Homology and Topology of Partially Ordered Fields
Abstract
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.
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BibTeX
@misc{bao2026homology,
title = {{Homology and Topology of Partially Ordered Fields}},
author = {Hanzhen Bao},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06261},
eprint = {2611.06261},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06261},
}
RIS
TY - UNPB
TI - Homology and Topology of Partially Ordered Fields
AU - Hanzhen Bao
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06261
UR - https://arxiv.org/abs/2611.06261
AB - 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.
ER -