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 $G$ in terms of piecewise linear homogeneous functions with integer coefficients, defined over the support $\vert\Sigma\vert$ of a fan $\Sigma$. A unimodular fan $\Delta$ over $\vert\Sigma\vert$ determines a Schauder basis of $G$: its elements are the minimal positive free generators of the pointwise ordered group of $\Delta$-linear support functions. Conversely, a Schauder basis $\mathbf{H}$ of $G$ determines a unimodular fan over $\vert\Sigma\vert$: its maximal cones are the domains of linearity of the elements of $\mathbf{H}$. 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, $G$ is finitely generated projective iff it can be presented by a purely lattice-theoretical word.