Equivalence Theory Of Finite Posets

Posets are important mathematical objects which have close relations with Algebra,Topology and Logic,etc;Deep study of it can enhance the understanding of itself and the associations with other subjects.By utilizing the concepts and methods developed in Algebra Topology,Algebra Geometry and Algebra Representations,we first depicted the concepts and results of Incidence Algebra which reflects the linear structure of underlying posets and Sheaf theory which reflects the topological structure of underlying poset in the framework of Category Theory.For detail,we studied the Incidence Algebra I defined by finite Partially Ordered Set X,and sheaf F based on the topology T_x defined by the partial order;By studying the relation between finite(right) module category mod I and the(Abelian Group) sheaf category Sh_x A,we proved that they are equivalent.During this part,specific forms of some theories of theaf in the case of posets were obtained by enhanced pondering,thus not only supported the process of this part,but also further paved the way for following study.Next we raised the question that equivalence of posets in the background of Derived Category based on deep study of derived category of sheaf category.To obtain some characteristics of posets which are equivalent in the derived category sense,we adopted Helix Theory.Helix Theory has formed an efficient procedure in establishing the strongly exceptional collection for sheaf in order to determine the relations between objects.Finally,through some ideas from Algebraic Geometry,we constructed strongly exceptional collections of derived sheaf categories,and solved the problem that equivalences between some posets.