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## Fundamenta Mathematicae

2010 | 208 | 1 | 1-21
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### Large semilattices of breadth three

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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.
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Tom
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1-21
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wydano
2010
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autor
• LMNO, CNRS UMR 6139, Département de Mathématiques, BP 5186, Université de Caen, Campus 2, 14032 Caen Cedex, France
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