Biology Reference
In-Depth Information
O
f
1
f
2
f
M
/
E
1
M
/
E
1
M
/
E
2
M
/
E
2
×
π
1
π
2
In the above diagram, replacing O by the partition M/E
1
∩
E
2
, we can always
define a map
f M/E
1
∩
E
2
→
M/E
1
×
M/E
2
This map associates with each equivalence class in partition M/E
1
∩
E
2
the
pair of classes of which it is the intersection. The map f is onto M/E
1
×
M/E
2
if the two observables are totally unlinked, i.e. if each E
equivalence class
intersects with every E
-class, and conversely. If in addition the intersection of
an E
-class with an E
-class contains exactly one element, then f is one-to-one
and M is said to be isomorphic to M/E
1
∩
E
2
.Iff is an isomorphism, we
obtain via individual observations of
1
and
2
information about M itself. If
1
and
2
are not related in that way, f is only onto (surjective) and what we
obtain is a representation of the quotient set of M; an embedding of M into a
product, and not a representation of M such a product.
An important observation is that we are now in a position to consider
more than one analysis of M, by the mutual refinement of equivalence rela-
tions, leading to a partial order of analyses
a
M. The category of analytical
models, denoted
M, that is the set of all analytical models, is shown in
Fig. 5. As can be seen from the figure, the objects of the category are ana-
lytical models, whereas the morphisms in that category are defined in terms
of an inclusion relation. One should not fail to notice that with the partial
M
/
E
Ξ
M
/
E
1
2
3
M
/
E
1
2
M
/
E
M
/
E
M
/
E
3
M
/
E
1
2
4
Figure 5
Extract from the category of analytic models,
M, generated by the product
of partitions that are induced from three totally unlinked observables,
1
2
3
.
Each node in which two nodes ,
below merge is defined as a product M/E
×
=
M/E
,
realising the mapping f M/E
∩
→
M/E
×
E
M/E
. A refinement model is said to
be
'bigger'
than the one shown.
