Database Reference
In-Depth Information
Proof
Let us show that for a given
DB
simple arrow
f
:
B
→
C
and hence, from
Theorem
1
, obtained from a schema mapping
M
BC
:
B
→
C
and an R-algebra
α
,
α
∗
(MakeOperads(
α
∗
(
such that
f
=
M
BC
))
with the simple databases
B
=
B
)
and
)
,thereisalsoanarrow
id
A
rn
f
in
DB
. In fact, let us consider the identity
schema mapping
α
∗
(
C
=
C
M
AA
:
A
→
A
and the sketch's mapping
MakeOperads(
M
BC
)
M
AA
)
rn
MakeOperads(
M
=
:
A
⊕
B
→
A
⊕
C
.
Then,
α
∗
MakeOperads(
M
BC
)
M
AA
)
rn
MakeOperads(
α
∗
MakeOperads(
M
AA
)
rn
α
MakeOperads(
M
BC
)
=
id
A
rn
f
A
rn
B
A
rn
B
=
:
→
A
rn
B
A
rn
B.
=
A
⊕
(f )
:
→
A
rn
B
and
A
A
rn
C
, we obtain the
DB
arrow
From the fact that
A
⊕
B
⊕
C
id
A
rn
f
:
A
⊕
B
→
A
⊕
C
.
It is easy to verify that for any two simple databases
A
and
B
the database
A
rn
B
is different from
A
∪
B
only in the fact that by renaming we can have the
copies of the same relations. Thus, the set of views (obtained by SPRJU queries) of
both databases are equal. Consequently,
T(A
rn
B)
=
T(A
∪
B)
=
T(TA
∪
TB)
=
T(TA
rn
TB)
, that is,
A
TB
, and
A
rn
B
⊕
=
⊕
⊕
B
TA
A
B
.
Now we can verify that
A
⊕
_ is an endofunctor. In fact, for any object
B
=
1
≤
i
≤
k
B
i
,
k
1 and its identity arrow
id
B
=
1
≤
i
≤
k
(id
B
i
:
≥
B
i
→
B
i
)
,
id
A
rn
id
B
i
A
⊕
(id
B
)
id
B
=
:
A
⊕
B
i
→
A
⊕
B
i
|
id
B
i
:
B
i
→
B
i
∈
A
.
id
B
=
⊕
(id
B
i
)
:
A
⊕
B
i
→
A
⊕
B
i
|
id
B
i
:
B
i
→
B
i
∈
Thus, for each simple point-to-point arrow
A
⊕
(id
B
i
)
:
A
⊕
B
i
→
A
⊕
B
i
, its flux is
(id
A
rn
id
B
i
)
=
T(A
rn
B
i
)
A
⊕
(id
B
i
)
=
id
A
rn
B
i
=
=
A
⊕
B
i
. Consequently, for the
identity arrows is valid the functorial property,
A
⊕
(id
B
i
)
=
id
A
⊕
B
i
:
A
⊕
B
i
→
A
⊕
B
i
.
From the fact that for any simple object (i.e., a database)
B
,
A
B
, each point-
to-point arrow, resulting by application of this endofunctor, contains a sub-arrow
id
A
. Thus, based on composition of operads in point 4 of Definition
8
(Sect.
2.4
),
⊆
A
⊕
