Partially Ordered Set Generated Tightly

The generalization of the descending chain condition in the lattice can get the definition of atom,and the generalization of the arising chain condition can be defined as a compact element.The results for them in lattice are almost perfect,the study of generalizing the relevant conclusions of the atoms in the lattice to the poset has also received widespread attention.However,there are relatively few studies to generalize the conclusions of compact elements in lattices to posets.This paper starts from the definition of compact elements in posets.In the third chapter,we first give three ways of defining compact elements.Based on this,the definitions of compactly generated posets,strongly compact elements and continuous are given.With the definitions of weak atom and descending chain conditions in the posets,we obtain the properties and theorems related to the compact element and compactly generated posets.Based on the concept of the compact element,the fourth chapter combines the predecessors in posets.The definitions of modular,semimodular,uniquely complement,and complete posets are discussed in detail.The relations between compact elements in posets,complete posets,and modular posets are further discussed.In the complete modular posets,five equivalent conditions related to compact elements are obtained.In Chapter 5,Ipropose a new method to prove the fourth step of Peirces theorem.Then,some equivalent conditions of distributivity in uniquely complement lattice are proposed..