  A 1984 problem of S.Z. Ditor asks whether there exists a lattice of
cardinality aleph two, with zero, in which every principal ideal is finite and
every element has at most three lower covers. We prove that the existence of
such a lattice follows from either one of two axioms that are known to be
independent of ZFC, namely (1) Martin's Axiom restricted to collections of
aleph one dense subsets in posets of precaliber aleph one, (2) the existence of
a gap-1 morass. In particular, the existence of such a lattice is consistent
with ZFC, while the non-existence of such a lattice implies that omega two is
inaccessible in the constructible universe. We also prove that for each regular
uncountable cardinal $\kappa$ and each positive integer n, there exists a
join-semilattice L with zero, of cardinality $\kappa^{+n}$ and breadth n+1, in
which every principal ideal has less than $\kappa$ elements.
