Database Reference
In-Depth Information
Finally, the update of the relation
r
∈
A
, which changes its extension from
t
RA
)
α(r)
=
r
#
into
α
1
(r)
=
o
U
(r
⊗
#
, is represented in the
DB
category
α
∗
(
α
1
(
as the composed morphism
f
U
=
Eval(o
U
)
◦
h
r
:
A
)
→
A
)
.
By generalization, the updates of
k
≥
1 different relations
r
1
,...,r
k
of
a given schema
A
such that for each of them we have the morphisms
h
r
i
, Eval(o
U
i
)
,
i
=
1
,...,k
can be represented in
DB
by a composed morphism
Eval(o
U
i
)
h
r
i
α
∗
(
α
1
(
f
U
=
◦
:
A
)
→
A
),
(5.1)
1
≤
i
≤
k
1
≤
i
≤
k
t
RA
i
)
=
r
i
#
,
α
1
(r
i
)
=
⊗
#
,for
i
=
where
α(r
i
)
o
U
i
(r
1
,...,k
, with the
t
RA
i
#
,for
i
r
1
,...r
k
}
)
and
α(r
i
)
B
=
{
∅
=
r
i
⊗
=
intermediate schema
(
,
1
,...,k
, and for each
r
A
∈
A
∈{
r
1
,...r
k
}
=
with
r
A
/
,
α
1
(r
A
)
α(r
A
)
.
(ii) The operation of inserting the set
{
d
11
,...,d
1
n
,...,
d
m
1
,...,d
mn
}
of
m
≥
1
tuples of a relation
r
I
into a given relation
r
(used in the operation of
update above) corresponds to the following term in
Σ
RE
∈
A
INSERT INTO
r
[
S
]
...,
in a nonempty relation) to the path-term,
(r
⊗
r
I
)
RE-
DUCE
S
1
TO
S
,ofthe
Σ
RA
algebra, with the leaf
r
equal to Cartesian
product
(r
⊗
r
I
)
, where
S
1
=
equivalent (if
r
nr
r
(n
+
1
),...,nr
r
(
2
n)
and
S
=
nr(r)
=
nr
r
(
1
),...,n
r
r
(
n)
(if
r
={}
is an empty relation, than this path-term
=
is
r
I
WHERE 1
1).
Hence, it corresponds to the arrow
o
I
=
r
I
)
:
⊗
→
_ REDUCE
S
1
TO
S
(r
r
I
)
in
RA
. That is, from Proposition
24
, for a schema
r
}
o
I
(r
⊗
B
=
(
{
,
∅
)
with
r
I
#
, we obtain the
DB
-morphism
Eval(o
I
)
α(r
)
α
∗
(
α
1
(
=
r
⊗
:
B
)
→
A
)
,
where
α
1
is the R-algebra obtained from
α
after this insertion (with
α
1
(r)
=
r
I
)
o
I
(r
⊗
#
and for each
r
A
∈
A
such that
r
A
=
r
,
α
1
(r
A
)
=
α(r
A
)
).
α
∗
(
α
∗
(
)
such that
α(r
)
is
Let us show that there is a morphism
h
r
:
A
)
→
B
the result of an inter-schema mapping
M
AB
:
A
→
B
. It is easy to show that
M
AB
={
Φ
}
, where
Φ
is equal to SOtgd:
x
r(
x
)
r
x
d
11
,...,d
1
n
∧···∧∀
x
r(
x
)
∀
⇒
;
r
x
d
m
1
,...,d
mn
,
⇒
;
α
∗
(
M
AB
)
α
∗
(
α
∗
(
so that we have the morphism
h
r
=
:
A
→
B
)
)
, where
M
AB
=
{
}
={
v
1
·
q
A,
1
,...,v
m
·
q
A,m
,
1
r
∅
}
MakeOperads(
Φ
)
for which the
R-algebra
α
is a mapping-interpretation,
q
A,i
∈
O(r,r
q,i
)
is an operation
O(r
q,i
,r
)
with
of expression
(
_
)
1
(
x
)
⇒
(
_
)(
x
;
d
i
1
,...,d
in
)
and
v
i
∈
α(r
)
is an injective function for each
i
α(v
i
)
:
α(r
q,i
)
→
=
1
,...,m
, so that
1
≤
i
≤
m
Im(α(v
i
))
=
1
≤
i
≤
m
α(r
q,i
)
r
I
#
.
Consequently, this insertion is represented in the
DB
category by the com-
posed morphism
f
I
=
α(r
)
=
=
r
⊗
α
∗
(
α
1
(
Eval(o
I
)
◦
h
r
:
A
)
→
A
)
.