Database Reference
In-Depth Information
By generalization, the insertion of the tuples into the
k
≥
1 different rela-
tions
r
1
,...,r
k
of a given schema
A
, by the set of INSERT INTO
r
i
[
S
i
]
...
,
for
i
=
1
,...,k
such that that for each of them we have the morphisms
h
r
i
, Eval(o
I
i
)
,
i
=
1
,...,k
, can be represented in
DB
by a composed morphism
Eval(o
I
i
)
h
r
i
α
∗
(
α
1
(
f
I
=
◦
:
A
)
→
A
),
(5.2)
1
≤
i
≤
k
1
≤
i
≤
k
r
ρ
where
α(r
i
)
=
r
i
#
,
α
1
(r
i
)
=
o
I
i
(r
i
⊗
I
i
)
#
,for
i
=
1
,...,k
, with the inter-
r
ρ
r
1
,...r
k
}
)
and
α(r
i
)
mediate schema
B
=
(
{
,
∅
=
r
i
⊗
I
i
#
,for
i
=
1
,...,k
,
α(r
A
)
.
(iii) The operation of deleting a set of tuples of a relation
r
I
corresponds to the
following term
t
R
of the
Σ
RE
-algebra:
and for each
r
A
∈
A
with
r
A
/
∈{
r
1
,...r
k
}
,
α
1
(r
A
)
=
DELETE FROM
r
WHERE
C.
It is equivalent to the
Σ
RA
path-term
r
WHERE
¬
C
, that is, to the atomic
arrow
o
D
=
¬
C
:
r
→
o
D
(r)
in
RA
, so that, from Proposition
24
,
its representation in
DB
is given by
_ WHERE
α
∗
MakeOperads
{
}
=
α
∗
{
q
A,
1
,
1
r
∅
}
:
α
∗
(
α
1
(
Eval(o
D
)
=
Φ
v
1
·
A
)
→
A
),
:
α(r)
→
α(r
q
)
is the function defined in Proposition
24
(by substituting
C
with
¬
=
#
and
α(q
A,
1
)
where
α(v
1
)
is the identity function for
α(r
q
)
o
D
(r)
C
), and
q
A,
1
∈
O(r,r
q
)
is the expression
((
_
)
1
(
x
)
∧¬
C(
x
))
⇒
(
_
)(
x
)
, ob-
r
(
x
)))
.
However, analogously to the previous cases, we can express this deleting
term
t
R
by an, equivalent to it, tree-term of
Σ
RA
algebra
r
MINUS (
r
WHERE
C
). Hence, it can be transformed into the path-term
((r
⊗
r
ρ
)
WHERE
C
ρ
)
DISJOINT
S
FROM
S
1
, where
C
ρ
is obtained from
C
by renaming column
names w.r.t.
ρ
of Cartesian product, and, equivalent to it, the following arrow
in
RA
:
tained from SOtgd
Φ
,
∀
x
((r(
x
)
∧¬
C(
x
)
⇒
_ WHERE
C
ρ
)
:
r
⊗
r
ρ
→
o
D
r
⊗
r
ρ
,
o
D
=
_ DISJOINT
S
FROM
S
1
◦
nr(r
ρ
)
,
S
1
=
where
S
=
nr(r)
. Consequently, by the functor
Eval
:
RA
→
DB
α
1
(
of Proposition
24
, we obtain the morphism
Eval(o
D
)
:
B
→
A
)
, where
B
=
r
ρ
r
}
)
where
r
is a
{
r
⊗
#
,
⊥}
is an instance database of a schema
B
=
(
{
,
∅
relational symbol for the leaf term
r
⊗
r
ρ
.
Let us show that there is a morphism
h
r
:
α
∗
(
r
ρ
A
)
→
B
such that
r
⊗
#
is the result of an inter-schema mapping
M
AB
:
A
→
B
and R-algebra
α
,ex-
tended to
r
∈
B
by
α(r
)
r
ρ
=
r
⊗
#
.
M
AB
={
}:
A
→
B
In fact, we can define a mapping
Ψ
such that
Ψ
is equal
x
1
((r(
x
0
)
∧
r(
x
1
))
⇒
r
(
x
0
&
x
1
))
.
Consequently,
to SOtgd
∀
x
0
∀
α
∗
MakeOperads(
M
AB
)
=
α
∗
{
q
A,
1
,
1
r
∅
}
:
α
∗
(
α
∗
(
h
r
=
v
1
·
A
)
→
B
)
