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⊥
0
, obtained from empty set of SOtgds between the schemas
1
={⊥}=⊥
A
and
B
).
Moreover, any two simple arrows
f,g
:
A
→
B
are equal (
f
≡
g
, in Defini-
tion
23
) iff they are observationally equivalent, that is, if
f
g
.
This relation of behavioral equivalence between the objects does not strictly cor-
respond to the notion of isomorphism in the category
DB
(see Proposition
8
), in
fact, we have:
≈
Corollary 14
Any two isomorphic objects A and B in
DB
are behaviorally equiv-
alent
,
that is
,
if A
B
,
but not vice versa
.
Only for the simple objects
(
databases
)
the behavioral equivalence corresponds
to the isomorphism in
DB
.
B then A
≈
Proof
If there is an isomorphism
A
B
, with an iso arrow
is
:
A
→
B
then, from
B
. Moreover, from duality, the arrow
is
OP
the proof of Theorem
6
, we obtain
A
:
B
→
A
is an isomorphism and hence
B
A
.
The converse does not hold: for example,
A
A
≈
A
,but
not A
A
A
.Fortwo
simple objects
A
and
B
,
A
B
means
TA
⊆
TB
. Thus
A
≈
B
means
TA
=
TB
,
so that
A
TA
=
TB
B
, i.e.,
A
B
.
Notice that a database
A
‡
A
, which is composed of two copies of a database
A
,
each one separated from another (each one has a different and independent DBMS),
is behaviorally equivalent to a database
A
but not isomorphic to
A
.
The behavioral equivalence is derived from the PO relation '
' so that it will have
an important role in the generation of the database complete lattice in Sect.
8.1.5
based on this ordering. It is well known that any two isomorphic objects in a given
category can be mutually substituted and, from the fact that they are also behav-
iorally equivalent, it means that in this database lattice two isomorphic object can
be substituted as well by preserving PO ordering in this lattice.
3.4.2 Weak Observational Equivalence for Databases
A database instance can also have some relations with tuples containing
Skolem
constants
. For example, the minimal Herbrand models of a Global (virtual) schema
of a Data integration system with incomplete information [
5
,
7
,
12
] will have the
Skolem constants for such a missed information. This example will be provided and
discussed in Sect.
4.2.4
. We consider (see Definition
1
for FOL and Sect.
1.4
in
the introduction) a recursively enumerable set of all Skolem constants as marked
(labeled) nulls
SK
={
ω
0
,ω
1
,...
}⊆
U
, disjoint from a database domain set
dom
⊆
U
of all values for databases. Moreover, we introduce a unary predicate
Va l(
_
)
such
that
Va l(d
i
)
is true iff
d
i
∈
dom
and hence
Va l(ω
i
)
is false for any
ω
i
∈
SK
. Thus,
we can define a new weak power-view operator for databases as follows:
