Finitely presented lattice-ordered abelian groups with order-unit

Let $G$ be an $\mathrm{\ell }$-group (which is short for ``lattice-ordered abelian group’'). Baker and Beynon proved that $G$ is finitely presented iff it is finitely generated and projective. In the category $\mathcal{U}$ of {\it unital} $\mathrm{\ell }$-groups—those $\mathrm{\ell }$-groups having a distinguished order-unit $u$—only the $(\Leftarrow )$-direction holds in general. Morphisms in $\mathcal{U}$ are {\it unital $\mathrm{\ell }$-homomorphisms,} i.e., hom-o-mor-phisms that preserve the order-unit and the lattice structure. We show that a unital $\mathrm{\ell }$-group $(G,u)$ is finitely presented iff it has a basis, i.e., $G$ is generated by an abstract Schauder basis over its maximal spectral space. Thus every finitely generated projective unital $\mathrm{\ell }$-group has a basis $\mathcal B$. As a partial converse, a large class of projectives is constructed from bases satisfying $\bigwedge \mathcal{B}≠0$. Without using the Effros-Handelman-Shen theorem, we finally show that the bases of any finitely presented unital $\mathrm{\ell }$-group $(G,u)$ provide a direct system of simplicial groups with 1-1 positive unital homomorphisms, whose limit is $(G,u)$.