Database Reference
In-Depth Information
Notice that for any two complex objects
A
and
B
such that
TA
⊆
TB
, these
≥
complex objects must have the
same
number
k
2 of simple databases, i.e., it must
be true that
A
=
1
≤
i
≤
k
A
i
and
B
=
1
≤
i
≤
k
B
i
with
A
i
⊆
B
i
,for1
≤
i
≤
k
, and
TA
=
1
≤
i
≤
k
TA
i
⊆
1
≤
i
≤
k
TB
i
=
TB
, and hence we obtain the monomorphism
in
TA
=
1
≤
i
≤
k
(in
TA
i
:
TA
i
→
TB
i
)
.
Thus, we are able to define all kinds of morphisms in the
DB
category as follows:
Proposition 8
The following properties for morphisms are valid
:
1.
Each arrow f
B such that f
TA
(
and in special case when f
:
A
→
=
TA
)
is
a monomorphism
(
monic arrow
).
2.
Each arrow f
B such that f
TB
(
and in special case when f
:
A
→
=
TB
)
is an epimorphism
(
epic arrow
).
3.
Each arrow f
:
A
→
B such that f
TA
TB
(
and in special case when
f
=
TA
=
TB
)
is an isomorphism
(
iso arrow
).
Proof
Claim 1. For a simple arrow it holds from Corollary
9
, so consider a strictly-
complex source object
A
. For a given complex arrow
f
:
1
≤
j
≤
m
A
j
→
1
≤
i
≤
k
B
i
,
f
. Then, from this isomorphism and the fact
that
f
is strictly complex, also
TA
let there be an isomorphism
TA
=
1
≤
j
≤
m
A
j
is strictly complex, so we obtain
f
|
f
→{
and a bijection
σ
:
m
=|
1
,...,m
}
such that for each
f
ji
:
A
j
→
B
i
,
f
ji
=
TA
j
. Thus, from above, this ptp (simple) arrow
f
ji
is monic.
For each pair of morphisms
g,h
TA
σ
(f
ji
)
=
:
1
≤
i
≤
k
X
i
→
g
A
(with
(g
lj
:
X
l
→
A
j
)
∈
=
h
TA
j
and
h
lj
⊆
∅
and
(h
lj
:
X
l
→
A
j
)
∈
=∅
, it holds from above that
g
lj
⊆
TA
j
)
f
◦
g
f
◦
h
if
f
, i.e., they
have the same ptp arrows, that is,
f
ji
◦
g
lj
=
f
ji
◦
h
lj
:
X
l
→
B
i
and hence they have
the same fluxes, i.e.,
f
ji
◦
g
lj
=
f
ji
◦
h
lj
. However,
f
ji
◦
g
lj
=
f
ji
∩
g
lj
=
TA
j
∩
g
lj
=
g
lj
and
f
ji
◦
h
lj
=
f
ji
∩
h
lj
=
TA
j
∩
h
lj
=
h
lj
, so that
h
lj
=
g
lj
, i.e.,
h
lj
=
g
lj
:
X
l
→
A
j
. It holds for all ptp arrows in
g
and
h
so that
g
=
h
and, consequently,
f
is an monomorphism. If
A
is not strictly complex, then we have an isomorphism
(which is monic as well)
is
◦
g
=
f
◦
h
:
X
→
B
then, from point 3 of Definition
23
,
=
A
(thus,
TA
TA
) where
A
is strictly complex,
:
A
→
f
◦
B
(where
f
:
A
→
so that
f
=
is
:
A
→
B
is a monomorphism above with
f
f
, i.e.,
f
TA
TA
) is a composition of two monomorphism, with
f
TA
and hence it is monic as well.
Claim 2. For a simple arrow it holds from Corollary
9
and so consider a strictly
complex target object
B
. For a given complex arrow
f
:
1
≤
j
≤
m
A
j
→
1
≤
i
≤
k
B
i
,
lettherebeanisomorphism
TB
f
. Then, from this isomorphism and the fact that
f
and also
TB
=
1
≤
j
≤
k
B
j
are strictly complex, we obtain
k
f
=|
|
and a bijection
such that for each
f
ji
:
A
j
→
B
i
,
f
ji
=
TB
σ
(f
ji
)
=
TB
i
. Thus,
from above, this ptp (simple) arrow
f
ji
is epic. For each pair of morphisms
g,h
:
f
σ
:
→{
1
,...,k
}
B
→
1
≤
i
≤
k
X
i
(with
(g
il
:
B
i
→
X
l
)
∈
g
h
=∅
and
(h
il
:
B
i
→
X
l
)
∈
=∅
,it
TB
i
and
h
il
⊆
holds from above that
g
il
⊆
TB
i
)if
g
◦
f
=
h
◦
f
:
A
→
X
then,
from point 3 of Definition
23
,
g
◦
f
h
◦
f
=
, i.e., they have the same ptp arrows, that is,
