Database Reference
In-Depth Information
It is easy to verify that the lattice
L
DB
is well defined (it is an algebraic complete
lattice from Proposition
52
in Sect.
8.1.5
) by the restriction of matching and merging
operators to simple
closed
objects
A
and
B
, given by
A
⊗
B
=
TA
∩
TB
=
A
∩
B
B)
and
A
⊕
(by Theorem
12
)
=
T(A
∩
B
=
A
⊕
B
=
T(A
∪
B)
(by Definition
57
)
and hence the PO relation '
' reduces to the subset relation '
⊆
'.
Ob
DB
sk
as a representative ele-
ment of an equivalence class of the instance databases (based on the equivalence
relation corresponding to the isomorphism of objects in
DB
category),
We can consider a simple closed object
TA
∈
[
A
]={
B
∈
Ob
DB
|
TB
=
TA
}
, with
TA
∈[
A
]
, and for any two
B,C
∈[
A
]
, they are behav-
iorally equivalent (hence, isomorphic)
B
C
in
DB
. Thus, in what follows we will
consider this “weak monoidal topos”-logic based on the DB-quotient (w.r.t. behav-
ioral equivalence) lattice
L
DB
,
with the meet lattice operator equal to
T
∩=∩
and
the join operator
T
, for the logical conjunction and disjunction, respectively. The
logical operators for negation and implication will be considered in the next sec-
tions.
The examination of the closure algebraic operators (the power-view operator
T
in our case) from a logical point of view and the investigation of the modal negation
operators based on the complete lattice of truth-values, provided by the Birkhoff
polarity, has been presented in [
16
]. In the next section, we will briefly present
these results and, successively, consider if some of them can feature the power-view
operator
T
in our
DB
-algebraic lattice
L
DB
in Definition
63
.
∪
9.2.1 Birkhoff Polarity over Complete Lattices
Generally, lattices arise concretely as the substructures of closure systems (intersec-
tion systems) where a closure system is a family
F
(W)
of subsets of a set
W
such
I
, then
i
∈
I
A
i
∈
F
that
W
(W)
. Then the repre-
sentation problem for general lattices is to establish that every lattice can be viewed,
up to an isomorphism, as a collection of subsets in a closure system (on some set
W
), closed under the operations of the system.
Closure operators
Γ
are canonically obtained by the composition of two maps
of Galois connections. The Galois connections can be obtained from any binary
relation
R
∈
F
(W)
and if
A
i
∈
F
(W),i
∈
W
2
onaset
W
[
2
] (i.e., Birkhoff
polarity
) in a canonical way:
If
(W,R)
is a set with a particular relation on a set
W
,
R
⊆
W
×
W
, with map-
pings
λ
R
:
P
⊆
(W)
OP
,
R
:
P
(W)
OP
(W)
→
P
→
P
(W)
, such that for any
U,V
∈
P
(W)
,
λ
R
U
={
a
∈
W
|∀
u
∈
U.((u,a)
∈
R)
}
,ρ
R
V
={
a
∈
W
|∀
v
∈
V.((a,v)
∈
R)
}
, where
P
(W)
is the
powerset poset
with the bottom element being the empty
(W)
OP
OP
set
∅
and the top element
W
, and
P
its dual (with
⊆
inverse of
⊆
), then
OP
V
iff
U
we obtain the induced Galois connection
λ
R
ρ
R
V
.
The following lemma is useful for the relationship of these set-based operators
with the operation of negation in the complete lattices.
ρ
R
, i.e.,
λ
R
U
⊆
⊆
Lemma 20
(I
NCOMPATIBILITY RELATION
[
2
])
Let (W,
≤
,
∧
,
∨
,
0
,
1
) be a com-
plete lattice with '
' the meet and join lattice operators
,
respectively
,
and
with
1
and
0
the top and bottom elements in W
.
Then we can use a binary relation
∧
' and '
∨
