Database Reference
In-Depth Information
B
∈
DB
f
Σ
D
B
from
K(B)
B
=
⊗
=
DB
I
(B,A)
B
ω
A
=
DB
I
(B,A)
⊗
Σ
D
B
B
∈
DB
f
&
B
ω
A
+
DB
I
(B,A)
⊗
Σ
D
B.
0
(zero object in
DB
)ifthere
is no (monic) arrow from
B
to
A
, we can continue as follows:
⊥
Since a hom-object
DB
I
(B,A)
is an empty database
Lan
K
(Σ
D
)(A)
B
ω
A
0
=
⊗
+⊥
DB
I
(B,A)
Σ
D
B
B
ω
A
(
∗
)
DB
I
(B,A)
⊗
Σ
D
B
(by Corollary
13
)
B
ω
A
in
B
⊗
=
Σ
D
B
(here
in
B
:
B
→
A
is a unique monic arrow into
A
)
B
ω
A
=
TB
⊗
Σ
D
B
B
ω
A
TB
for finite
B
,
Σ
D
(B)
T(B)
=
TB
⊗
=
B
ω
A
=
TB
T(B)
ω
A
l.u.b. of compact elements of directed set
=
|
B
}
{
B
|
B
ω
A
=
T(A),
from the fact that, by Corollary
28
, the poset
DB
I
is a complete algebraic lattice
[
17
]
(
DB
I
,
)
with the meet and join operators
⊗
and
⊕
, respectively, and with
compact elements
TB
for each
finite
database
B
.
Remark
Let us consider now which kind of interpretation can be associated to the
tensor product (see (*) above):
B
∈
DB
f
B
ω
A
DB
I
(B,A)
DB
I
(B,A)
⊗
Σ
D
B
⊗
Σ
D
B
and its
B
-components (for
B
ω
A
, that is,
B
⊆
ω
TB
⊆
TA
),
DB
I
(B,A)
⊗
Σ
D
B
, in the enriched lfp database category
DB
:
The second component
Σ
D
B
cannot be a set of signature operators, just because
an object in
DB
is not a set of functions, and it is not interesting in this interpretation:
in fact, it can be omitted from
B
-component because it is equal to
TB
which is the
