宽度为3的大半格
Large semilattices of breadth three
摘要：
1984年的S.Z.问题迪托尔问是否存在一个格的基数aleph 2，零，其中每一个主理想是有限的，每一个元素有最多三个下盖。我们证明了这样的格的存在性遵循已知独立于ZFC的两个公理之一，即（1）马丁公理限于前校准者aleph 1的偏序集中aleph 1稠密子集的集合，（2）gap-1沼泽的存在。特别地，这种晶格的存在与ZFC是一致的，而不存在这种晶格意味着在可构造宇宙中ω 2是不可接近的。证明了对每个正则不可数基数$κ$和每个正整数n，存在基数为$κ^{+n}$，宽度为n+1的并半格L，其中每个主理想的元素小于$κ$.
英文摘要：
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 $κ$ and each positive integer n, there exists a join-semilattice L with zero, of cardinality $κ^{+n}$ and breadth n+1, in which every principal ideal has less than $κ$ elements.
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