有限半模格的结构及其应用

Many researchers all over the world have done a lot of work for the structures of finite semimodular lattices and their applications on the representation of finite distributive lattices by using congruence relations. But so far, the characterizations of the optimal extension geometric lattices of finite semimodular lattices and the upper and lower bounds of semimodular lattices in the representation are still open. The content of this project mainly includes: Giving a method to construct all optimal extension geometric lattices of finite semimodular lattices, further understanding the structures of finite semimodular lattices; Investigating the characterizations of the congruence lattices of semimordular lattices and then obtaining an upper and lower bound of the semimodular lattices; At last, this project will optimize the upper and lower bounds of plane semimodular lattices in the representation of congruence of finite distributive lattices. The main ideal is to promote a deep study of the problems mentioned as above and solve them.

国内外广大研究者围绕有限半模格结构的刻画及其在有限分配格的同余表示中的应用做了大量的工作, 但至今为止有限半模格的最优扩张几何格的结构及同余表示有限分配格问题中半模格的上、下界仍是一个开问题. 本项目的研究内容主要包括: 给出有限半模格的最优扩张几何格的构造方法, 认识有限半模格的结构特征; 讨论有限半模格的同余格的特征, 给出同余表示有限分配格问题中半模格的上、下界; 最后讨论同余表示有限分配格问题中平面半模格的上、下界优化问题. 本项目旨在推动这些问题的深入研究和解决.

国内外广大研究者围绕格结构的刻画及格上的一些算子的构造做了大量的工作. 本项目主要从扩张和子结构两个方向给出了有限格的等价刻画. 一方面给出了有限半模格的最优扩张几何格的构造方法, 另一方面运用子格等价刻画了有限伪补格. 通过对格结构的进一步认知, 给出了有界格上一致模的一些构造方法及优化了同余表示有限分配格问题中半模格的上、下界.