Large semilattices of breadth three

Friedrich Wehrung. "Large semilattices of breadth three." Fundamenta Mathematicae 208.1 (2010): 1-21. <http://eudml.org/doc/286567>.

@article{FriedrichWehrung2010,
abstract = {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.},
author = {Friedrich Wehrung},
journal = {Fundamenta Mathematicae},
keywords = {ladders; Kurepa tree; normed lattice; principal ideal; lower covers; Martin's Axiom; precaliber; gap-1 morass},
language = {eng},
number = {1},
pages = {1-21},
title = {Large semilattices of breadth three},
url = {http://eudml.org/doc/286567},
volume = {208},
year = {2010},
}

TY - JOUR
AU - Friedrich Wehrung
TI - Large semilattices of breadth three
JO - Fundamenta Mathematicae
PY - 2010
VL - 208
IS - 1
SP - 1
EP - 21
AB - 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.
LA - eng
KW - ladders; Kurepa tree; normed lattice; principal ideal; lower covers; Martin's Axiom; precaliber; gap-1 morass
UR - http://eudml.org/doc/286567
ER -