Database Reference
In-Depth Information
is a pullback square (the commutative diagram on the right is its logic dual with
false(
A
which is repre-
sented in this category by the injective function (a monomorphism)
in
B
:
∗
)
=
0). The PO relation in
Set
is a set-inclusion
B
⊆
→
B
A
,
where the characteristic function
C
in
B
represents the image of this injective func-
tion
in
B
.
In order to generalize this idea for
DB
, for any two database instances
A
=
A
j
,m
≥
1
and
B
=
B
i
,k
≥
1
,
1
≤
j
≤
m
1
≤
i
≤
k
we have to use the PO relation '
'inthe
DB
category (see Theorem
6
in Sect.
3.2.5
)
and hence
B
is a subobject of
A
iff
B
A
, defined by Definition
19
with a mapping
σ
:{
1
,...,k
}→{
1
,...,m
}
(if
A
and
B
are simple objects, i.e.,
m
=
k
=
1, then
the subobject relation reduces to the set inclusion
TB
⊆
TA
), which is represented
f
={
by a monomorphism
f
:
B
→
A
with
f
iσ(i)
:
B
i
→
A
σ(i)
|
1
≤
i
≤
k
}
(see
Theorem
6
).
The image of a monic arrow
f
in
DB
corresponds to its information flux
f
transferred into the target object
A
, that is, the characteristic mapping
C
f
:
A
→
Ω
in
DB
must have exactly the information flux of the monomorphism
f
. Thus, for this
subobject monomorphism, we obtain
f
=
f
1
≤
i
≤
k
f
iσ(i)
=
1
≤
i
≤
k
TB
i
=
TB
, so that from the fact that
TB
B
, and from the fact that two isomorphic
objects are also behaviorally equivalent, we obtain
f
≈
TB
≈
B
and hence the
subobject relation
f
A
. That is, instead of the subobject
B
of
A
, we can
use the equivalent characteristic-mapping subobject
f
≈
B
A
(in the case of simple
database, we obtain
f
=
TB
⊆
TA
). From the extreme case of the monomorphism
Ω
,
C
f
f
=
id
Υ
:
Υ
→
Υ
we obtain that for its characteristic mapping
C
f
:
Υ
→
f
=
id
Υ
=
Υ
which is a top object of the complete
algebraic database lattice
L
DB
=
(Ob
DB
,
,
⊗
,
⊕
)
in Proposition
51
of Sect.
8.1.5
,
and hence we can use the bottom and top objects of this complete lattice for the
false and true logical values in
DB
( as in the case of the many-valued logics with
complete lattice of truth-values).
The constant “true” morphism in
DB
for the subobject classifier with the tar-
get object
Ω
Υ
. Consequently, we obtain
Ω
=
=
Υ
is its identity morphism
true
=
id
Υ
:
Υ
→
Ω
. The opposite
(
ω
⊥
1
1
)
“false” morphism in
DB
is the empty morphism
false
Ω
.
This interpretation for the true and false morphisms in
DB
is supported by this
considerations: the false logical value has to be associated by the empty flux
=⊥
∪
:
Υ
→
