In [2], Chen and Yan introduced the notion of Eulerian stratified spaces and found a correspondence between such spaces and partially ordered sets under some conditions. They further found that these conditions are highly related to Dehn-Sommerville equations after defining a boundary of weight on the poset. I pick some of those conditions and the boundary operator to define Euler characteristic structure equipped on and weight homology of the poset respectively. In this thesis, I study properties of the Euler characteristic structure as well as weight homology in detail....[ Read more ]

In [2], Chen and Yan introduced the notion of Eulerian stratified spaces and found a correspondence between such spaces and partially ordered sets under some conditions. They further found that these conditions are highly related to Dehn-Sommerville equations after defining a boundary of weight on the poset. I pick some of those conditions and the boundary operator to define Euler characteristic structure equipped on and weight homology of the poset respectively. In this thesis, I study properties of the Euler characteristic structure as well as weight homology in detail.

In the Euler characteristic structure, I find all constraints of choosing d and χ in the structure. These constraints even restrict the attention of certain posets. In the weight homology, I find several properties such as torsion, long exact sequence and duality. Moreover, several examples and an inductive algorithm in calculating the homology are included. Furthermore, after knowing the homology is a direct sum of Ζ₂, I find the necessary and sufficient condition for the homology taking the maximal number of Ζ₂.