Simplicial geometry of unital lattice-ordered abelian groups

By an $\ell$-group $G$ we mean a lattice-ordered abelian group. This paper is concerned with the category $\FP$ of finitely presented {\it unital} $\ell$-groups, those $\ell$-groups having a distinguished order-unit $u$. Using the duality between $\FP$ the category of rational polyhedra, we will provide (i) a construction of finite limits and co-limits in $\FP$; (ii) a Cantor-Bernstein-Schr\"oder theorem for finitely presented unital $\ell$-groups; (iii) a geometrical characterization of finitely generated subalgebras of free objects of $\FP$.