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In-Depth Information
=
1
≤
j
≤
m
C
j
and
B
Claim 4. For any two strictly-complex objects
C
=
1
≤
i
≤
k
B
i
,let
is
is
:
C
→
B
be an isomorphism. Thus,
{
dom(is
i
)
|
is
i
∈
}=
is
{
C
1
,...,C
m
}
and
{
cod(is
i
)
|
is
i
∈
}={
B
1
,...,B
k
}
(from the fact that
is
◦
is
1
=
id
B
=
1
≤
i
≤
k
id
B
i
id
A
=
1
≤
j
≤
m
id
A
j
. Then, from point 1
of this proposition, each point-to-point arrow
is
i
∈
and
is
−
1
◦
is
=
is
is an isomorphism with
is
|
k
. Let us suppose that
m>k
, so that
we have two different isomorphisms
is
i
, is
j
∈
|≥
max
(m,k)
. Let us show that
m
=
is
with
dom(is
i
)
=
dom(is
j
)
and
is
OP
i
, is
O
j
]=
is
OP
i
is
O
j
=
=
=
is
i
, is
j
◦[
is
i
◦
, is
j
◦
cod(is
i
)
cod(is
j
)
B
l
. Thus,
id
B
l
, id
B
l
=
id
B
l
. Hence
is
cannot be an isomorphism, which is a contradiction.
Analogously, if we suppose that
m<k
, we again obtain a contradiction. Conse-
quently, it must be that
m
=
k
.
0
,themor-
Remark
Claim 2 of this proposition explains the fact that when
C
=⊥
phism
k
C
where both point-to-point arrows are identities
(thus, isomorphisms) is not an isomorphism (but only a monic and epic arrow, as
established by Proposition
5
). While, for non-strictly-complex objects, we have the
isomorphisms when
m
=[
id
c
, id
c
]:
C
C
→
0
=
k
as well: let us consider the isomorphism
C
⊥
C
,
0
) and
k
when
m
=
2(
A
1
=
C
,
A
2
=⊥
=
1 with
B
1
=
C
.
We obtain the following important results for the
monomorphisms
in
DB
:
Proposition 7
The following monomorphisms in
DB
category are derivable from
the morphisms and from the inclusions of the objects
:
1.
The conceptualized object f
=
B
T
(f )
,
obtained from a morphism f
:
A
−→
B
,
is a closed object in
DB
such that f
T f
B
.
This PO relation may
be represented internally in
DB
by the monomorphism in
=
A
⊗
:
f
→
id
A
⊗
id
B
and its derivations in
A
:
f
→
id
A
and in
B
:
f
→
id
B
where id
A
=
TA
=
1
≤
j
≤
m
TA
j
,
m
=
1
≤
i
≤
k
TB
i
,
k
1,
and id
B
=
≥
TB
≥
1,
with ptp arrows
:
in
ji
={
,
f
∈
f
ji
}:
f
ji
id
R
|
R
→
S
ji
|
(f
ji
:
A
j
→
B
i
)
∈
in
in
A
in
B
where S
ji
=
TA
j
∩
TB
i
for
,
S
ji
=
TA
j
for
,
and S
ji
=
TB
i
for
.
2.
For any two objects TA
⊆
TB there exists a monomorphism in
TA
:
TA
→
TB
.
Proof
Claim 1. Each conceptualized object (information flux) of an atomic arrow
is a closed object. In fact, from Definition
13
, for a simple atomic morphism
f
=
α
∗
(
M
AB
)
B
, both databases
A
and
B
are simple objects (sets of relations)
and so are
TA
and
TB
.Thus,if
f
:
A
→
0
0
0
. Otherwise,
=⊥
(empty flux) then
T
⊥
=⊥
if
f
0
=⊥
then
T
T
π
x
i
e
(
_
)
j
/r
i,j
1
≤
j
≤
k
A
|
q
i
∈
T f
=
O
r
i,
1
,...,r
i,k
,r
i
is equal to
e
(
_
)(
t
i
)
∈
M
AB
⇒
