  A poset is called upper homogeneous, or "upho," if every principal order
filter of the poset is isomorphic to the whole poset. We study (finite type
$\mathbb{N}$-graded) upho lattices, with an eye towards their classification.
Any upho lattice has associated to it a finite graded lattice called its core,
which determines its rank generating function. We investigate which finite
graded lattices arise as cores of upho lattices, providing both positive and
negative results. On the one hand, we show that many well-studied finite
lattices do arise as cores, and we present combinatorial and algebraic
constructions of the upho lattices into which they embed. On the other hand, we
show there are obstructions which prevent many finite lattices from being
cores.
