We have three somewhat independent sets of results.
Our first results are a mixed blessing.
We show that Morava $K$-theories don't see $k$-invariants for homotopy
commutative $H$-spaces which are finite Postnikov systems, i.e.
for those with only a finite number of homotopy groups.
Since $k$-invariants are what holds the space together, this
suggests that Morava $K$-theories will not be of much use
around such spaces.
On the other hand, this gives us the Morava $K$-theory of a wide
class of spaces which is bound to be useful.
In particular, this work allows the recent work in \cite{RWY}
to be applied to compute the Brown-Peterson cohomology
of all such spaces.
Their Brown-Peterson cohomology
turns out to be all in even degrees (as is their Morava $K$-theory)
and flat as a $BP^{*}$ module for the category of finitely presented
$BP^{*}(BP)$ modules.
Thus these examples have extremely nice
Brown-Peterson cohomology which is as
good as a Hopf algebra.
Our second set of results produces a large family of spaces which
behave as if they were finite Postnikov systems from the point
of view of Morava $K$-theory even though they are not.
This allows us to apply the above results to an even wider class
of spaces than finite Postnikov systems.
These examples come from spaces in omega spectra with certain
properties.
There are many well known examples with these properties.
In particular, we compute the $K(n)$ homology of all the spaces
in the $\Omega$-spectra for $P(q)$ and $k(q)$ where $q > n$.
In order to prove our results on finite Postnikov systems we
need our third set of results; a beginning of an analysis of
bicommutative Hopf algebras over $K(n)_*$.