Canonical Lattices and Integer Relations Associated to Rational Fans

TL;DR

This paper develops a canonical lattice theory for rational fans, introducing invariants that capture local and global linear relations, and proves a generation theorem linking local relations to the global structure.

Contribution

It introduces the ray and relation lattices as functorial invariants and establishes a generation theorem connecting local and global relations in rational fans.

Findings

Global relation lattice generated by local relations on cones of codimension ≥ 1

Filtration sensitive to the fan's facial structure

Subdivisions do not increase filtration depth

Abstract

We propose a canonical local-to-global lattice theory for rational fans. We define the $\textit{ray lattice } L_{\mathrm{rays}}(\Sigma)$ and the $\textit{relation lattice } L_{\mathrm{rel}}(\Sigma)$ as invariants functorial under fan isomorphisms. We introduce $\textit{star-local relation lattices}$, defined via the relation lattice of the localized quotient fan, which capture the linear dependencies visible within local neighborhoods. We define a $\textit{codimension filtration}$ on the global relation lattice and prove a generation theorem: the global lattice is generated by local relations supported on the stars of cones of codimension at least 1. This filtration is sensitive to the facial structure of $\Sigma$; explicit examples and a conjecture suggest that subdivisions can only preserve or lower filtration depths, distinguishing fans with different combinatorial topologies.