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A generalization of incidence relations in abstract polytope has beenexplored, and parameterized surfaces are used as primers. The abstractorientable incidence structure is defined as an algebraic model of incidencerelations, in which some algebraic properties in abtract polytope theory aregeneralized. The geometric interpretation of abstract orientable incidencestructure are also discussed. The orientable incidence structure in asemi-regular normal CW complex are briefly investigated.

Abstract Orientable Incidence Structure and Algorithms for Finite Bounded Acyclic Categories. I. Incidence Structure