Database Reference
In-Depth Information
μ
A
◦
(μ
A
⊗
id
A
)
◦
α
A,A,A
=
μ
A
◦
(id
A
⊗
μ
A
)
is valid, and
β
A
=
μ
A
◦
(η
A
⊗
id
A
)
,
γ
A
=
μ
A
◦
(id
A
⊗
η
A
)
.
={
A
1
,...,A
n
}
We generalize the merging operator for objects for the set
S
of
more than two databases by
⊗
S
=
A
1
⊗···⊗
A
n
.
8.1.2 Merging Operator
Merging of an instance-database
A
with another instance-database is a unary al-
gebraic operation, similar to the binary operation Data Federation of two database
instances (denoted by
rn
in Sect.
3.3.1
, i.e., the union
∪
with the renaming (
rn
)of
A
B
the relational symbols in
that have originally the same names, by unifying
these two databases, that are in two different DB machines, for example, under the
common DBMS. Consequently, given a connection
and
A
⊕
B
of two schemas and an
R-algebra
α
(which is their model),
A
rn
B
α
∗
(
)
).
In fact, the merging of two databases is defined as the database obtained by ap-
plying the power-view operator
T
to data federation of these two databases. That is,
the data federation of two databases is isomorphic to the database obtained by their
merging, from the behavioral point of view. Any view which can be obtained from
the data federation (union with renaming) of two databases can also be obtained
from the merging of these two databases, and vice versa.
In what follows, similarly to the matching tensor products which, for any two
given databases, return a closed object, also the merging operator will return a closed
object. As we will see, these two operators will provide the meet and joint operators
of the complete algebraic database lattice, where
=
A
⊕
B
0
⊥
and
Υ
are the bottom and top
elements (i.e., the instance-databases), respectively.
Theorem 13
For any fixed simple database A
∈
Ob
DB
we define the parameterized
“merging with A” operator as an endofunctor A
⊕
_
:
DB
−→
DB
,
as follows
:
=
1
≤
i
≤
k
B
i
,
k
1.
For any object B
≥
1,
the object A
⊕
B is a merging of A
,
defined
by
:
T(A
rn
B)
=
T(A
∪
B) if k
=
1
,B
=
B
1
;
A
⊕
B
≡⊕
(A,B)
1
≤
i
≤
k
(A
⊕
B
i
)
otherwise.
→
1
≤
j
≤
m
C
j
,
m
2.
Fo r a n y f
:
B
≥
1,
the arrow A
⊕
(f )
:
A
⊕
B
→
A
⊕
C is
defined by the canonical representation
(
Lemma
8
)
of
A
⊕
(f )
id
A
rn
f
ij
:
A
⊕
B
i
→
A
⊕
C
j
|
f
ij
:
B
i
,
id
A
rn
f
ij
1
.
f
→
C
j
∈
=⊥
