Large semilattices of breadth three
Volume 208 / 2010
Abstract
A 1984 problem of S. Z. Ditor asks whether there exists a lattice of cardinality $\aleph_2$, 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 $\aleph_1$ dense subsets in posets of precaliber $\aleph_1$, (2) the existence of a gap-$1$ morass. In particular, the existence of such a lattice is consistent with ZFC, while the nonexistence implies that $\omega_2$ is inaccessible in the constructible universe. We also prove that for each regular uncountable cardinal $\kappa$ and each positive integer $n$, there exists a $(\lor,0)$-semilattice $L$ of cardinality $\kappa^{+n}$ and breadth $n+1$ in which every principal ideal has fewer than $\kappa$ elements.
