  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.

Metric Geometry;Geometric Topology