Database Reference
In-Depth Information
DB Sch (G) , that is, we can have the
functors from Sch (G) into DB such that they are not the mapping-interpretations
(as provided by Definition 11 ), for example:
But we have the graphs G such that Int(G)
Example 25
Let us consider the graph G composed of two schemas
A =
(
{
r 1 ,r 2 }
,
) and
B =
(
{
r 3 }
,
) , where ar(r i )
=
2, i
=
1 , 2 , 3, with unique mapping
M ={
Φ
}: A B
where Φ is the SOtgd
x,y,z((r 1 (x,y)
(y,z))
r 3 (a,z)) .
We can define another graph G 1 with the same schemas
A
and
B
but with the map-
ping
M 1 ={
Ψ
}: A B
where Ψ is the SOtgd
x,y,z,w((r 1 (x,y)
(w,z))
α (
r 3 (a,z)) . Let us consider an R-algebra α such that A
=
A
)
={
R 1 ,R 2 ,
⊥}
and
α (
B
=
B
)
={
R 3 ,
⊥}
with
= (a 1 ,b 1 ),(a 1 ,b 2 ) ,
R 1 =
α(r 1 )
= (b 1 ,c 1 ),(b 3 ,c 2 ) ,
R 2 =
α(r 2 )
= (a,c 1 ),(a,c 2 ) ,
R 3 =
α(r 3 )
α (
and g
=
M
)
={
α(q 1 )
·
α(v 1 ),q
}:
A
B where α(v 1 )
:
α(r q )
R 3
is an injection and f
=
α(q 1 )
:
R 1 ×
R 2
α(r q ) is the function with the
graph
{
((a 1 ,b 1 ),(a,c 1 )),((a 1 ,b 1 ),(a,c 2 )),((a 1 ,b 2 ),(a,c 1 )),((a 1 ,b 2 ),(a,c 2 ))
}
,
and where α(r q )
R 3 (i.e., image of f ).
It is easy to verify that α is not a mapping-interpretation of
=
G . In fact,
in order for it to be a mapping-interpretation, the graph of f has to be equal to
{
M
((a 1 ,b 1 ),(a,c 1 ))
}
.But α is a well defined functor from the fact that f
Mor DB ,
and other arrows in Sch (G) are only the identity arrows of
.Infact,we
have that α is a mapping-interpretation of the graph G 1 , so that from Theorem 1 ,
f
A
and
B
α (MakeOperads(
=
M 2 ))
:
A
B , so that f is a well defined morphism in DB .
It is easy to verify that there is at most one arrow between any given two nodes in
the obtained sketch Sch (G) . For any object
in Sch (G) , we have its identity arrow
and possibly an integrity-constraint arrow T AA .
A
4.1.3 Models of a Database Mapping System
There is a fundamental functorial- interpretation relationship between the schema
mappings and their models in the instance-level category DB . It is based on the
Lawvere categorial theories [ 1 , 9 ], where he introduced a way of describing alge-
braic structures using categories for theories, functors into base category Set (which
we substitute by a more adequate category DB ) and natural transformations for mor-
phisms between models. For example, Lawvere's seminal observation is that the
theory of groups is a category with a group object, that a group in Set is a prod-
uct preserving functor, and that a morphism of groups is a natural transformation of
functors. This observation was successively extended to define the categorial seman-
tics for different algebraic and logic theories. This work is based on the theory of
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