Database Reference
In-Depth Information
q
B,
1
t
1
∧···∧∀
y
m
q
C,k
+
m
(
y
m
)
⇒
q
B,m
t
m
,
⇒
(3.2)
Consequently, this mapping can be equivalently represented by the graph:
where
M
CA
=
∃
f
A
∀
x
1
q
C,
1
(
x
1
)
⇒
q
A,
1
(
t
1
)
∧···∧∀
x
k
q
C,k
(
x
k
)
⇒
q
A,k
(
t
k
)
and
M
BC
=
∃
f
B
∀
y
1
q
B,
1
(
y
1
)
q
C,k
+
1
t
1
∧···∧∀
y
m
q
B,m
(
y
m
)
⇒
q
C,k
+
m
t
m
.
⇒
If we again introduce the mappings
M
1
={∧{∀
x
i
(r
Ai
(
x
i
)
⇒
r
Ai
(
x
i
))
|
r
Ai
∈
A
}}
and
M
2
={∧{∀
y
i
(r
Bi
(
y
i
)
⇒
r
Bi
(
y
i
))
|
r
Bi
∈
B
}}
then
is a dual graph that can be seen as a cone diagram for the schema database mappings.
It is important to notice that differently from the mapping graphs, where the
schema mappings are expressed by SOtgds, the small sketch categories derived
from such a graph will have the equivalent schema mappings but represented al-
gebraically by the sets of operad's operations, in order to be able to represent each
R-algebra
α
as a
functor
from this sketch category (at the schema-level) into the de-
notational (instance-level)
DB
category. For the mappings that involve the composed
source schema (by separation operator †)
M
={
Φ
}:
A
†
B
→
C
, we cannot directly
use the algorithm
MakeOperads(
)
because we would obtain operad's operations
possibly with relations of both schemas. Consequently, we have to divide this SOtgd
Φ
into a conjunction
Φ
A
∧
M
Φ
B
, where
Φ
A
is an SOtgd whose left-hand sides of im-
plications contain only relational symbols in
A
and
Φ
B
only relational symbols in
