Database Reference
In-Depth Information
to be maintained in the case when they are equal as well. For example, a complex
object
C
C
has to be represented formally as an indexed complex object
A
1
A
2
with
A
1
=
C
.This“
indexing by position
” is fundamental for the formalization
of complex morphisms between such complex objects. It will be demonstrated that a
disjoint union of two objects (databases) is isomorphic to a separation-composition
of these objects and it will be considered in more details in Sect.
3.3.2
.
For any two simple morphisms
f
A
2
=
α
∗
(
M
AC
)
=
={
α(q
1
),...,α(q
k
),q
⊥
}:
A
→
α
∗
(
M
BD
)
α(q
1
),...,α(q
m
),q
⊥
}:
C
and
g
=
={
B
→
D
, we define a complex mor-
phism
(f
g)
:
A
B
→
C
D
by the disjunctive union of them, that is, by the set
1
,α(q
i
)
|
M
AC
∪
2
,α
q
j
|
M
BD
∪{
q
j
∈
α
∗
(
M
AC
q
i
∈
q
⊥
}=
M
BD
).
Thus, from Definition
15
, its flux is equal to
f
α
∗
(
M
AC
g
=
M
BD
)
=
f
=
Flux(α,
M
AC
)
Flux(α,
M
BD
)
g
∈
Ob
DB
.
Thus, we can consider the following extension of morphisms in
DB
, due to Theo-
rem
1
, based on Definition
15
:
Definition 20
Complex arrows in
DB
presented by Theorem
1
are obtained by the
structural-operations _
_,
[
_
,
_
]
,
_
,
_
and
_
,
_
(all
A,B,C
and
D
are simple
objects), as follows:
1. Morphism
f
g
:
A
B
→
C
D
for any
f
:
A
→
C
and
g
:
B
→
D
;
[
]:
→
:
→
:
→
2. Morphism
f,g
A
B
C
for any
f
A
C
and
g
B
C
;
3. Morphism
f,g
:
A
→
C
D
for any
f
:
A
→
C
and
g
:
A
→
D
;
4. Morphism
f
◦
k,g
◦
l
=[
f,g
]◦
k,l
:
A
→
B
D
→
C
with simple arrows
C
such that
f
0
and
g
0
f
◦
k
:
A
→
B
→
C
and
g
◦
l
:
A
→
D
→
◦
k
=⊥
◦
l
=⊥
k
if
k
if
k
0
0
;
1
and with reductions:
k,g
=
= ⊥
and
g
=⊥
k,g
=⊥
=
0
.
g
=⊥
Hence, a
simple
arrow (or morphism) is an arrow between two simple (non-
separation-composed) objects. Notice that a complex arrow
:
→
C
exists
only as a composition of two complex arrows defined in point 4. It means that
both simple arrows
k,l
k,l
A
:
A
→
C
are composed arrows with
different
intermediate
databases.
We recall that an arrow
1
={
q
⊥
:⊥→⊥}:
A
→
C
, which represents an
empty function, means that, in fact, we have no mapping between
A
and
B
(the
information flux of
⊥
is empty, i.e.,
⊥
1
0
). This fact explains why we
are using only the arrows with nonempty fluxes in point 4 in a representation of
a set of arrows between two fixed objects by
⊥
1
=⊥
={⊥}
. Based on Definitions
15
and
20
for binary compositions, we have the following cases for composition of complex
morphisms (all
A
i
,B
i
,C
i
,
i
,
=
1
,
2, and
A,B,C
are simple objects):
