Database Reference
In-Depth Information
Proposition 17
Let
Sch
(G) be a sketch category generated from a schema map-
ping graph G by applying the method in Proposition
16
and let α
∗
be a functor
α
∗
:
DB
Sch
(G)
).
This functor is a
model
of a database mapping system defined by the graph G if
each atomic sketch's mapping-operad arrow
M
AB
={
v
1
·
q
A,i
,...,v
n
·
q
A,n
,
1
r
∅
}:
A
→
B
DB
(
that is
,
a mapping-interpretation α
∗
∈
Sch
(G)
−→
Int(G)
⊆
n are the injective functions
.
Consequently
,
the category of models of a database schema mapping graph G is
a subcategory of functors
DB
Sch
(G)
and is denoted by Mod(
Sch
(G))
.
For a given model
(
i
.
e
.,
a functor
)
α
∗
∈
in
Sch
(G)
,
all α(v
i
),
1
≤
i
≤
Mod(
Sch
(G)) of a graph G
,
its image in
DB
composed of only atomic sketch's arrows is called a DB-mapping system and is
denoted by
M
G
.
Proof
It is easy to verify based on general theory for sketches [
1
]. Each arrow in a
sketch (obtained from a schema mapping graph
G
) is a mapping-operad that can be
converted by R-algebra
α
of a given functor
α
∗
∈
Int(G)
into a morphism in
DB
.
The functorial property for the identity mappings follows from Proposition
15
.For
any two atomic mappings
M
AB
:
A
−→
B
and
M
BC
:
B
−→
C
and their atomic
morphisms in
DB
,
f
=
α
∗
(
M
AB
)
and
g
=
α
∗
(
M
BC
)
,
α
∗
(
M
BC
◦
M
AB
)
=
g
◦
f
.
It remains to show that such a functor satisfying the conditions specified in this
proposition is a
model
of a database mapping system given by a graph
G
. In fact,
such a mapping interpretation
α
∗
∈
Int(
Sch
(G))
is a model of
G
if it is a model of
each database schema (a vertex in
G
) and hence the schema integrity-constraints
of all schemas in
G
are satisfied. i.e., all integrity-constraints mapping-operads in
Proposition
16
are satisfied as well. From Corollary
4
and Definition
18
for atomic
morphisms, all these mapping-operads are satisfied if all
α(v
i
)
are injective func-
tions, that is, for each
q
i
=
v
i
·
q
A,i
∈
O(r
1
,...,r
k
,r)
, with
q
A,i
∈
O(r
1
,...,r
k
,r
q
)
,
v
i
∈
O(r
q
,r)
, and the function
f
i
=
α(q
A,i
)
:
α(r
1
)
×···×
α(r
k
)
→
α(r
q
)
,
im(f
i
)
⊆
α(r)
.
The same condition is valid for each atomic inter-schema mapping
M
AB
:
A
→
B
M
AB
)
(in point 2 of
Proposition
16
) and hence they are satisfied as well. Consequently, this mapping-
interpretation functor
α
∗
is a model of a database mapping system given by a
graph
G
.
and its mapping-operad
M
AB
=
MakeOperads(
Based on these results, for each model of a database mapping system given by a
graph
G
, we may omit the integrity-constraint arrows (that have the empty informa-
tion flux) from the DB-mapping system
M
G
, as well as identity arrows (which are
always satisfied for each mapping-interpretation
α
), so that such a reduced instance-
level DB-mapping system
M
G
will have
the same
user-meaningful structure as
the specification schema-level mapping system of the graph
G
. More about arrows
(natural transformations) and their compositions in
Mod(
Sch
(G))
, in the case of
schema-graphs is presented in Sect.
6.2.2
(Proposition
30
) for the database transac-
tions, and in the chapter dedicated to the operational semantics.
