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