Database Reference
In-Depth Information
phism
h
B
are 'indexed by position' is fundamental for the definition (in-
dexing) of its ptp arrows. For example, the complex arrow
h
:
A
→
=[
id
C
, id
C
, id
C
]:
C
C
C
→
C
has to be represented formally by
h
:
A
1
A
2
A
3
→
B
1
, with
h
A
1
=
A
2
=
A
3
=
B
1
=
C
, so that
={
h
11
=
id
C
:
A
1
→
B
1
,h
21
=
id
C
:
A
2
→
B
1
,h
31
=
id
C
:
A
3
→
B
1
}
is the set of three ptp arrows and not the singleton
{
. This requirement is fundamental for the definition of ptp arrows
because it takes into consideration the mutually separated databases in the complex
source object
A
and mutually independent morphisms from them into the target
database
B
.
Based on Definitions
15
,
20
and
21
,for
h
id
C
:
C
→
C
}
{
f
i
|
we obtain
h
h
=∅
f
i
∈
}
(that is,
h
is not equal to
h
but only
isomorphic
to it). This isomorphism (instead
of an equality) is based on the fact that
(A,B)
=
A
B
B
A
=
(B,A)
(formally demonstrated by point 2 of the following Lemma
9
) and ptp arrows in
h
,
as explained in Definition
21
, are ordered while the set
is unordered: when
is
h
applied to a set then
any
ordering can be taken in this set.
Notice that the mapping
B
T
for complex morphisms continues to generate the
closed objects. In fact, we can show it recursively. Let
f
B
T
(g)
be
closed objects (i.e., from the fact that power-view operator
T
(Sect.
1.4.1
)isidem-
potent,
T f
=
B
T
(f ),
g
=
=
f
,
T
g
). Then,
T(f
T f
=
f
g
=
g)
=
T
g
g
and hence closed
as well.
Remark
As we have specified in Definition
22
, the path-arrows with the same
simple source and target objects are fused (by union) into a single morphism.
For example, for given two simple databases
A
and
B
, the complex arrows
α(
M
(
1
)
α(
M
(
2
)
:
→
=
=
f,g
A
B
, where
f
AB
)
and
g
AB
)
(with sketch's arrows
M
(n)
(n)
AB
)
(
1
)
AB
={
AB
=
MakeOperads(
M
:
A
→
B
,
n
=
1
,
2, and
M
Φ
}:
A
→
B
,
(
2
)
M
AB
={
Ψ
}:
A
→
B
), will be fused into a single arrow
k
:
A
→
B
, where
α(
M
AB
)
is obtained by fusing the sketch's mappings
M
(
1
)
and
M
(
1
)
AB
k
=
f
∪
g
=
AB
into
M
AB
=
(
M
(
1
)
M
(
1
)
. Thus, based on Definition
13
, the flux of
f
∪
g
is equal to (as in point 3 of Definition
15
):
AB
∪
AB
)
:
A
→
B
T
Δ
α,
AB
AB
∪
Δ
α,
(
1
)
(
2
)
=
f
f,g
∪
g
=
M
M
T
T
Δ
α,
AB
AB
∪
T
Δ
α,
(
1
)
(
2
)
=
M
M
T
Flux
α,
AB
AB
∪
Flux
α,
(
1
)
(
2
)
=
M
M
T
f
g
=
f
=
∪
g.
B
may represent the
two composed independent mapping paths
f
and
g
with different intermediate
databases. Thus,
Generally, when
g
=
f
, the complex arrow
f,g
:
A
→
T(f
∪
f)
T f
=
f
(a closed object), that is, this pair of
f,g
=
=
arrows
f,g
transfers the same information flux from the source to target database
