Homology and Topology of Partially Ordered Fields

We develop a unified homology theory for directed partial orders (DPOs) and lattice-ordered positive cones on fields of characteristic zero. With the underlying field and its operations fixed, we define order?carrying spaces independently by topological conditions, establish a bijection with DPO positive cones, and characterize lattice orders by intersections of minimal open neighborhoods. A unified comparison theorem identifies graded Ext for region ideals with sheaf cohomology and multiplicative Tor with the reduced singular homology of truncated regions, giving a topological interpretation of the existence and preservation of reduced decompositions. We obtain canonical cohomological obstructions to lattice ordering, prove that a coefficient module for positive multiplication is flat if and only if the multiplier has a positive inverse, and show that every strict extension of positive cones on the same field produces a nonzero first Tor obstruction. Finally, we place the concrete lattice recently constructed by Bao-Zhang within this framework.