HAL : hal-00272111, version 2

arXiv : 0804.1781

DOI : 10.4064/fm208-1-1

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Fundamenta Mathematicae 208, 1 (2010) 1--21

Versions disponibles :

v1 (10-04-2008)

v2 (21-12-2009)

Large semilattices of breadth three

Friedrich Wehrung 1

(2010)

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.

1 :	Laboratoire de Mathématiques Nicolas Oresme (LMNO)
	CNRS : UMR6139 – Université de Caen

Domaine

:

Mathématiques/Mathématiques générales Mathématiques/Catégories et ensembles

Mots Clés : Poset – lattice – breadth – lower cover – lower finite – ladder – Martin's Axiom – precaliber – gap-1 morass – Kurepa tree – normed lattice – preskeleton – skeleton

hal-00272111, version 2

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Contributeur : Friedrich Wehrung <>

Soumis le : Lundi 21 Décembre 2009, 14:53:55

Dernière modification le : Mercredi 30 Mars 2011, 09:53:33