Database Reference
In-Depth Information
Proposition 3
For each schema mapping
M
AB
:
A
→
B
with
(Φ
E
,Φ
M
)
=
DeCompose(
M
AB
),
let us define the mapping-operads
MakeOperads
{
Φ
E
}
and
MakeOperads
{
Φ
M
}
.
M
E
=
M
M
=
Let S
={
χ
1
,...,χ
m
}
M
AB
.
If
for a mapping-interpretation α
(
with A
=
α
∗
(
A
) and B
=
α
∗
(
B
)
)
such that for each
implication χ
i
∈
be the set of implications of the normalized SOtgd of
S
,
q
Ai
(
x
i
)
⇒
r
i
(
t
i
)
,
and its derived corresponding implications
q
Ai
(
x
i
)
⇒
r
q
i
(
y
i
) in S
E
and (r
q
i
(
y
i
)
∧
ψ
i
(
x
i
))
⇒
r
i
(
t
i
) in S
M
(
in the algorithm
Decompose
),
we have α(r
q
i
)
=
π
y
i
q
Ai
(
x
i
)
A
,
then
:
Flux
α, MakeOperads(
M
AB
)
=
Flux(α,
M
E
)
∩
Flux(α,
M
M
).
For such a mapping-interpretation α with the instance-mapping
f
=
α
∗
MakeOperads(
M
AB
)
:
A
→
B,
α
∗
(
M
E
)
we denote the decomposition of f by two instance-mappings ep(f )
=
:
α
∗
(
M
M
)
→
=
:
→
C
=
∅
) and S
C
=
A
TC and in(f )
TC
B
,
with schema
(S
C
,
q
i
∈
M
E
∂
1
(q
i
)
α
∗
(S
C
)
.
={
r
q
1
,...,r
q
m
}
and C
=
Proof
Let
M
AB
=
MakeOperads(
M
AB
)
={
q
1
,...,q
m
,
1
r
∅
}
with
q
i
=
v
i
·
q
A,i
∈
O(r
i,
1
,...,r
i,k
,r
i
)
,for1
q
1
,...,q
m
,
1
r
∅
}
≤
≤
={
i
m
, so that
MakeOperads(Φ
E
)
,
with
q
i
=
v
i
·
q
A,i
∈
O(r
i,
1
,...,r
i,k
,r
q
i
)
,for1
≤
i
≤
m
, and
MakeOperads(Φ
M
)
=
q
1
,...,q
m
,
1
r
∅
}
O(r
q
i
,r
i
)
.
First of all, from the fact that
α(r
q
i
)
=
π
y
i
q
Ai
(
x
i
)
A
,for1
with
q
i
∈
{
≤
i
≤
m
, it follows,
from Definition
11
for a mapping-interpretation, that (a.1.) the functions
α(v
i
)
and
α(v
i
)
are equal. Then, from Definition
13
,
(i)
Flux(α,
M
E
)
q
i
=
v
i
·
q
A,i
=
=
Flux(α,
M
AB
)
=
T
{
π
y
i
e
[
(
_
)
j
/r
i,j
]
1
≤
j
≤
k
A
|
M
E
,q
i
∈
(e
⇒
(
_
)(
y
i
))
∈
O(r
i
1
,
1
,...,r
i
k
,k
,r
q
i
)
and
y
i
= ∅} =
T
{
π
y
i
q
A,i
(
x
i
)
A
|
1
≤
i
≤
m
and
y
i
=∅}=
T
{
α(r
q
i
)
|
1
≤
i
≤
m
and
y
i
=
TC
(because
α(v
i
)
is an identity function and hence an
injection
as well,
∅}=
for 1
≤
i
≤
m
such that
y
i
=∅
).
q
i
=
v
i
·
q
C,i
=
(ii)
Flux(α,
M
M
)
=
T
{
π
y
i
e
[
(
_
)
1
/r
q
i
]
C
|
(e
⇒
(
_
)(
t
i
))
∈
M
M
,q
i
∈
O(r
q
i
,r
i
)
and
y
i
=∅}=
T
{
π
y
i
r
q
i
(
y
i
)
∧
ψ
i
(
x
i
)
C
|
1
≤
i
≤
m
and
y
i
=∅}=
m
and
y
i
=∅}=
(since
ψ
i
(
d
)
is true when
q
A,i
(
d
)
is true, and
q
A,i
(
x
i
)
{
π
y
i
C
|
≤
≤
T
r
q
i
(
y
i
)
1
i
⇒
r
q
i
(
y
i
)
is satisfied
by
α
)
=
T
{
α(r
q
i
)
|
if
α(v
i
)
is an injection for all 1
1
≤
i
≤
m
and
y
i
=∅}
≤
i
≤
m
0
such that
y
i
=∅;⊥
otherwise,
if
α(v
i
)
(equal to
α(v
i
)
)is
=
(from (a.1.))
=
T
{
α(r
q
i
)
|
1
≤
i
≤
m
and
y
i
=∅}
0
an injection for all 1
≤
i
≤
m
such that
y
i
=∅;⊥
otherwise,
=
T
{
π
y
i
q
Ai
(
x
i
)
A
|
1
≤
i
≤
m
and
y
i
=∅}
if
α(v
i
)
is an injection for all
