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However, the isomorphism between the two sends null
⊥
to constant 1, and constant 5 to
⊥
.
Likewise, in this section, homomorphisms are not necessarily the identity on constants
(and can, for instance, send constants to nulls), unless they are explicitly required to be
such.
We say t ha t
null
M
is
defined
by a finite set
Σ
st
of st-tgds if for each pair (
S
,
T
)
∈
I
NST
(R
s
)
×
I
NST
(R
t
),wehave
(
S
,
T
)
∈M ⇔
(
S
,
T
)
|
=
Σ
st
.
There are several desirable structural properties that one would like the mappings to
satisfy. A prototypical example of such a desirable property for a schema mapping is the
existence of universal solutions. In this chapter, we consider this property together with
some other desirable features. More precisely, assume that
M
is a mapping from a source
schema R
s
to a target schema R
t
.Then
• M
admits universal solutions
if every source instance
S
of R
s
has a universal solution
under
M
;
• M
is
closed under target homomorphisms
if for every (
S
,
T
)
∈M
and every homo-
T
that is the identity on constants, we have (
S
,
T
)
morphism
h
:
T
→
∈M
;
reflects source homomorphisms
reflecting if for every pair
S
,
S
of instances of R
s
,
and every pair
T
,
T
of instances of R
t
such that
T
,
T
are universal solutions for
S
,
S
under
• M
S
can be extended to a homomor-
M
, respectively, every homomorphism
h
:
S
→
phism
h
:
T
T
.
→
Notice that in the third property above, neither
h
nor
h
are required to be the identity on
constants. For the class of mappings specified by st-tgds, the following holds.
Proposition 21.1
Every mapping specified by a finite set of st-tgds admits universal so-
lutions, is closed under target homomorphisms, and reflects source homomorphisms.
Clearly, every mapping specified by a finite set of st-tgds admits universal solutions
and is closed under target homomorphisms. We leave it as an exercise for the reader to
prove that these mappings also reflect source homomorphisms (see
Exercise 22.3
). In the
following section, we study the problem of characterizing the class of mappings defined
by st-tgds by using the above structural properties.
21.2 Schema mapping languages characterizations
Proposition 21.1
shows three fundamental properties satisfied by all mappings given by st-
tgds. Thus, it is natural to ask whether these properties characterize this class of mappings.
We give a negative answer to this question in the following proposition, where we also
identify another fundamental property that mappings given by st-tgds satisfy.
Proposition 21.2
There exists a mapping
M
that is closed under target homomorphisms,
