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 nonexistence 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 and breadth n + 1 in which every principal ideal has fewer than κ elements.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm208-1-1,
author = {Friedrich Wehrung},
title = {Large semilattices of breadth three},
journal = {Fundamenta Mathematicae},
volume = {209},
year = {2010},
pages = {1-21},
zbl = {1202.03053},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm208-1-1}
}
Friedrich Wehrung. Large semilattices of breadth three. Fundamenta Mathematicae, Tome 209 (2010) pp. 1-21. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm208-1-1/