Lattice-ordered Abelian groups and Schauder bases of unimodular fans

Baker-Beynon duality theory yields a concrete representation of any finitely generated projective Abelian lattice-ordered group in terms of piecewise linear homogeneous functions with integer coefficients, defined over the support of a fan . A unimodular fan over determines a Schauder basis of : its elements are the minimal positive free generators of the pointwise ordered group of -linear support functions. Conversely, a Schauder basis of determines a unimodular fan over : its maximal cones are the domains of linearity of the elements of . 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, is finitely generated projective iff it can be presented by a purely lattice-theoretical word.