Finitely presented lattice-ordered abelian groups with order-unit

TL;DR

This paper characterizes finitely presented unital lattice-ordered abelian groups using bases, linking algebraic and spectral properties, and constructs these groups via direct limits of simplicial groups.

Contribution

It establishes that finitely presented unital $ ext{l}$-groups are characterized by the existence of a basis, and constructs these groups as direct limits without relying on the Effros-Handelman-Shen theorem.

Findings

Finitely presented unital $ ext{l}$-groups are characterized by the existence of a basis.

Every finitely generated projective unital $ ext{l}$-group has a basis.

Finitely presented unital $ ext{l}$-groups can be constructed as direct limits of simplicial groups.

Abstract

Let $G$ be an $\ell$-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} $\ell$-groups---those $\ell$-groups having a distinguished order-unit $u$---only the $(\Leftarrow)$-direction holds in general. Morphisms in $\mathcal U$ are {\it unital $\ell$-homomorphisms,} i.e., hom\-o\-mor\-phisms that preserve the order-unit and the lattice structure. We show that a unital $\ell$-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 $\ell$-group has a basis $\mathcal B$. As a partial converse, a large class of projectives is constructed from bases satisfying $\bigwedge\mathcal B\not=0$. Without using the Effros-Handelman-Shen theorem, we finally show that the bases of any finitely presented unital $\ell$-group $(G,u)$ provide a direct system of simplicial groups with 1-1 positive unital homomorphisms, whose limit is $(G,u)$.