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iff
b
φ
K .(b,a)
R
λ R
K .
iff a
φ
Thus,
¬ φ K = λ R
( φ K ) .
In fact, as follows from this example, the negation operator in intuitionistic logic
(i.e., Heyting algebra) has a Kripke semantics base on the accessibility relation
which is a PO relation '
' between the possible worlds. This fact, together with
the fact that in the autoreferential semantics we use the elements of W as possi-
ble worlds, explains why we have a special interest to consider the incompatibility
relation equal to the PO relation used in a complete lattice (W,
) .
The resulting Galois connection on this partial order is the familiar Dedekind-
McNeile Galois connection of antitonic mappings λ and ρ , with Γ
=
ρλ (by
a ,
{
|
}
{
|
}
we denote the ideal
b
W
b
a
and by
a , we denote
b
W
a
b
):
= a
a) ,
= a
v) .
λU
W
|∀
u
U.(u
V
W
|∀
v
V.((a
=↓ (S) (the ' '
It is easy to verify that for every subset S
W , we obtain Γ(S)
is extension of the binary join operator
of the lattice to any subset S of elements
in W , that is, it is the l.u.b. of S in W , i.e., (S)
W ).
In fact, for each singleton U
={
b
}
, λU
={
a
W
|∀
u
∈{
b
}
.(u
a)
}=↑
b .
= {
{
}|
Thus, from the additivity (point 1 of the lemma above), we obtain λS
λ
b
}=↑ (S) . Analogously, from the multiplicativity (point 2
of the lemma above), we obtain ρS =↓ where ' ' is the extension of the binary
meet operator
}= {↑
b
S
b
|
b
S
of the lattice to any subset S of elements in W , that is, it is the g.l.b.
of S in W .
Consequently, λ
=↑ is a kind of an additive modal negation and ρ
=↓ is a
=↓ =
( from
kind of a multiplicative modal negation, so that Γ
=
ρλ
=↓ from the logical point of view is a composition of these two modal
negations.
Notice that Γ(
=
id W )
)
=
Γ(
{
0
}
)
={
0
}
. Hence, from the monotone property of Γ and
from
∅⊆
U (for any U
P
(W) ), we obtain
{
0
}=
Γ(
)
Γ(U) . That is, any stable
(closed) set contains the bottom element 0
W , and the minimal closed set is
{
0
}
and not the empty set
while, naturally, the maximal closed set is W . Notice that
we have similar facts for the database complete lattice L DB =
(Ob DB sk ,
,
,T
)
0
with the bottom object (a singleton)
={⊥}=∅
and with a similar property for
0 )
0
the power-view closure operator, T(
=⊥
and T( Υ )
=
Υ for the bottom and
top objects in Ob DB sk , respectively.
Moreover, each closed set is a downclosed ideal and there is the bijection between
the set of closed sets and the set of algebraic truth-values in W , so that we are able
to define the representation theorem for a complete lattice (W,
) as follows:
Proposition 68 (A UTOREFERENTIAL R EPRESENTATION FOR C OMPLETE I NFINI -
TA RY D ISTRIBUTIVE L ATTICES ) Let (W, ) be a complete infinitary distribu-
tive lattice . We define the Dedekind-McNeile coalgebra
↓: W P (W) , where
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