The Euler characteristic of a polyhedron as a valuation on its coordinate vector lattice

TL;DR

This paper extends Hadwiger's theorem by characterizing the Euler characteristic as a unique valuation on finitely presented unital vector lattices associated with polyhedra, linking geometric invariants with lattice-theoretic structures.

Contribution

It provides an analogue of Hadwiger's theorem for vector lattices, identifying the Euler characteristic as a unique valuation on these algebraic structures related to polyhedra.

Findings

The Euler characteristic is the unique vl-valuation assigning one to each vl-Schauder hat.

All finitely presented unital vector lattices arise from continuous piecewise linear functions on polyhedra.

The paper establishes a correspondence between geometric invariants and algebraic valuations in vector lattices.

Abstract

A celebrated theorem of Hadwiger states that the Euler-Poincar\'e characteristic is the the unique invariant and continuous valuation on the distributive lattice of compact polyhedra in R^n that assigns value one to each convex non-empty such polyhedron. This paper provides an analogue of Hadwiger's result for finitely presented unital vector lattices (i.e. real vector spaces with a compatible lattice order, also known as Riesz spaces). The vector lattice of continuous and piecewise (affine) linear real-valued functions on a compact polyhedron, with operations defined pointwise from the vector lattice R, is a finitely presented unital vector lattice; and it is a non-trivial fact that all such vector lattices arise in this manner, to within an isomorphism. Each function in such a vector lattice can be written as a linear combination of a subset of distinguished elements that we call vl-Schauder hats. We prove here that the functional that assigns to each non-negative piecewise linear function on the polyhedron the Euler-Poincar\'e characteristic of its support is the unique vl-valuation (a special class of valuations on vector lattices) that assigns one to each vl-Schauder hat of the vector lattice.