Database Reference
In-Depth Information
T
π
x
i
e
(
_
)
j
/r
i,j
1
≤
j
≤
k
A
|
q
i
∈
=
O
r
i,
1
,...,r
i,k
,r
i
is equal to
e
(
_
)(
t
i
)
∈
M
AB
⇒
=
f
⊆
TA.
From the fact that this information flux is transferred into
B
,
f
TB
, so that
f
⊆
TA
∩
TB
, i.e., by Definition
19
,
f
A
⊗
B
. Each arrow is a composition of a
number of atomic arrows and the intersection of closed objects (their information
fluxes) is always a closed object.
If
f
:
1
≤
j
≤
m
A
j
→
1
≤
i
≤
k
B
i
is a complex morphism with a set of more than
one point-to-point arrow in
f
⊆
f
,
f
ji
⊆
then, for each
f
ji
:
A
j
→
B
i
∈
TA
j
∩
TB
i
.
Hence
f
f
f
B
and
f
(from Definition
22
,
f
), so that
f
A
⊗
B
. From Lemma
9
,
id
A
=
A
⊗
TA
and, from a Definition
19
and
T(TA)
=
TA
,
id
A
⊗
id
B
and we will show that it can be internally represented in
DB
by a monomorphism
in
=
id
A
⊗
id
B
. Thus, we obtain the PO relation
f
A
⊗
B
=
TA
⊗
TB
→
id
A
⊗
id
B
. The fact that
in, in
A
and
in
B
are the morphisms defined by
Theorem
1
and hence they can be obtained from particular schema mappings and a
given R-algebra
α
will be shown in Theorem
4
.
Let us consider a morphism
f
:
f
:
1
≤
j
≤
m
A
j
→
1
≤
i
≤
k
B
i
,
k,m
≥
1, with point-
f
:
f
→
to-point (simple) arrows
(f
ji
:
A
j
→
B
i
)
∈
. Let us define the morphism
in
in
∈
f
ji
}:
f
ji
TA
⊗
TB
with
={
in
ji
={
id
R
:
R
→
R
|
R
→
TA
j
∩
TB
i
|
(f
ji
:
f
f
in
:
f
ji
→
A
j
→
B
i
)
∈
}
. Thus, there is a bijection
σ
:
→
with
in
ji
=
σ(f
ji
)
in
f
TA
j
∩
TB
i
and
in
jk
=
f
jk
for any ptp arrow, and hence
in
f
.
:
1
≤
l
≤
n
X
l
→
f
of morphisms (with
(g
i,ji
:
X
l
→
f
ji
)
∈
For each pair
g,h
g
X
l
→
f
ji
)
T f
ji
=
h
=∅
and
(h
l,ji
:
∈
=∅
so that, from Proposition
6
,
g
l,ji
⊆
f
ji
and
h
l,ji
⊆
T f
ji
=
f
ji
and hence
σ(f
ji
)
in
◦
g
in
◦
h
◦
g
l,ji
∈
and
σ(f
ji
)
◦
h
l,ji
∈
)
in
g
=
◦
in
◦
h
such that
in
.
Thus, they have the same ptp arrows, that is,
σ(f
ji
)
◦
g
l,ji
=
σ(f
ji
)
◦
h
l,ji
:
X
l
→
TA
j
∩
◦
g
=
in
◦
h
:
X
→
B
, i.e., from point 3 of Definition
23
,
TB
i
so that they have the same fluxes, i.e.,
σ(f
ji
)
◦
g
l,ji
=
σ(f
ji
)
◦
h
l,ji
.
g
l,ji
=
σ(f
ji
)
g
l,ji
=
f
ji
∩
However,
σ(f
ji
)
◦
∩
g
l,ji
=
g
l,ji
and, analogously,
σ(f
ji
)
∩
h
l,ji
=
f
ji
∩
h
l,ji
=
h
l,ji
so that
h
l,ji
=
g
l,ji
, i.e., ptp arrows
h
l,ji
=
g
l,ji
:
X
l
→
f
ji
are equal. This holds for all ptp arrows in
g
and
h
so that
g
=
h
. That is,
for each
g,h
such that
in
◦
g
=
in
◦
h
:
X
→
B
, we obtained that
g
=
h
, so that
:
f
→
id
A
∩
id
B
is a monomorphism and, analogously,
(i)
in
in
A
:
f
→
id
A
=
TA
and
in
B
:
f
→
id
B
=
(ii)
TB
.
Claim 2. Let us show that for any two simple objects
A
and
B
,
TA
⊆
TB
implies
the existence of a monomorphism
in
TA
:
TA
→
TB
.
If
TA
⊆
TB
then we have a morphism
f
={
id
R
:
R
→
R
|
R
∈
TA
}:
TA
→
TB
with
f
TA
. Then, from (ii), we have a monomorphism
in
TA
:
f
→
=
id
TB
=
TB
so, by substituting
f
by
TA
, we obtain
in
TA
:
T(TB)
=
TA
→
TB
.
