Database Reference
In-Depth Information
}:
A
→
A
is obtained from the schema mapping
that
is a simple atomic mapping as well. All other atomic sketch's mappings in Defi-
nition
17
are obtained from the atomic schema mappings. For complex
M
A
A
={∀
x
(q(
x
)
⇒
r
q
(
x
))
-atomic
sketch's schema mappings it holds from point 5 in Definition
15
.
Let us show the following important property for a schema database mapping
system represented by a graph
G
=
(V
G
,E
G
)
in Definition
14
:
Proposition 4
For a given non-empty schema database mapping system specified
by a graph G
=
(V
G
,E
G
)
(
Definition
14
),
for each graph edge
M
AB
∈
E
G
,
the
M
AB
) in the sketch category
Sch
(G) is an atomic map-
morphism MakeOperads(
ping
.
Proof
From Definition
14
, each edge in a schema mapping graph
G
is obtained by
the set of tgds or by the set of egds, so that by
MakeOperads
algorithm applied to
such an edge we obtain an atomic view-mapping or an atomic integrity-constraint
mapping in Definition
17
.
Notice that in the sketch categories we introduce the schema-separation compo-
sitions
-atomic sketch's schema mappings in Definition
15
between
composed database schemas. As usual in the category theory (sketches are small
categories), the composition of
A
†
B
and the
-
atomic
sketch's schema mappings can generate
any kind of
-complex sketch's schema mappings as explained in Definition
15
and
will be presented by more details in the next section.
3.2
DB (Database) Category
Based on an observational point of view for relational databases and on the theory of
R-algebras for mapping-operads, presented in Sect.
2.4
, Definition
11
, we are able
to introduce a category
DB
[
14
] for instance-databases and view-based mappings
between them, with the set of its objects
Ob
DB
and the set of its morphisms
Mor
DB
.
For the instance-database mappings, in Sect.
2.4.1
and Definition
10
, we con-
sidered the R-algebras that transform the set of schema-mapping operad's opera-
tions
M
AB
=
MakeOperads(
M
AB
)
={
q
1
,...,q
k
,
1
r
∅
}
, of a given schema mapping
, into the set of functions
α
∗
(
M
AB
)
={
α(q
1
),...,α(q
k
),q
⊥
}
M
AB
:
A
→
B
. Con-
sequently, each morphism in the base denotational
DB
category has to be a set of
functions from an instance-database
A
into an instance-database
B
(of schemas
A
and
, respectively).
However, we need to characterize such an instance-mapping morphism also by its
information flux (introduced in Definition
13
) corresponding to this set of functions
obtained from a mapping-interpretation
α
(in Definition
11
) of operad's operations
q
i
above.
In order to facilitate the presentation of the properties of this category, we will
begin with simple objects and simple arrows, and then extend all to complex objects
and complex arrows. In this section, we will establish the principal properties for the
B
