Database Reference
In-Depth Information
3.3
Basic Operations for Objects in
DB
In this section, we will introduce the set of basic operations for instance-databases
used during the definition of database mappings between two (or more) databases.
They are corresponding operators in this base
DB
category for two schema basic
operations, separation † and connection
⊕
, introduced in Sect.
3.1
,usedinorderto
define the schema data mappings.
Consequently, from the fact that the schema separations and schema connections
will be present in the sketch's categories obtained from the schema mapping graphs,
we need, in order to satisfy the functorial semantics for database mapping, their
corresponding operations at the instance-level (i.e.,
DB
category) as well.
As we will see, these operations correspond to important database integration
concepts used in practice.
3.3.1 Data Federation Operator in
DB
The basic database operation is the DBMS's
Data federation
of two database in-
stances
A
and
B
. A federated database system is a type of meta-database manage-
ment system (DBMS) which transparently integrates multiple autonomous database
systems into a
single federated database
. The constituent databases are intercon-
nected via a computer network, and may be geographically decentralized. Since the
constituent database systems remain autonomous, a federated database system is a
contrastable alternative to the (sometimes daunting) task of merging together several
disparate databases. A federated database, or virtual database, is the fully-integrated,
logical composite of all constituent databases in a federated database system. In this
way, we are able to compute the queries with the relations of
both
databases. In fact,
Data Federation technology is just used for such an integration of two previously
separated databases.
Consequently, given any two databases (objects in
DB
)
A
and
B
, the federation
of them (under the common DBMS) corresponds to a
union
, denoted by
rn
(i.e., the
union
that have orig-
inally the same names) of them under the same DBMS, thus, equals to the database
A
rn
B
.
That is, for any two schemas
∪
with the renaming (
rn
) of the relational symbols in
A
and
B
A
and
B
and an interpretation
α
, such that
A
=
α
∗
(
A
)
and
B
=
α
∗
(
B
)
, we have that
A
rn
B
=
α
∗
(
A
⊕
B
)
, and:
Proposition 11
For any two simple databases A and B
,
their federation A
rn
B is
isomorphic to their simple union A
∪
B
.
Proof
Notice that for any relation in
A
B
we have possibly a set of copies of the
same relation, but only with different name: thus,
T(A
rn
B)
∪
=
T(A
∪
B)
and hence
is
rn
:
A
rn
B
→
A
∪
B
obtained from the isomor-
we have the isomorphism
is
−
1
◦
A
rn
B
T(A
rn
B)
and
is
phisms
is
rn
:
→
:
A
∪
B
→
T(A
∪
B)
(from Lemma
9
).
