A Lattice is a partially ordered set where both least upper bound and greatest lower bound of any pair of elements are unique and exist within the set. K\"{o}tter and Kschischang proved that codes in the linear lattice can be used for error and erasure-correction in random networks. Codes in the linear lattice have previously been shown to be special cases of codes in modular lattices. Two well known classifications of modular lattices are geometric and distributive lattices. We have identified the unique criterion which makes a geometric lattice distributive, thus characterizing all finite geometric distributive lattices. Our characterization helps to prove a conjecture regarding the maximum size of a distributive sublattice of a finite geometric lattice and identify the maximal case. The Whitney numbers of the class of geometric distributive lattices are also calculated. We present a few other applications of this unique characterization to derive certain results regarding linearity and complements in the linear lattice.

中文翻译：

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有限几何分布格的刻画

格是一个偏序集合，其中任何一对元素的最小上界和最大下界都是唯一的并且存在于集合中。K\"{o}tter 和 Kschischang 证明了线性格中的代码可用于随机网络中的错误和纠删。线性格中的代码以前已被证明是模格中代码的特殊情况。两个很好模格的已知分类是几何格和分布格。我们已经确定了使几何格具有分布性的唯一标准，从而表征了所有有限几何分布格。我们的表征有助于证明关于有限分布亚格的最大尺寸的猜想几何格子并确定最大情况。还计算了几何分布格的类的惠特尼数。我们介绍了这种独特表征的一些其他应用，以推导出有关线性晶格中线性和互补的某些结果。