Database Reference
In-Depth Information
7. For
f
i
:
A
→
B,
1
≤
i
≤
n
and
g
:
B
→
C
,
1
◦
f
1
,...,f
n
=
:
→
|
|≥
; ⊥
:
→
g
S
A
C
if
S
1
A
C
otherwise
f
i
|
g
0
,
1
(where, from Definition
20
,
S
={
g
◦
◦
f
i
=⊥
≤
i
≤
n
}
);
8. For
g
i
:
B
→
C,
1
≤
i
≤
n
and
f
:
A
→
B
,
1
g
1
,...,g
n
◦
=
:
→
|
|≥
; ⊥
:
→
f
S
A
C
if
S
1
A
C
otherwise
|
g
i
◦
0
,
1
={
g
i
◦
=⊥
≤
≤
}
(where, from Definition
20
,
S
f
f
i
n
).
By these rules of composition of complex arrows, we obtain that the resulting
arrow is a composition of structural-operators
(
_
,..,
_
)
,
[
_
,..,
_
]
,
_
,..,
_
, and
_
,..,
_
. The 'point-to-point' (ptp) arrow is a
nonempty
(with flux different from
0
) simple arrow (a path of only simple atomic arrows) between two simple objects
that compose the source and target database of this complex arrow: if we have more
than one simple arrow between the same source and target simple objects then we
fuse them (by union) into a single ptp arrow, as follows:
⊥
Definition 22
For any complex arrow
h
:
A
→
B
between 'indexed by position'
=
1
≤
j
≤
m
A
j
and
B
=
1
≤
i
≤
k
B
i
,m,k
complex objects
A
≥
1, we define its set
of ptp morphisms by:
h
ji
=
f
l
:
h
A
j
→
B
i
|
f
l
is a composition of simple arrows in
h
k
.
0
=∅|
such that
f
l
=⊥
1
≤
j
≤
m,
1
≤
i
≤
=
h
is equal (up to
Then we extend the mapping
B
T
to complex arrows, too.
B
T
(h)
isomorphism) to the object
⎧
⎨
{
B
T
(h
ji
)
|
h
ji
∈
}=
{
h
ji
|
h
ji
∈
h
h
h
}
if
|
|≥
2
;
h
=
h
ji
h
B
T
(h
ji
)
if
={
h
ji
};
⎩
0
⊥
otherwise
.
Notice that for a given complex arrow
h
each of its ptp arrows has a nonempty
information flux (different from
1
arrow.
In the previous example for composition of complex arrows, the resulting arrow is
composed of point-to-point arrows. For example, in rule 5, we obtained the complex
arrow
0
) so that a ptp arrow is different from
⊥
⊥
f
◦
k,g
◦
k
:
A
1
→
C
1
C
2
where
f
◦
k
:
A
1
→
C
1
and
g
◦
k
:
A
1
→
C
2
are two point-to-point simple arrows. In rule 3, we obtained the complex arrow
[
f
◦
k,g
◦
l
]:
A
1
A
2
→
C
where
f
◦
k
:
A
1
→
C
and
g
◦
l
:
A
2
→
C
are two
point-to-point simple arrows, etc.
This property extends by induction to any composition of complex arrows.
The fact that the source and target complex objects
A
and
B
of a complex mor-
