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 σ(j) =
T Υ so that T(A
Υ) j
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
=
. 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.
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