作为分配格上模函数极小元集的分解闭子格
Decomposition-Closed Sublattices as Minimizer Sets of Modular Functions over Distributive Lattices
AI总结：
该研究刻画有限分配格中可作为模函数极小元集的子集，证明分解闭子格与模函数极小元集等价，给出构造性证明并指出该性质不适用于非分配格。
AI中文摘要：
我们刻画有限分配格中可作为模函数极小元集的子集。显然，任何模函数的极小元集都是分解闭子格。我们的主定理证明了其逆命题成立：每个分解闭子格都是某一模函数的极小元集。证明具有构造性：利用Birkhoff表示，沿给定子格任意极大链确定的区间分解问题，关键在于对连通差集赋予权重，生成恰在各区间端点处取零值的局部模函数。我们还给出示例表明，该刻画对非分配格不成立。
英文摘要：
We characterize the subsets of a finite distributive lattice that arise as the minimizer sets of modular functions. It is immediate that the minimizer set of any modular function is a decomposition-closed sublattice. Our main theorem establishes the converse: every decomposition-closed sublattice is the minimizer set of a modular function. Our proof is constructive. Using Birkhoff's representation, we decompose the problem along the intervals determined by an arbitrary maximal chain of the given sublattice. The key ingredient is a weight assignment on connected difference posets that yields local modular functions vanishing precisely at the endpoints of each interval. We also provide an example showing that this characterization fails for nondistributive lattices.
