Database Reference
In-Depth Information
used in the next two chapters dedicated to the categorial RDB-machines and to the
operational semantics for the database mappings.
5.4
DB Category and Relational Algebras
From the introduction to the symmetric categories and Corollary
1
(Sect.
1.5.1
)
and the results presented in Chap.
3
, dedicated to the definition of the
DB
cate-
gory, we obtain that its algebra
Alg
DB
=
((Ob
DB
, Mor
DB
),Σ
DB
)
has the signature
Σ
DB
={
dom, cod, id,
◦}∪
S
DB
, with the additional two operators
∗
,
B
T
∈
S
DB
,
Ob
2
DB
→
where
Ob
DB
and the unary operator
B
T
∈
Σ
DB
is also a homomor-
phism between two algebras
B
T
:
(Mor
DB
,
◦
)
→
(Ob
DB
,
∗
)
, and the following op-
erators
∗:
{
,
rn
,
,
}⊂
S
DB
for the composition of objects for any two simple
database instances
A,B
∈
Ob
DB
:
1. Data-separation (i.e., (Co)product, disjoint union),
×
(A,B)
=
(A,B)
;
2. Data-federation (union with renaming of the relations),
rn
(A,B)
;
3. Power-view composition,
TA
T(A)
;
4. Matching (monoidal product),
(A,B)
=
=
(T (A),T (B))
(to be introduced
formally in Sect.
8.1.1
);
5. Merging,
(A,B)
=
(T (A),T (B))
(to be introduced formally in Sect.
8.1.2
).
Another set of operators in
S
DB
was introduced for the composition (pairing) of the
arrows in Chap.
3
.
What we can discover, for the composition of the objects in
DB
, is the fact that all
of them are the standard set-like operators (as the intersection, union, union with re-
naming and disjoint union). The only specifically relational DB operator is the unary
power-view operator
T
Ob
DB
and, as it was demonstrated at the end of
Sect.
5.1.1
, the object (i.e., instance-database)
TA
is obtained from the instance-
database
A
by the set
:
Ob
DB
→
T
P
X
(where the set of variables
X
=
A
is the set of relational
symbols of the database schema
A
)oftermsof
Σ
R
(SPJRU) relational algebra
over the schema
A
(obtained by the unique surjective extension
α
#
:
T
P
X
→
TA
of the assignment (R-algebra)
α
:
X
→
TA
which defines the instance-database
A
).
It means that all objects in
DB
are constructed by the standard set-like operators
above and the Codd's relational (SPRJU) algebra
Σ
R
in Definition
31
.
Similarly, all arrows in
DB
are constructed by simple arrows and by the set of
composition (pairing) operations in
S
DB
⊂
={
α(r)
|
r
∈
X
=
A
}
Σ
DB
, however, the open question is if
the internal structure of the simple arrows based on the R-operads can be expressed
equivalently by some extension of the Codd's relational algebra
Σ
R
.Herewewill
try to provide a proper answer to this question.
For each simple arrow
f
={
q
1
,...,q
n
,q
⊥
}:
A
→
B
in
DB
with
q
i
∈
O(r
i
1
,...,
r
i
k
,r
B
)
,
r
i
m
∈
A
,
1
, we can obtain its “concep-
tualized” object (the information flux),
f
=
B
T
(f )
⊆
TA
∩
TB
, which is a closed
object, that is,
f
≤
m
≤
k
,for1
≤
i
≤
n
, and
r
B
∈
B
TB
. Thus, the conceptualization of every
simple arrow in
DB
is an instance database obtained by the set of
Σ
R
terms over a
schema
=
TC
, where
C
⊂
TA
∩
C
with relations in
C
. Based on this conceptualization and duality between
