Lattice-ordered Abelian groups and Schauder bases of unimodular fans
Corrado Manara, Vincenzo Marra, Daniele Mundici · Transactions of the American Mathematical Society · 2006
Baker-Beynon duality theory yields a concrete representation of any finitely generated projective Abelian lattice-ordered group G G in terms of piecewise linear homogeneous functions with integer coefficients, defined over the support | Σ | |\Sigma | of a fan Σ \Sigma . A unimodular fan Δ \Delta over | Σ | |\Sigma | determines a Schauder basis of G G : its elements are the minimal positive free generators of the pointwise ordered group of Δ \Delta -linear support functions. Conversely, a Schauder basis H \mathbf {H} of G G determines a unimodular fan over | Σ | |\Sigma | : its maximal cones are the domains of linearity of the elements of H \mathbf {H} . The main purpose of this paper is to give various representation-free characterisations of Schauder bases. The latter, jointly with the De Concini-Procesi starring technique, will be used to give novel characterisations of finitely generated projective Abelian lattice ordered groups. For instance, G G is finitely generated projective iff it can be presented by a purely lattice-theoretical word.