Large semilattices of breadth three

• Autorzy: Friedrich Wehrung
• Języki publikacji: EN

Abstrakty

EN

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 $κ^{+n}$ and breadth n + 1 in which every principal ideal has fewer than κ elements.

Kategorie tematyczne

• 03E05: Other combinatorial set theory
• 03E35: Consistency and independence results
• 03C55: Set-theoretic model theory
• 06A07: Combinatorics of partially ordered sets

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2010
• Tom: 208
• Numer: 1
• Strony: 1-21

Daty

wydano

2010

Twórcy

autor

Friedrich Wehrung

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

Identyfikatory

DOI

10.4064/fm208-1-1