Database Reference
In-Depth Information
algebraic lattice of closed databases
(
C
,
⊆
)
. Let us show now that it is also algebraic:
in fact, for each
A
∈
Ob
DB
we have
≈
A
TA
(from the algebraic property of
T
)
T
=
{
TB
|
B
⊆
ω
TA
}≈
{
TB
|
B
⊆
ω
TA
}
B
is finite and hence
B
=
Comp C
is compact element
∈
TB
T
B
∈
A
B
=
|
Comp C
sup
B
∈
A
,
B
=
Comp C
|
so that this lattice is compactly generated and hence algebraic as well. Consequently,
we obtain the surjective homomorphism
T
:
(Ob
DB
,
,
⊗
,
⊕
)
→
(
C
,
⊆
,
∩
,T
∪
)
.
Now we can extend the lattice
(
C
,
⊆
)
of only closed objects of
DB
into a lattice
of all objects of
DB
category:
,
⊕
Corollary 28
PO subcategory
DB
I
⊆
DB
,
with
DB
I
=
L
DB
=
(Ob
DB
,
,
⊗
)
,
is an algebraic lattice
.
Proof
In Proposition
52
, it was shown that
T
is an algebraic closure operator for all
simple
objects in
DB
. Let us show that this closure operator (for all objects in
DB
)
is algebraic for the complex objects
A
=
1
≤
j
≤
m
A
j
,j
≥
2 as well. In fact,
T
1
A
j
T(A)
=
≤
j
≤
m
=
T(A
j
)
(from the fact that each
A
j
is a simple object)
1
≤
j
≤
m
T
A
j
|
A
j
⊆
ω
A
j
=
1
≤
j
≤
m
j,T
A
j
|
m
A
j
⊆
ω
A
j
,
1
=
≤
j
≤
A
j
⊆
ω
A
j
T
A
j
=
1
≤
j
≤
m
A
j
T
A
j
A
j
⊆
ω
1
=
1
≤
j
≤
m
1
≤
j
≤
m
≤
j
≤
m
T
A
|
A
⊆
ω
A
.
=
