Database Reference
In-Depth Information
Definition 26
The full subcategory of
DB
composed of only simple objects will
be denoted as
DB
, with the power-view endofunctor equal to the restriction of
T
to
simple objects and arrows only.
The bottom object in
DB
is that of
DB
, equal to
0
⊥
(zero object), while the top
object is equal to
Υ
=
A
∈
Ob
DB
A
⊂
Υ
=
A
∈
Ob
DB
A
.
The exact relationship between
Υ
and the top complex object
Υ
is given
in Proposition
55
, Sect.
8.1.5
, dedicated to algebraic database lattice,
Υ
=
Υ
∪
ω
Υ
Υ
···
)
. Let us show that the objects
Υ
and
(
ω
Υ
)
=
Υ
∪
(
0
⊥
are the top
and bottom objects in
DB
w.r.t. the PO relation
.
Proposition 10
For any object A
∈
Ob
DB
(
for A
=
Υ as well
)
the following are
valid
:
1.
A
⊗
Υ
Υ and A
Υ
Υ
.
0
2.
Υ and
⊥
are the top and bottom objects
,
respectively
,(
up to isomorphism
)
in
DB
.
Proof
Claim 1. From Definition
19
,wehave
A
⊗
Υ
=
TA
j
∩
Υ
i
1
≤
j
≤
m
&
i
=
1
,
2
,...
=
TA
j
∩
Υ
1
≤
j
≤
m
&
i
=
1
,
2
,...
=
TA
j
1
≤
j
≤
m
&
i
=
1
,
2
,...
=
(T A
1
···
TA
m
)
(T A
1
···
TA
m
)
···
.
Thus, we have an identity mapping
σ
:{
1
,
2
,...
}→{
1
,
2
,...
}
such that
T(A
⊗
Υ)
j
∈{
A
1
,...,A
m
}
and
TΥ
σ(j)
=
T
Υ
so that
T(A
⊗
Υ)
j
⊆
TΥ
σ(j)
for all
j
≥
1
and hence
A
⊗
Υ
Υ
.
Hence
A
Υ
=
A
1
···
A
m
Υ
and for the identity mapping
σ
:{
1
,
2
,...
}→
{
1
,
2
,...
}
,for1
≤
j
≤
m
,
T(A
Υ)
j
=
TA
j
⊆
TΥσ(j)
=
T
Υ
, and for
j>m
,
T(A
Υ)
j
=
T
Υ
⊆
TΥσ(j)
=
T
Υ
. Thus,
A
Υ
Υ
.
Claim 2. For any object
A
in
DB
,
TA
⊆
B
∈
Ob
DB
B
=
Υ
=
TΥ
. Thus,
A
Υ
and
TA
Υ
, i.e., the closed object
Υ
is a top object. Notice that from point 2 of
Proposition
7
there exist a monomorphism
in
TA
:
TA
→
Υ
, and
in
TA
◦
is
A
:
A
→
Υ
is a composition of two monomorphisms (
is
A
:
TA
is an isomorphism, thus
monic arrow as well) and hence a monomorphism as well.
From the fact that
A
→
0
⊥∈
A
for any object
A
in
DB
and the fact that
⊥
={⊥}
,
0
0
0
⊥
⊆
A
, so that
⊥
A
, we obtain that
⊥
is a bottom object.