Database Reference
In-Depth Information
Corollary 33
For any simple object
(
database
)
A
,
gfp
A
T
P
=
gfp
T
P
=
⊥
0
,
T
(A)
=
T
∞
∞
and hence the diagram
(
c
.1)
above can be substituted by
0
)
is the set of all (also infinite) term-trees with leafs equal to
Note that
T
∞
(
⊥
0
, that is, it is a gfp obtained by the union of the following
⊥∈⊥
the empty relation
chain:
∅⊆
T
P
(
∅
)
⊆
T
P
T
P
(
∅
)
⊆
T
P
T
P
T
P
(
∅
)
⊆···
,
T
P
(
0
)
.
∅
⊆
⊥
and hence
)
T
∞
(
Example 41
Let us consider a Data Integration framework, described in Sect.
4.2.3
,
with the foreign key constraints (FK) in the global schema
given by Example
27
(Sect.
4.2.4
). The database mapping graph of this GAV data integration system,
at the logical sketch's level, can be represented by the graph composed of two
mapping-operad arrows,
M
SG
T
=
G
MakeOperads(TgdsToSOtgd(
M
))
:
S
−→
G
T
,
where
G
T
=
(S
G
,
∅
)
is used for the retrieved (from the source database
S
only)
data and, consequently, has the same global database schema as
but with empty
set of integrity constraints, and the new inter-schema mapping-operad
M
G
T
G
=
q
i
|
G
q
i
=
(
_
)
1
(
x
i
)
O
r
i
,r
i
for each
r
i
∈
G
T
and the same relation
r
i
∈
G
∪{
(
_
)(
x
i
)
∈
⇒
1
r
∅
}:
G
T
−→
G
that represents the inclusion between retrieved global database
ret(
I
,D)
and the
complete canonical global database
can(
,D)
(obtained by satisfaction of the in-
tegrity constraints
Σ
G
for the foreign key constraints).
The integrity-constraints mapping operad
I
MakeOperads
EgdsToSOtgd
Σ
egd
G
T
GA
=
TgdsToCanSOtgd
Σ
tgd
G
:
G
→
A
∧
is satisfied because we assumed that all primary key-integrity constraints for a global
schema are satisfied by the retrieved database
ret(
I
,D)
, and foreign-key constraints
are satisfied by an insertion of the new tuples (with the Skolem constants) in the
relations as explained in Sect.
4.2.4
.
