Database Reference
In-Depth Information
R
i
k
=
α(r
i
k
)
=
r
i
k
#
∈
A
,
k
=
1
,...,n
, so that
R
=
t
R
#
=
R
i
1
×···×
R
i
n
(here
is its
operational symbol used for relational symbols). Let us introduce a relational
symbol
r
such that
ar(r)
×
is the Cartesian product (a function) of the relations, while
⊗
=
1
≤
k
≤
n
ar(r
i
1
)
, with the schema
B
=
(
{
r
}
,
∅
)
and
B
=
α
∗
(
B
)
=
Eval(t
R
)
={
R,
⊥}
, so that SOtgd
Ψ
equal to
x
i
n
r
i
1
(
x
i
1
)
∧···∧
r
i
n
(
x
i
n
)
⇒
r(
x
i
1
,...,
x
i
n
)
∀
x
i
1
...
∀
(6.1)
corresponds to the Cartesian product in
t
R
.
Hence, we have the mapping-operad
MakeOperads
{
}
={
M
=
Ψ
q
1
,
1
r
∅
}:
A
→
B
,
with
q
1
∈
O(r
i
1
,...,r
i
n
,r)
and, consequently, the arrow
=
α(q
1
),q
⊥
:
α
∗
(
M
)
f
q
=
A
→
B,
so that we obtain the surjective mapping function
α(q
i
)
:
R
i
1
×···×
R
i
n
α(r)
=
R
=
t
R
#
,
that is, we have the composition
Eval(f
R
)
◦
f
q
:
A
→
Eval(f
R
(t
R
))
⊆
TA
. Con-
sequently, we obtain
is
−
A
◦
f
DB
=
in
◦
Eval(f
R
)
◦
f
q
:
A
→
A
1
,
with
A
1
=
A,
(6.2)
where
is
−
A
:
TA
A
is an isomorphism in
DB
and a monomorphism
in
:
R
,
TA
, with
R
=
Eval(f
R
(t
R
))
→
TA
(the inclusion
Eval(f
R
(t
R
))
={
⊥}⊆
f
R
(t
R
)
#
), with
in
={
id
R
,q
⊥
}
.
Let us consider the following program for the categorial RDB machine
M
R
,
where the Application Plane is composed of
P
n
,n
3 embedded compiled SQL
statements in the host program, where
P
1
is the plan for COMMIT,
P
2
is the plan
for ROLLBACK, and the rest of the plans are the embedded statements (
n
-operators,
composed of the variable-binding operators, obtained from the arrows of category
≥
