  This paper introduces the seed order, a partial order of the class of uniform
countably complete ultrafilters that generalizes the Mitchell order on normal
measures. Like that order, the seed order is consistently a linear ordering
even under strong large cardinal assumptions. In fact, the linearity of the
seed order is a feature of all known canonical inner models for large cardinal
axioms. We develop the theory of the seed order under the assumption that it is
linear. Augmented by large cardinal hypotheses currently out of reach of inner
model theory (namely supercompactness), this linearity assumption, which we
call the Ultrapower Axiom, has surprisingly strong consequences reminiscent of
the structure theory of the canonical inner models.
