Database Reference
In-Depth Information
Lemma 8
Let us define the function
:
Mor
DB
→
Mor
DB
such that for any arrow
f
:
A
→
B where A
=
1
≤
j
≤
m
A
j
and B
=
1
≤
i
≤
k
B
i
,
m,k
≥
1,
h
m
1
,...,h
mk
:
(f )
h
11
,...,h
1
k
,...,
A
→
B,
f
1
otherwise
.
This mapping is an identity for the simple arrows
.
For any complex arrow it
returns an equal arrow that contains the maximal number of simple arrows in its
structural representation
.
For a given set S
where h
ji
=
f
ji
if (f
ji
:
A
j
→
B
i
)
∈
;
⊥
f
=
of ptp arrows between A and B
,
by
(A,B,S)
(f ) we denote a canonical arrow obtained from S
.
Proof
There are the following cases:
1. The case of
m
f
={
=
k
=
1, so that
f
:
A
1
→
B
1
is a simple arrow. Then
f
}
if
f
0
;
=⊥
otherwise.
Consequently,
∅
f
={
f
=∅
1
(f )
=
f
if
f
}
,or
(f )
=⊥
if
, but from the
= ⊥
fact that in this case
f
0
=⊥
1
, from Definition
23
, we obtain again that
f
.
2. The case of
m
(f )
=
2 (a complex arrow) and
f
·
k
≥
=∅
. Let us consider the following
cases:
2.1.
m
=
1. Then,
(f )
=[
h
11
,...,h
1
k
]=
(in
[
_
]
we have only one element)
1
,...,k
, so that
(f )
f
1
=
h
11
,...,h
1
k
where
h
1
i
=⊥
=
=∅=
for all
i
and hence
(f )
=
f
is maximal with 1
·
k
=
k
≥
2 simple arrows.
1
2.2.
k
=
1. Then,
(f )
=[
h
11
,...,
h
m
1
]=[
h
11
,...,h
m
1
]
where
h
j
1
=⊥
1
,...,m
, so that
(f )
f
for all
j
=
=∅=
and hence
(f )
=
f
is maximal
with
m
·
1
=
m
≥
2 simple arrows.
2.3. Both
m,k
≥
2. Then,
(f )
=[
h
11
,...,h
1
k
,...,
h
m
1
,...,h
mk
]
where
1
,...,k
, so that
(f )
f
1
h
ji
=⊥
for all
j
=
1
,...,m
and
i
=
=∅=
and
hence
(f )
=
f
is maximal
k
·
m
≥
2 simple arrows.
f
=∅
3. The case of
m
. Then, as in case 2, we
have tree possible cases, and in each of them we obtain that
(f )
·
k
≥
2 (a complex arrow) and
f
=
and hence
(f )
=
f
is maximal
k
·
m
≥
2 simple arrows.
1
in
this canonical structural representation where, for each simple object
A
j
in a
A
,we
have a full cone of simple arrows
It is easy to see that for any
f
,
inserts only the new empty arrows
⊥
h
j
1
,...,h
jk
:
A
j
→
B
from
A
j
into each simple
object in
B
i
for
i
=
1
,...,k
. For example, consider
(k)
for
k
=
k
11
[
k
2
,
2
,k
3
,
2
]:
A
1
B
2
.
Moreover, we are able to construct a well-defined complex arrow between any
two complex objects by specifying the set
S
of ptp arrows. This is useful also when
the set
S
of ptp arrows is empty. For example,
A
2
A
3
→
B
1
1
,f
21
]:
(A
1
A
2
,B
1
,
{
f
21
}
)
=[⊥
B
1
,or
1
,
1
A
1
B
2
.
We can show the following important isomorphisms for (also complex) instance-
databases in
DB
:
A
2
→
(A
1
,B
1
B
2
,
∅
)
=⊥
⊥
:
A
1
→
B
1
Lemma 9
Fo r a n y A,B,C
∈
Ob
DB
,
we have the following isomorphisms
:
