Fans and generators of free abelian l-groups

Abstract

Let $t_1,\ldots,t_n$ be $\ell$-group terms in the variables $X_1,\ldots,X_m$. Let $\hat t_1,\ldots,\hat t_n$ be their associated piecewise homogeneous linear functions. Let $G $ be the $\ell$-group generated by $\hat t_1, \ldots,\hat t_n$ in the free $m$-generator $\ell$-group $\mathcal A_m.$ We prove: (i) the problem whether $G$ is $\ell$-isomorphic to $\mathcal A_n$ is decidable; (ii) the problem whether $G$ is $\ell$-isomorphic to $\mathcal A_l$ ($l$ arbitrary) is undecidable; (iii) for $m=n$, the problem whether $\{\hat t_1,\ldots,\hat t_n\}$ is a {\it free} generating set is decidable. In view of the Baker-Beynon duality, these theorems yield recognizability and unrecognizability results for the rational polyhedron associated to the $\ell$-group $G$. We make pervasive use of fans and their stellar subdivisions.