Database Reference
In-Depth Information
(iii)
TA
=↓
R
A
,
where R
A
=
(A)
∈
Υ
.
Proof
This PO is well defined since a singleton
{
R
}
is equivalent to a database,
and
T(
)
is the set of all views that can be obtained from it in SPRJU relational
algebra, with
{
R
}
0
⊥∈
T(
{
R
}
)
∈
Ob
DB
.
T
{⊥}={⊥}=⊥
⊆
A
⊆
TA
for each database
A
in
DB
, so that
T
R
.
From point 2 of Proposition
69
, for any simple database
A
,
TA
{⊥}⊆
T
{
R
}
for any
R
∈
Υ
, i.e.,
⊥≤
=
T(
{
(A)
}
)
,
which is valid when
A
is a singleton set of relations as well. Let
R
1
≤
R
2
, i.e.,
T
{
R
1
}⊆
T
{
R
1
}
. Then,
T
T
T
T
T(R
1
∧
R
2
)
=
{
R
1
}∩
T
{
R
2
}
=
{
R
1
}
=
T
{
R
1
}
,
so that we obtain the lattice equality
R
1
∧
R
2
R
1
. While
T
T
R
2
}
=
T
T
R
2
}
=
T(R
1
∨
R
2
)
=
{
R
1
}∪
T
{
{
T
{
R
2
}
,
so that we obtain the lattice equality
R
1
∨
R
2
R
2
. Consequently, the meet and
join operations of this lattice are well defined. The finitary relation
(A)
is the
∈
{
}⊆
=
l.u.b. of all elements in
TA
. In fact, for each view (relation)
v
TA
,
T
v
TA
{
}
≤
∈
T(
(A)
)
, so that
v
(A)
and, consequently,
(A)
TA
is the l.u.b. of relations
=
(T A)
and
TA
∈
=
in
TA
. Thus, for each closed database
TA
Ob
DB
sk
,
(T A)
↓
(
(A))
.
Consequently, from the fact that
Υ
=
T
Υ
is a closed object in
DB
, we obtain that
the top element of this complete lattice is 1
=
(T
Υ
)
=
R
Υ
=
=
=
(
Υ
)
(T
Υ
)
(
Υ
)
.
From the fact that
(
{
∨
(R
1
,R
2
)
T
{
R
1
}
,T
{
R
2
}}
)
, we obtain for the supre-
mum
T
A
A
(A),
(from the set-equality
T(
{
(
{
T
{
R
}|
R
∈
A
}
)
}
)
=
T(
{
(A)
}
)
) and hence the
supremum in this lattice corresponds to the relation
R
Υ
=
(
Υ
)
=
(
Υ
)
.
{
R
}|
R
∈
We recall that a Boolean algebra is a distributive lattice (which satisfies the equa-
tions of points 1-6 in Sect.
1.2
) with additional negation operator
¬
, such that it
satisfies the following additional equations:
a
∧
0
=
0
,
a
∨
1
=
1
,
a
∧¬
a
=
0
,
a
∨¬
a
=
1
.
The lattice of relations
L
Υ
in Lemma
22
and the lattice
L
v
in Example
44
above
have the following properties:
Proposition 70
The complete distributive lattice L
v
in Lemma
21
is a Boolean
,
Υ
) with negation operation defined by
:
algebra (
Υ
,
∧
v
,
∨
v
,
¬
v
,
⊥
=
R
¬
∈
R
¬
v
R
a
|
a
∈
U
,
a
/