Sperner理论的质因子分解问题

Prime Factorization of Sperner Theory

摘要: Sperner理论是建立在偏序集上的极值理论，在运筹学、计算机、超图理论等领域有很多的应用。然而原始的Sperner定理对集合限制颇大。本文的主要工作是借助于数论的方法，给出Sperner定理在自然数域上推广的一个可选择的证明。将Sperner中集合与不定方程的解对应起来，把复杂的集合结构简化为解的结构，得到了良好的性质。推广中还运用对称链分解辅助说明，凭借架构对称链数量上的一一对应，证明了推广的部分。

Abstract: Sperner theory is one of the most marvelous branches in extremal set theory. It has many applica-tions in the field of operation research, computer science, hypergraph theory and so on. The original Sperner theorem is brilliant; however, there are quite a few constraints. Using number theory method, an alternative proof of Sperner theorem was obtained. As an application, we correspond subsets of Sperner to the roots of indefinite equations, simplifying the complex conformation of the set of solutions and getting nicer properties. The utilization of symmetric chain decomposition plays a great role in promotion, by establishing numerical correspondence between symmetric chain structure and integer collection. The symmetric chain decomposition method also supports the promotion. We build a connection between chains and collections.