Database Reference
In-Depth Information
objects and morphisms in
DB
and, after that, we will consider in dedicated sections
the other particular properties.
Theorem 1
Let us define
,
for a fixed enough big universe
U
=
dom
∪
SK where
dom
is a finite set of values and SK
an infinite set of indexed Skolem
constants
,
the sets of objects Ob
DB
and morphisms Mor
DB
of this
DB
category
:
1.
A simple object is an instance-database A
,
composed of a set of n-ary
(
finite
n
={
ω
0
,ω
1
,...
}
≥
0)
relations R
i
=
a
i
∈
A
,
i
=
1
,
2
,... also called the “elements of A” with
0
0
unique nullary empty relation
.
Complex objects are obtained by disjunctive union of simple objects
.
A
closed object
is an instance-database A such that A
⊥∈
A such that A
∩⊥
=⊥
={⊥}
=
TA
.
=
A
∈
Ob
DB
A
.
By Υ we denote the top-closed object in
DB
such that Υ
2.
Let us define the set
α
∗
(
M
AB
), Flux(α,
M
AB
)
|
S
DB
=
A,B
∈
Ob
DB
M
AB
:
A
→
B
is a simple atomic sketch's mapping
or a complex
-atomic sketch's mapping,
and α is a mapping-interpretation
such that A
=
α
∗
(
A
),B
=
α
∗
(
B
)
.
The set of morphisms includes atomic morphisms in π
1
(S
DB
) and all other mor-
phisms that can be obtained by well-defined compositions of atomic morphisms
.
3.
We define recursively the mapping B
T
:
Mor
DB
→
Ob
DB
such that for any sim-
:
→
ple morphism f
A
B
:
f
if (f, f)
B
T
(f )
∈
S
DB
;
B
T
(g)
∩
B
T
(h) otherwise
when f
dom(g)
.
=
g
◦
h,h,g
∈
Mor
DB
with cod(h)
=
These sets of objects and morphisms define the principal part of the database cate-
gory
DB
.
The extension of the mapping B
T
to complex arrows
(
and their composi-
tions
)
obtained by structural-operators _
_
,
[
_ , _
]
,
_, _
and
_ , _
is given in
this section by Definition
20
.
α
∗
(
Proof
For each simple object
A
=
A
)
, we have its identity morphism
id
A
:
A
→
A
where (from Lemma
5
)
α
∗
MakeOperads
∀
r
i
∈
A
x
i
r
i
(
x
i
)
r
i
(
x
i
)
|
id
A
=
⇒
α
∗
1
r
i
∈
r
i
∈
A
∪{
1
r
∅
}
=
O(r
i
,r
i
)
|
=
id
r
i
:
r
i
∈
A
∪{
α(r
i
)
→
α(r
i
)
|
q
⊥
}
