Large semilattices of breadth three

TL;DR

This paper investigates the existence of large lattices with specific properties, showing their consistency with ZFC under certain axioms and constructing semilattices with controlled breadth and size.

Contribution

It proves the existence of such lattices under axioms independent of ZFC and constructs semilattices with prescribed breadth and cardinality.

Findings

Existence of certain lattices is consistent with ZFC under specific axioms.

Construction of join-semilattices with given breadth and size for any regular uncountable cardinal.

Non-existence implies large cardinal inaccessibility in the constructible universe.

Abstract

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.