Database Reference
In-Depth Information
Definition 9
For any given schema mapping
M
AB
:
A
→
B
and an
atomic
schema
mapping
M
BC
:
B
→
C
, we can define the corresponding mapping-operads
=
q
1
,...,q
n
,
1
r
∅
:
A
→
B
M
AB
=
M
AB
)
MakeOperads(
,
MakeOperads(
M
BC
)
=
q
1
,...,q
m
,
1
r
∅
:
B
→
C
,
M
BC
=
and their composition
M
BC
◦
M
AB
q
i
=
q
i
·
q
i
1
,...,q
ij
|
q
i
∈
{
1
r
∅
}
O(r
B,i
1
,...,r
B,ij
,r
C,i
)
∈
M
BC
1
≤
i
≤
m
and
q
ik
∈
O(r
k
1
,...,r
kl
,r
B,ik
)
if
q
ik
∈
otherwise
1
≤
k
≤
j
.
M
AB
;
q
ik
=
1
r
B,ik
Note that each abstract operation
q
i
in the composed operad mapping
M
AC
=
M
BC
◦
M
AB
is represented by the operation composition
q
B
·
(q
1
,...,q
n
)
where
q
B
M
BC
and each
q
j
,for1
∈
≤
j
≤
n
, is an operation in
M
AB
or an identity
operation for relations in
. Let us show that the transformation of the SOtgd of a
given mapping into operads is well defined and that the properties of the mapping-
operads obtained by the algorithm
MakeOperad
satisfy the general properties of
operads in Definition
8
.
B
Proposition 1
The transformation by the algorithm MakeOperads of SOtgds of the
schema mappings into the mapping-operads is well defined and satisfies the general
properties of operads in Definition
8
.
Proof
Let us show the following mapping-operads properties required by Defini-
tion
8
:
1. There exist the identity mappings that are transformed into the identity operad's
operations for each relational symbol (as required by point 3 in Definition
8
).
In fact, for any relational symbol
r
we can define a database schema
A
=
(
{
r
}
,
∅
)
with only this relation and its identity mapping
Id
A
:
A
→
A
, where
Id
=
∀
x
(r(
x
)
⇒
r(
x
))
. Then, the identity operad's operation 1
r
∈
O(r,r)
is defined by
MakeOperad(Id
A
)
={
(
_
)
1
⇒
(
_
),
1
r
∅
}={
1
r
,
1
r
∅
}
.
·
2. Let us define the composition of mapping-operads '
' so that it satisfies the prop-
erties in point 2 of Definition
8
, namely for any
f
∈
O(r
1
,...,r
k
,r)
and any
g
1
∈
O(r
11
,...,r
1
i
1
,r
1
),...,g
k
∈
O(r
k
1
,...,r
ki
k
,r
k
)
, an element
f
·
(g
1
,...,g
k
)
∈
O(r
11
,...,r
1
i
1
,...,r
k
1
,...,r
ki
k
,r)
.
Define the database schemas
A
=
(S
A
,
∅
)
with
S
A
=
1
≤
j
≤
k
{
r
j
1
,...,r
ji
j
}
,
B
=
∅
)
with
S
B
={
r
1
,...,r
k
}
C
=
{
}
∅
)
, with the mappings
M
BC
=
(S
B
,
, and
(
r
,
{
f,
1
r
∅
}:
B
→
C
and
M
AB
={
g
1
,...,g
k
,
1
r
∅
}:
A
→
B
.
Hence, the operad's operation composition '
·
', based on Definition
9
, is de-
fined by
{
f
·
(g
1
,...,g
k
),
1
r
∅
}=
M
BC
◦
M
AB
.
