Large semilattices of breadth three

Friedrich Wehrung¹

¹ LMNO, CNRS UMR 6139 Département de Mathématiques, BP 5186 Université de Caen Campus 2 14032 Caen Cedex, France

Fundamenta Mathematicae, Tome 208 (2010) no. 1, pp. 1-21

Résumé

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.

Détail

DOI : 10.4064/fm208-1-1

Keywords: problem ditor asks whether there exists lattice cardinality nbsp aleph zero which every principal ideal finite every element has three lower covers prove existence lattice follows either axioms known independent nbsp zfc namely nbsp martins axiom restricted collections nbsp aleph dense subsets posets precaliber nbsp aleph nbsp existence gap morass particular existence lattice consistent nbsp zfc while nonexistence implies nbsp omega inaccessible constructible universe prove each regular uncountable cardinal nbsp kappa each positive integer nbsp there exists lor semilattice cardinality nbsp kappa breadth which every principal ideal has fewer nbsp kappa elements