Database Reference
In-Depth Information
Then this mapping interpretation
α
defines the following functions:
1. The function
α(q
A,
1
)
:
α(
EmpAcme
)
×
α(
Local
)
→
α(r
q
1
)
such that for any
tuple
a
∈
α(
EmpAcme
)
and
b
∈
α(
Local
)
,
α(q
A,
1
)
=
a,I
T
f
1
(a)
if
a
a
,
b
=
b
;
otherwise.
And for any
a,b
∈
α(r
q
1
)
,
α(v
1
)(
a,b
)
=
a,b
if
a,b
∈
α(
Office
)
;
otherwise.
2. The function
α(q
A,
2
)
:
α(
EmpAjax
)
×
α(
Local
)
→
α(r
q
2
)
such that for any
tuple
a
∈
α(
EmpAjax
)
and
b
∈
α(
Local
)
,
α(q
A,
2
)
=
a,I
T
f
2
(a)
if
a
a
,
b
=
b
;
otherwise.
And for any
a,b
∈
α(r
q
2
)
,
α(v
2
)(
a,b
)
=
a,b
if
a,b
∈
α(
Office
)
;
otherwise.
3. The function
α(q
A,
3
)
:
α(
EmpAcme
)
×
α(
Over65
)
→
α(r
q
3
)
such that for any
tuple
a
∈
α(
EmpAcme
)
and
b
∈
α(
Over65
)
,
α(q
A,
3
)
a
,
b
=
a
if
a
=
b
;
otherwise.
And for any
a
∈
α(r
q
3
)
,
α(v
3
)(
a
)
=
a
if
a
∈
α(
CanRetire
)
;
other-
wise.
4. The function
α(q
A,
4
)
:
α(
EmpAjax
)
×
α(
Over65
)
→
α(r
q
4
)
such that for any
∈
∈
tuple
a
α(
EmpAjax
)
and
b
α(
Over65
)
α(q
A,
4
)
=
a
,
b
a
if
a
=
b
;
otherwise.
And for any
a
∈
α(r
q
4
)
,
α(v
4
)(
a
)
=
a
if
a
∈
α(
CanRetire
)
;
other-
wise.
From the fact that the mapping-interpretation satisfies the schema mappings, based
on Corollary
4
, all functions
α(v
i
)
,for1
≤
i
≤
4, are injections.
The second example is a continuation of Example
9
:
Example 14
For the operads defined in Example
9
, let a mapping-interpretation
(an R-algebra)
α
be an extension of Tarski's interpretation
I
T
of the source schema
A
=
(S
A
,Σ
A
)
that satisfies all constraints in
Σ
A
and defines its database instance
α
∗
(S
A
)
A
=
={
α(r
i
)
|
r
i
∈
S
A
}
and, analogously, an interpretation of
C
.
M
AC
by Tarski's interpretation for the
functional symbol
f
1
in this SOtgd (denoted by
I
T
(f
1
)
).
Then we obtain the relations
α(
Local
)
,
α(
Office
)
and
α(
CanRetire
)
.The
interpretation of
f
Over
65
is the characteristic function of the relati
on
α(
Over65
)
in
the instance
B
Let
α
satisfy the SOtgd of the mapping
α
∗
(S
B
)
of the database
=
B
=
(S
B
,Σ
B
)
so that
f
Over
65
(a)
=
1if
α(
Over65
)
and, analogously, the interpretatio
n
of
f
Emp
is the characteristic
function of the relation
α(
Emp
)
in the instance
B
with
f
Emp
(a)
a
∈
=
1if
a
∈
α(
Emp
)
.
Then this mapping interpretation
α
defines the following functions:
