Biology Reference
In-Depth Information
classes of E
1
with all equivalence classes of E
2
. E
1
2
has more equivalence
classes than E
1
or E
2
, and may thus be considered a refinement of E
1
and
E
2
. With only two unlinked observables we have three analyses of M, namely
M/E
1
, M/E
2
and M/E
1
2
, where the latter is a refinement of the former. The
more refined a partition, the more equivalence classes define it. Each class of
M/E
1
2
is a subset of M/E
1
, respectively M/E
2
, which means we can relate
the partitions by some mapping that is defined in terms of a subset relation. In
other words, the set of models M/E
1
M/E
2
M/E
1
2
is a partially ordered
set (poset).
Note that for Eqn 1 to hold, every E
-equivalence class intersects every other
E
-equivalence class, and vice versa. In this case, the two observables are said
to be totally unlinked. The partition M/E can be considered a base space and an
equivalence class p
E
of this partition as a fibre, consisting of all p
∈
M that
project onto p. For any number of observables on M we define an analysis of
M in terms of a direct product:
a
M
=
i
i
M
(2)
where
i
i
M
M/
∩
E
i
For the discussion that follows, we shall denote the set of all observables as
HM
=
for which M
→
Note that we can only unambiguously decode back from the product of the
partitions if the observables are unlinked, i.e. if the value of one observable is not
in some way determined by the other. Fig. 4 illustrates this point by considering
four observables and their partitions. We find in case of
1
and
2
whatever
pair rr
r
∈
1
Mr
∈
2
M, is chosen, always every E
1
-class intersects
every E
2
-class; the pair is thus unlinked. The same is true for
1
4
,
2
4
,
1
3
and
2
3
but not for
3
4
, if the circle of M/E
4
is a subset of the
inner rectangle of M/E
3
.
Considering the partition M/E
i
, an analytical model of M, as a single math-
ematical object, we can form a category of models. In this setting, the product
of factors
=
i
P
M/E
i
is an object for which there exists a family of morphisms that are surjective
projections onto the factors
i
P
→
M/E
i
