Database Reference
In-Depth Information
(ii)
i
(A
∩
B)
=
i
(A)
∩
i
(B) and
i
(
Υ
)
=
Υ
,
for i
=
1
,
2
,
3.
Proof
It is easy to verify that for each closed object
A
=
TA
∈
Ob
DB
sk
,
A
∩−
A
=
0
and
A
∪−
A
=
Υ
, and hence
BA
is a Boolean algebra and complete because its
lattice is complete as well. Moreover, for any two
A
=
TA,B
=
TB
in
Ob
DB
sk
⊥
we
have:
(a)
−−−→
=
A
B
,for
∈{∩
∪
\}
,
from the fact that J
A
⊆
A
, and
A
=∪
J
A
, and J
A
B
=
J
A
J
B
(for
A
B
,
,
=∪
it holds
directly, for
=∩
it holds from the distributivity between
∩
and
∪
, and analogously
=\
for the set difference
). Consequently,
J
A
J
B
J
A
B
=
(
J
A
J
B
)
−−−→
A
=
A
B.
B
=
=
The mapping _
.
:
Ob
DB
sk
→
Υ
is well defined and idempotent, for each
A
=
TA
∈
=
R
A
R
∈
A
R
Ob
DB
sk
,
A
Υ
with the bottom
⊥=⊥
and the top unary
relation
Υ
. Let us show that it satisfies the homomorphism properties for all other
(non-nullary) algebraic operators:
∈
−−→
−−→
−−−→
A
A
∩
B
(A)
(B)
=
A
∧
v
B
B
(from (a))
∩
=
=
∩
;
−−→
−−→
−−−→
A
A
∪
B
(A)
(B)
=
A
∨
v
B
B
(from (a))
∪
=
=
∪
;
−−−−−−−→
⊥
−→
⊥
−−−→
(
\
A)
={⊥}∪
a
|
a
∈
Υ
,
a
∈
A
(
Υ
A)
(from (a))
0
0
−
A)
=
∪
(
Υ
\
=
∪
−−→
={⊥}∪
(A)
={⊥}∪
¬
v
A)
a
|
a
∈
U
,
a
∈
/
(
=
∪
¬
v
A)
=¬
v
A,
(
from the fact that an empty tuple
is a tuple of every relation.
It is easy to verify that the free multiplicative modal algebraic operators that
extend the Boolean database algebra
BA
into the
N
-modal algebra
BA
+
are defined
for each
A
=
TA
∈
Ob
DB
sk
by:
=
R
Υ
R,R
∈
A
,
R
∈
R
i
implies
R
∈
i
(A)
∈
|∀
Υ
,
R
2
=
R
−
1
where
R
1
=
R
−
1
and
R
3
=
R
≤
are derived from the poset
R
≤
(or
≤
)of
≤
≤
the complete lattice
L
Υ
=
(
Υ
,
≤
,
∧
,
∨
,
⊥
,R
Υ
)
in Lemma
22
.
The embedding of the database Heyting algebra
=
Ob
DB
sk
,
0
L
alg
=
F
(
Υ
)
∩
,
,
⇒
,
⊥
