Database Reference
In-Depth Information
elements
(in the diagram of the composed morphism
h
, the elements
b
4
and
b
5
are
a hidden relations).
In such a representation, for the consideration of the information fluxes, we “for-
got” the parts of the tree
g
T
◦
f
T
that are not involved in a real information con-
tribution of composed mappings from the source into the target object. In fact, the
hidden relations do not appear on the left-hand side of SOtgd's implications of a
given schema mapping, but only as the characteristic functions of these hidden rela-
tions. The hidden relations appear always in the operad's operations and as a part of
the domain of the set of functions that compose a given morphism
f
B
. Con-
sequently, operad's based representation of morphisms are equivalent to the logical
representation based on the SOtgds.
Based on Definition
13
, the information flux for a composition of simple mor-
phisms is given by the set-intersection of the information fluxes of all its simple
morphisms.
:
A
→
Remark
The information flux
f
=
B
T
(f )
of a given morphism (instance-database
mapping)
f
B
is an instance-database as well, thus, an object in
DB
:the
minimal information flux is equal to the bottom object
:
A
−→
0
so that, given any two
database instances
A
and
B
in
DB
, there exists at least an arrow (morphism) be-
tween them
f
⊥
B
such that
f
0
1
:
−→
=⊥
⊥
:
→
A
(for example,
A
B
).
Definition 19
of simple
instance-databases and hence each query over such a complex object can have the
relational symbols belonging only to one of these simple databases. Let
A
A complex object in
DB
is expressed by a disjoint union
=
A
1
A
m
=
1
≤
j
≤
m
A
j
={{
···
1
}×
A
1
}∪···∪{{
m
}×
A
m
}
if
m
≥
2;
A
1
otherwise,
=
1
≤
i
≤
k
B
i
={{
and
B
2;
B
1
otherwise, where
all
A
j
and
B
i
are simple (non-composed) databases. Consequently, from the fact
1
}×
B
1
}∪···∪{{
k
}×
B
k
}
if
k
≥
that all simple databases are separated,
T(
1
≤
j
≤
m
A
j
)
=
1
≤
j
≤
m
TA
j
.
0
1. We say that
A
is
strictly complex
if
A
j
=⊥
for all 1
≤
j
≤
m
.
⊗
⊗
=
⊗
=
2. We can define the new composition
by
Υ
A
A
Υ
TA
and
TA
∩
TB
if
m
=
k
=
1
;
A
⊗
B
1
≤
j
≤
m
&1
≤
i
≤
k
(T A
j
∩
TB
i
)
otherwise
.
3. We define an “observational” PO relation
such that
Υ
B
iff
B
=
Υ
and
=
1
≤
j
≤
m
A
j
=
=
1
≤
i
≤
k
B
i
iff there exists a mapping
σ
for
A
Υ
,
A
B
:
{
1
,...,m
}→{
1
,...,k
}
such that for each 1
≤
j
≤
m
,
TA
j
⊆
TB
σ(j)
.
The equivalence in this ordering (i.e.,
A
B
and
B
A
) we denote by
A
≈
B
.
Notice that
is a partial order (PO):
A
A
(for
σ
an identity function) and if
=
σ
2
·
A
σ
1
). For a given
a tuple of objects (i.e., databases) in
DB
category
(S
1
,...,S
n
),n
≥
B
(with
σ
1
) and
B
C
(with
σ
2
) then
A
C
(with
σ
2, its disjoint
union (from Definition
19
)is
1
≤
i
≤
n
S
i
=
(S
1
,...,S
n
)
=
1
≤
i
≤
n
{
(i,a)
|
a
∈
S
i
}
(the union
indexed by positions
of the sets in a given tuple), denoted also by
S
1
S
2
···
S
n
. This indexing of simple databases that compose a complex object has
