Database Reference
In-Depth Information
For example, the subset
[
0
,
1
]
of the real numbers is a complete lattice, but it is not
algebraic.
The
finite objects
in
DB
are the databases with a finite number of
n
-ary relations
(
n
is a finite number
n
∈
ω
, and the nullary relation
is an element of each object in
DB
category); the
extension
of relations is not necessarily finite—in such a case, for
a finite object
A
in
DB
, the object
TA
is composed of infinite number of relations,
that is,
TA
is an infinite object.
We will demonstrate that the database lattice is an
algebraic lattice
.Onewayof
producing, and recognizing, complete algebraic lattices is through
closure
operators.
We can use this approach because the power-view operator
T
⊥
:
Ob
DB
→
Ob
DB
is a
closure operator (by Theorem
6
in Sect.
3.2.5
).
However, the complete lattice derived from a given closure operator must have
the meet operator '
' equal to the set-intersection operator, and, in our case of a
complete database lattice
L
DB
=
(Ob
DB
,
,
⊗
,
⊕
)
, the meet operator is the match-
ing operator: only for the
simple objects
in
DB
it is equal to set-intersection '
∧
∩
'.
Consequently, we are able to use the closure operator
T
in order to demonstrate that
the sublattice of
L
DB
composed of only simple databases is algebraic.
It is easy to show that the principal properties of the objects and morphisms
in the
DB
category are derivable using the properties of the simple objects and
simple morphisms between them. For example, a complex object
A
B
represents
the separation-composition of the simple objects (databases)
A
and
B
, and hence
the algebraic properties of the complex objects are derivable from the algebraic
properties of the simple objects that compose them. The same holds for complex
arrows.
The complex arrow
h
=[
f,g
]:
A
B
→
C
,
h
=
f
g
:
A
B
→
C
D
,
h
=
f,g
:
A
→
C
D
and
h
=
f,g
:
A
→
C
are representable in
DB
by the set of
their simple ptp arrows
f
and
g
.
Consequently, in order to investigate the algebraic database lattice, we can begin
from the full subcategory
DB
of
DB
composed of
only simple
objects, introduced
by Definition
26
in Sect.
3.2.5
with the object
Υ
equal to the union of all simple
objects (databases).
By definition, a closed-set system is
algebraic
if
C
is closed under unions of
,
S
∈
C
upward directed subsets, i.e., for every
S
⊆
C
. Equivalently, the closure
operator
onaset
Υ
is algebraic if it satisfy the following “finitary” property: for
any subset
X
J
=
{
J
(X
)
X
⊆
ω
X
, where
X
⊆
ω
X
means that
X
is
⊆
Υ
,
J
(X)
|
}
a
finite
subset of
X
.
Proposition 52
Ob
DB
be the set of all closed objects
(
w
.
r
.
t
.
the
power-view closure operator T
)
of the
DB
category
.
The following properties for a
database closure are valid
:
•
Let
C
=
Ob
DB
sk
⊂
A closed-set system (
Υ
,
C
) consists of the “total” closed object
(
i
.
e
.,
the top
database instance
)
Υ
∈
C
C
and the set
which is closed under intersections of
,
K
arbitrary subsets
.
That is
,
for any K
⊆
C
∈
C
.
•
The closure operator T is algebraic on the set
Υ
.
