Database Reference
In-Depth Information
Proof
Let us show that for any two simple arrows
f
D
in
the
DB
category, also the arrow
f
⊗
g
is a
DB
category arrow (that is, it satisfies
Theorem
1
).
From the fact that
f
and
g
are the arrows in
DB
, it holds from Theorem
1
that
there exist the schema mappings
:
A
→
C
and
g
:
B
→
M
AC
:
A
→
C
M
BD
:
B
→
D
, and an R-algebra
α
such that
f
=
α
∗
(MakeOperads(
M
AC
))
and
g
=
α
∗
(MakeOperads(
M
BD
))
with
A
,
α
∗
(
α
∗
(
α
∗
(
α
∗
(
=
)
.
Hence, we can specify the
α
-intersection schema mapping (from Definition
16
),
M
AC
α
M
BD
={
A
),B
=
B
),C
=
C
)
and
D
=
D
}:
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
,α(r)
∈
M
BD
)
.
Flux
α, MakeOperads(
∩
Consequently,
MakeOperads
M
AC
α
M
BD
M
=
1
r
|
Flux
α, MakeOperads(
M
AC
)
∈
A
α
B
=
r
,α(r)
∈
Flux
α, MakeOperads(
M
BD
)
∩
∪{
1
r
∅
}
1
r
|
r
∈
A
α
B
,α(r)
∈
f
∩
g
=
∪{
1
r
∅
}
A
α
B
B
α
D
is a sketch category arrow from the schema
into the schema
, with
α
∗
α
∗
A
α
B
C
α
D
=
TA
∩
TB
=
A
⊗
B,
=
TC
∩
TD
=
C
⊗
D,
so that
α(
1
r
)
g
∈
A
α
B
α
∗
(
M
)
∈
f
=
=
id
R
:
R
→
R
|
r
,R
=
α(r)
∩
∪{
q
⊥
}
=
f
⊗
g
:
A
⊗
B
→
C
⊗
D.
C
,
f
:
Let us show that for any four simple arrows
f
:
A
→
C
→
E
,
g
:
B
→
D
f
={
and
g
:
, and analogously
for all other arrows), where all
A,B,C,D,E
and
F
are simple objects,
f
◦
B
→
F
(with, from Definition
22
in Sect.
3.2
,
f
}
f
⊗
g
◦
g
=
f
⊗
g
◦
(f
⊗
g)
:
A
⊗
B
→
E
⊗
F.
