Database Reference
In-Depth Information
The number of Skolem constants must be infinite also when
dom
is a finite set
because in the foreign key constraint
π
2
(R
1
)
π
1
(R
1
)
in
Example
27
, the non-key attributes are formed by means of fresh (new) marked
values in SK (in the place of existentially quantified predicate's variables (relational
attributes)) and we cannot insert the Skolem values that are previously used because
in that case we would introduce the equivalences in the extensions of relations (for
their quantified attributes) which are not defined in the database schemas.
Consequently, for each fixed domain
dom
we will obtain a particular simple-top
object
Υ
and hence a specific
DB
category.
Hence, we have to deal with a class of database categories
DB
for the class
of all possible domains
dom
and, consequently, a class of the complete algebraic
lattices
(
⊆
π
1
(R
2
)
and
π
1
(R
2
)
⊆
0
,
Υ
)
in Proposition
52
, which will be used to
define a class of Heyting algebras and a particular intermediate (superintuitionistic)
propositional logic in Sect.
9.1.3
.
C
,
⊆
)
=
(Ob
DB
sk
,
⊆
,
⊗
,
⊕
,
⊥
8.2
Enrichment
It is not misleading, at least initially, to think of an enriched category as being a cat-
egory in which the hom-sets carry some extra structure (partial order
of algebraic
sublattice
(Ob
DB
,
)
in our case) and in which that structure is preserved by com-
position. The notion of enriched category [
10
] is more general and allows for the
hom-objects (“hom-sets”) of the enriched category to be objects of some monoidal
category, traditional called
V
.
Let us demonstrate that
DB
is a
monoidal closed
category:
=
1
≤
i
≤
k
B
i
,k
=
1
≤
l
≤
n
C
l
,n
Definition 58
≥
1, the set of all arrows
DB
(B,C)
from
B
into
C
can be represented by unique
principal arrow
h
For any two objects
B
≥
1 and
C
:
B
→
C
(Proposition
46
) such that its set of ptp arrows is
h
il
=
f
il
:
DB
(B,C)
h
f
B
i
→
C
l
|
f
il
∈
,f
∈
=∅|
1
≤
i
≤
k,
n
.
1
≤
l
≤
The object
C
B
is equal to the information flux of this principal arrow. Thus, we
define the hom-object
h
C
B
h
.
Remark
For any two
simple
objects
B
and
C
(when
n
=
k
=
1), the set of all arrows
S
={
f
1
,f
2
,...
}:
B
→
C
, from
B
into
C
, can be represented by a unique com-
plex arrow,
S
:
B
→
C
. Thus, from Definition
22
and Lemma
7
in Sect.
3.2
,we
obtain the hom-object
C
B
S
=
{
f
j
|
f
j
∈
DB
(B,C)
}
. From Proposition
7
,
