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We investigate the ways in which fundamental properties of the weak Bruhatorder on a Weyl group can be lifted (or not) to a corresponding highest weightcrystal graph, viewed as a partially ordered set; the latter projects to theweak order via the key map. First, a crystal theoretic analogue of thestatement that any two reduced expressions for the same Coxeter group elementare related by Coxeter moves is proven for all lower intervals in a simply ordoubly laced crystal. On the other hand, it is shown that no finite set ofmoves exists, even in type A, for arbitrary crystal graph intervals. In fact,it is shown that there are relations of arbitrarily high degree amongst crystaloperators that are not implied by lower degree relations. Second, for crystalsassociated to Kac-Moody algebras it is shown for lower intervals that theMobius function is always 0, 1, or -1, and in finite type this is also provenfor upper intervals, with a precise formula given in each case. Moreover, theorder complex for each of these intervals is proven to be homotopy equivalentto a ball or to a sphere of some dimension, despite often not being shellable.For general intervals, examples are constructed with arbitrarily large Mobiusfunction, again even in type A. Any interval having Mobius function other than0, 1, or -1 is shown to contain within it a relation amongst crystal operatorsthat is not implied by the relations giving rise to the local structure of thecrystal, making precise a tight relationship between the Mobius function andthese somewhat unexpected relations appearing in crystals.  New properties of the key map are also derived. The key is shown to bedetermined entirely by the edge-colored poset-theoretic structure of thecrystal, and a recursive algorithm is given for calculating it.

From the weak Bruhat order to crystal posets