2012/09/14 by Andrea Pedrini, Pedrini, Andrea · 2 citations
Computer Science · Mathematics · #06F20 #46A40 #52B45 #Advanced Algebra and Logic #Advanced Topics in Algebra #FOS: Mathematics #Geometric Topology (math.GT) #Holomorphic and Operator Theory #Metric Geometry (math.MG) #math.GT #math.MG #msc:06F20 #msc:46A40 #msc:52B45
paper · pdf · doi:10.48550/arxiv.1209.3248
13 pages
arxiv created 2012/09/14 · openalex publication_date 2012/09/14 · arxiv updated 2012/09/17 · openalex created_date 2017/05/26 · openalex updated_date 2026/07/28
A celebrated theorem of Hadwiger states that the Euler-Poincaré characteristic is the the unique invariant and continuous valuation on the distributive lattice of compact polyhedra in Rn 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-Poincaré 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.