Finitely presented lattice-ordered abelian groups with order-unit

Let G be an ℓ-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 U of unital ℓ-groups, those ℓ-groups having a distinguished order-unit u, only the (⇐)-direction holds in general. We show that a unital ℓ-group (G,u) is finitely presented iff it has a basis. A large class of projectives is constructed from bases having special properties.