AI 中文总结
研究有限全域\(U\)的划分格\(Part(U)\)的极大布尔子格，通过树的子树划分等方法构建，证明其含最小和最大元，刻画含\(0\)布尔子格，表明超图为超树时可扩展，得出高度至少为三的划分格有极大布尔子格的结论。
AI 中文摘要
我们研究有限全域\(U\)的划分格\(Part(U)\)的极大布尔子格。首先，\(Part(U)\)的任何最大规模布尔子格可由其块是顶点集为\(U\)的树的子树的所有划分构成。证明了\(Part(U)\)的极大布尔子格总是包含\(Part(U)\)的最小元和最大元。含\(0\)的\(Part(U)\)的布尔子格\(B\)由对与\(B\)在集合\(U\)上的原子对应的线性超图的圈施加的特定条件来刻画。我们表明，当且仅当由其原子诱导的超图是超树时，\(B\)可扩展为最大规模的\(Part(U)\)的布尔子格。当\(B\)由其块在广义上是区间的所有划分构成时就是这种情况。主要结果表明，所有高度至少为三的划分格对于所有至少为三的可能维度都有极大布尔子格。
英文摘要
We investigate maximal Boolean sublattices of the partition lattice Part(U) of a finite universe U. First, any largest size Boolean sublattice of Part(U) can be formed using all partitions whose blocks are subtrees of a tree with vertex set U. It is shown that a maximal Boolean sublattice of Part(U) always contains the least and the largest elements of Part(U). Boolean sublattices B of PartU containing 0 are characterized by a certain condition imposed on the cycles of a linear hypergraph corresponding to the atoms of B on the set U. We show that B can be extended to a Boolean sublattice of Part(U) with a largest size, if and only if the hypergraph induced by its atoms is a hypertree. This is the case when B is formed by all the partitions whose blocks are intervals in a generalized sense. The main result states that all partition lattices of height at least three have maximal Boolean sublattices for all possible dimensions at least three
Comments20 pages, 3 figures
