Semilattices of Breadth Three

A 1984 problem of S. Z. Ditor asks whether there exists a lattice of cardinality א2, 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 א1 dense subsets in posets of precaliber א1, (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 ω2 is inaccessible in the constructible universe. We also prove that for each regular uncountable cardinal κ and each positive integer n, there exists a (∨, 0)-semilattice L of cardinality κ and breadth n+1 in which every principal ideal has less than κ elements.