Database Reference
In-Depth Information
1
α
∗
(
M
A,B
)
introduces, for each two simple objects
A
and
B
, the morphism
⊥
=
where
M
A,B
=
MakeOperads(
{
r
∅
⇒
r
∅
}
)
={
1
r
∅
}
(see Example
7
in Sect.
2.4
), with
the empty information flux
⊥
0
and
Υ
are the bottom and the top closed objects in
DB
(much more about them and the
lattice of all databases (objects in
DB
) is presented in Sect.
8.1.5
).
From definitions of morphisms, each morphism
f
0
1
=⊥
⊥
(from Definition
13
). The objects
∈
Mor
DB
is a
set
of func-
tions in
α
∗
(
M
AB
)
such that
α(q
i
)
α(v
i
)(α(q
A,i
))
is a function that satisfies the
mapping-interpretation conditions of Definition
11
. Consequently,
DB
is different
from the
Set
category (where each morphism is a single function) as we previ-
ously established in Sect.
3.1
and Lemma
6
. As we have seen in the algorithm
MakeOperads
, each operad's operation
q
i
of a schema mapping between a schema
A
=
is generally composed of two components: the first one cor-
responds to a conjunctive query
q
A,i
over a source database
and a schema
B
that defines this
view-based mapping and the second component
v
i
defines which contribution of
this mapping is transferred into the target relation, i.e., a kind of Global-or-Local-
As-View (GLAV) mapping (sound or exact) [
12
].
Based on the atomic schema mappings introduced by Definition
17
, we can in-
troduce the
atomic morphisms
for instance-mappings in
DB
category as follows:
A
Definition 18
(A
TOMIC MORPHISMS
in
DB
category)
An atomic morphism in
α
∗
(
M
AB
)
with the information flux
f
DB
is a set of functions
f
=
=
B
T
(f )
such that
M
AB
={
is an
atomic sketch's
schema mapping
(Definition
17
) and
α
is a mapping-interpretation (Definition
11
) with
A
q
1
,...,q
k
,
1
r
∅
}:
A
→
B
α
∗
(
=
A
),
B
=
α
∗
(
B
)
.
Let the schema mapping
M
AB
=
InverseOperads(
M
AB
)
:
A
→
B
be satis-
fied by
α
. Then for each operad's operation
q
i
=
v
i
·
q
A,i
∈
M
AB
, with
q
A,i
∈
O(r
i
1
,...,r
im
,r
q
)
and
v
i
∈
, the function
α(v
i
)
is an injection
(from Corollary
4
) used to distinguish sound and exact assumption on the views
as follows:
1.
Sound
case, i.e., when
α(r
q
)
O(r
q
,r
i
),r
i
∈
B
α(r
i
)
;
2.
Exact
case, i.e., special inclusion case when
α(r
q
)
⊆
α(r
i
)
.
We extend the operators
∂
0
and
∂
1
to
DB
morphisms as well, so that
=
∂
k
(f )
=
∂
k
(q
i
),
for
k
=
0
,
1
.
α
∗
(
M
AB
)
q
i
∈
A
complete morphism
(c-morphism)
f
:
A
→
B
satisfies the condition
∂
0
(f )
⊆
A
.
Every atomic morphism is a complete morphism. Thus, each view-map
q
A
i
, i.e.,
an atomic morphism
f
={
q
A
i
,q
⊥
}:
−→
TA
, is a complete morphism (the case
when
B
=
TA
and
α(v
i
)
belongs to the “exact case”).
As we have seen from Definition
15
for the
A
-atomic sketch's mappings,
α
∗
(
M
DA
1
,...,
M
DA
n
)
=
α
∗
(
M
DA
1
),...,α
∗
(
M
DA
n
)
,
and hence they are compositions of atomic simple morphisms as well.
