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Then,
F
sk
(g)(F
sk
(f )(S))
=
F
sk
(g)(S
1
)
={
(l,R)
|
(i,R)
∈
S
1
,R
∈
g
il
,(g
il
:
f
}={
f
∈
f
ji
,(f
ji
:
B
i
→
C
l
)
∈
(l,R)
|
(j,R)
∈
S,R
A
j
→
B
i
)
∈
,R
∈
f
}={
∈
f
ji
∩
g
il
,(g
il
:
B
i
→
C
l
)
∈
(l,R)
|
(j,R)
∈
S,R
g
il
,(g
il
◦
f
ji
:
g
◦
f
}={
∈
g
il
◦
A
j
→
C
l
)
∈
(l,R)
|
(j,R)
∈
S,R
f
ji
,(g
il
◦
f
ji
:
A
j
→
g
◦
f
}=
f )(S)
.
Consequently, the compositional property of this functor is valid for the arrows,
F
sk
(g
C
l
)
∈
F
sk
(g
◦
F
sk
(f )
.
3. Let us show that
F
sk
(f )
◦
f)
=
F
sk
(g)
◦
F
sk
(h)
implies
f
=
=
h
, i.e.,
F
sk
is a faithful functor,
as follows:
3.1.
m
=
k
=
1. Thus
f
=
f
1
,...,f
n
:
A
1
→
B
1
,
n
≥
1 and
g
=
g
1
,...,g
n
1
:
f
A
1
→
B
1
,
n
1
≥
1. Then
is a singleton set composed of a unique ptp ar-
row
f
11
:
A
1
→
B
1
which is obtained by fusing (Lemma
7
) all arrows
f
l
,l
=
g
1
,...,n
, and, analogously,
is a singleton set composed of a unique ptp
arrow
g
11
:
A
1
→
B
1
, obtained by fusing all arrows
g
l
,l
=
1
,...,n
1
.They
are defined by their information fluxes,
f
11
={
R
|
R
∈
A
1
,F
sk
(f )(
{
R
}
)
=
∅}={
R
|
R
∈
A
1
,F
sk
(g)(
{
R
}
)
=∅}=
g
11
, and hence from Definition
23
,
f
11
=
g
11
, thus
f
g
={
f
11
}={
g
11
}=
and hence
f
=
g
.
f
≥
=
1. Thus each ptp arrow
f
j
1
:
A
j
→
B
1
∈
3.2.
m
2
,k
is defined by
its information flux,
f
j
1
={
R
|
R
∈
A
j
,F
sk
(f )(
{
(j,R)
}
)
=∅}={
R
|
R
∈
A
j
,F
sk
(g)(
{
(j,R)
}
)
=∅}=
g
j
1
, and hence from Definition
23
,
f
j
1
=
g
j
1
:
m
, thus
f
g
A
j
→
≤
≤
=
B
1
. It holds for all ptp arrows with 1
j
and hence
f
=
g
.
f
3.3.
m
=
1
,k
≥
2. Thus each ptp arrow
f
1
i
:
A
1
→
B
i
∈
is defined by
its information flux,
f
1
i
={
R
|
R
∈
A
1
,(i,R)
∈
F
sk
(f )(
{
R
}
)
}={
R
|
R
∈
A
1
,(i,R)
∈
F
sk
(g)(
{
R
}
)
}=
g
1
i
, and hence from Definition
23
,
f
1
i
=
g
1
i
:
f
g
A
1
→
B
i
. It holds for all ptp arrows with 1
≤
i
≤
k
, thus
=
and hence
f
=
g
.
f
3.4.
m,k
is defined by its in-
formation flux,
f
ji
={
R
|
R
∈
A
j
,(i,R)
∈
F
sk
(f )(
{
(j,R)
}
)
}={
R
|
R
∈
A
j
,(i,R)
∈
F
sk
(g)(
{
(j,R)
}
)
}=
g
ji
, and hence from Definition
23
,
f
ji
=
g
ji
:
≥
2. Thus each ptp arrow
f
ji
:
A
j
→
B
i
∈
A
j
→
B
i
. It holds for all ptp arrows with 1
≤
j
≤
m
and 1
≤
i
≤
k
,
thus
f
g
and hence
f
=
g
.
Consequently, also
F
DB
=
F
sk
◦
T
is a faithful functor. Thus,
DB
sk
and
DB
are
concrete categories.
=
8.1.5 Algebraic Database Lattice
This section is an extension of Sect.
3.2.5
dedicated to the partial ordering '
'for
databases with an introduction of the PO subcategory
DB
I
.
We recall that the symbol '
=
' is used to express the fact that both sides name
the same objects, whereas '
' is used to build equations which may or may not be
true for particular elements. Note that in the case of equations between arrows in
≈
