤
,
∧
,
∨
,,
¬
,
⊥
,R
Υ
)
∈
HA
DB
,
is defined by Proposition
72
by reduction of this Heyting algebra to the underlying
poset
H
)
. Hence, we obtain an
autoreferential duality
(in the place of the
standard duality (*)):
=
(
Υ
,
≤
Proposition 73
(A
UTOREFERENTIAL
D
UALITY
)
For any complete Heyting al-
gebra
H
=
(
Υ
,
≤
,
∧
,
∨
,,
¬
,
⊥
,R
Υ
)
∈
HA
DB
we obtain the following isomor-
phism
:
1.
(
H
)
∗
=
F
(
Υ
)
.
Dually
,
for each autoreferential frame
F
=
↓:
H
≤
∈
K
DB
we obtain the equality
(
Υ
,
)
(
F
∗
)
.
2. F
=
Proof
From Proposition
72
, we obtain
L
alg
=
Ob
DB
sk
,
0
,
Υ
.
(
H
)
∗
=
,
¬
F
(
Υ
)
=
⊆
,
∩
,T
∪
,
⇒
,
⊥
Let us show that
↓
is indeed an isomorphism between the Heyting algebra
H
and
F
(
Υ
)
, that is,
F
0
,
Υ
.
,
¬
↓:
(
Υ
,
≤
,
∧
,
∨
,,
¬
,
⊥
,R
Υ
)
(
Υ
),
⊆
,
∩
,T
∪
,
⇒
,
⊥
Based on the isomorphism of their underlying complete lattices (Proposition
68
), it
is enough to show the homomorphic property for their relative pseudo-complements.
In fact, for any
a,b
∈
Υ
,
(
b)
=
↓
a)
⇒
(
↓
{↓
c
|↓
c
∩↓
a
⊆↓
b
}
(from the homomorphic property for meets, Proposition
68
)
{
}
=
=
{↓
c
|
c
∧
a
≤
b
}
=
c
|
c
∧
a
≤
b
ab.
Thus,
(
b)
=
Γ
(
b)
↓
=↓
↓
⇒
↓
↓
⇒
↓
(a b)
a)
(
a)
(
(the closure of a closed object is equal to itself)
=
(
↓
a)
⇒
(
↓
b),
so that the homomorphic property is satisfied.
