  We investigate the ways in which fundamental properties of the weak Bruhat
order on a Weyl group can be lifted (or not) to a corresponding highest weight
crystal graph, viewed as a partially ordered set; the latter projects to the
weak order via the key map. First, a crystal theoretic analogue of the
statement that any two reduced expressions for the same Coxeter group element
are related by Coxeter moves is proven for all lower intervals in a simply or
doubly laced crystal. On the other hand, it is shown that no finite set of
moves exists, even in type A, for arbitrary crystal graph intervals. In fact,
it is shown that there are relations of arbitrarily high degree amongst crystal
operators that are not implied by lower degree relations. Second, for crystals
associated to Kac-Moody algebras it is shown for lower intervals that the
Mobius function is always 0, 1, or -1, and in finite type this is also proven
for upper intervals, with a precise formula given in each case. Moreover, the
order complex for each of these intervals is proven to be homotopy equivalent
to a ball or to a sphere of some dimension, despite often not being shellable.
For general intervals, examples are constructed with arbitrarily large Mobius
function, again even in type A. Any interval having Mobius function other than
0, 1, or -1 is shown to contain within it a relation amongst crystal operators
that is not implied by the relations giving rise to the local structure of the
crystal, making precise a tight relationship between the Mobius function and
these somewhat unexpected relations appearing in crystals.
  New properties of the key map are also derived. The key is shown to be
determined entirely by the edge-colored poset-theoretic structure of the
crystal, and a recursive algorithm is given for calculating it.
