  We show that locally solvable subgroups of PLo(I) are countable. Then for
each countable ordered set, we construct a locally solvable subgroup of
Thompson's Group F. We develop machinery for understanding embeddings from
solvable subgroups into solvable subgroups. Finally, we apply this machinery to
show the ordered sets used in our construction are invariant under isomorphisms
between the groups constructed. Therefore, we effectively distinguish the
groups and provide uncountably many non-isomorphic locally solvable, hence
elementary amenable, subgroups of Thompson's Group F.
