Database Reference
In-Depth Information
4. The composition of last two types of complex morphisms is a mapping-operad
from
D
into
C
:
M
DC
=[
M
A
1
C
,...,
M
A
n
C
]◦
M
DA
1
,...,
M
DA
n
=
M
A
1
C
◦
M
DA
1
,...,
M
A
n
C
◦
M
DA
n
:
D
→
A
1
†
···
†
A
n
→
C
,
with a set of non-empty mappings
M
A
i
C
◦
M
DA
i
:
D
→
C
,
i
=
1
,...,n
, enclosed
by structural-operator
, all with the same source and target schemas, so
that the resulting information flux is a flux of the union of all these mappings:
_
,...,
_
T
Flux(α,
M
A
i
C
◦
n
.
Flux(α,
M
DC
)
=
M
DA
i
)
|
1
≤
i
≤
5. Each schema mapping above, with structural operators
,
[
,
]
,
,
and
,
,is
denominated as a
complex
-atomic sketch's schema mapping.
For a given mapping-interpretation
α
, we obtain the instance-database complex
morphisms, for example,
α
∗
(
α
∗
(
M
A
1
C
),...,α(
M
A
n
C
)
.It
will be analyzed in details in Sect.
3.3
. We also introduce another kind of binary
symmetric operations for a composition of the schemas and schema mappings, de-
noted by
α
, where
α
is an R-algebra introduced in Sect.
2.4.1
, Definition
10
:
[
M
A
1
C
,...,
M
A
n
C
]
)
=[
]
Definition 16
For each R-algebra
α
, we define an
α
-intersection symmetric binary
operator
α
for the schema mappings, as follows:
1. For any two simple schemas
A
and
B
, their
α
-intersection is defined by:
=
r
T
α
∗
(
)
∩
T
α
∗
(
)
.
A
α
B
=
(S,
∅
),
where
S
∈R|
α(r)
∈
A
B
2. For any two simple schema mappings
M
AC
:
A
→
C
and
M
BD
:
B
→
D
,the
α
-intersection of these two mappings is defined by:
M
AC
α
M
BD
={
}:
A
α
B
→
C
α
D
Φ
,
where
Φ
is a tgd
x
r(
x
)
r(
x
)
|
Flux
α, MakeOperads(
M
AC
)
∈
A
α
B
∀
⇒
∈
r
and
α(r)
M
BD
)
.
Flux
α, MakeOperads(
∩
It is easy to show that for a simple schema
A
and R-algebra
α
, with the instance-
A
α
A
α
∗
(
database
A
=
A
)
, the schema of the instance-database
TA
is equal to
, that
is,
TA
=
α
∗
(
A
α
A
)
.
More over, for
A
α
∗
(
α
∗
(
α
∗
(
A
α
B
=
A
)
and
B
=
B
)
,
TA
∩
TB
=
)
.
