Rational polyhedra and projective lattice-ordered abelian groups with order unit

TL;DR

This paper characterizes finitely generated projective unital lattice-ordered abelian groups using algebraic topology and toric variety theory, extending classical results in lattice-ordered group theory.

Contribution

It combines algebraic topology and toric geometry to describe finitely generated projective unital l-groups, providing new insights beyond classical algebraic results.

Findings

Finitely generated projective unital l-groups are characterized using topological and geometric methods.

The paper extends classical algebraic results by integrating algebraic topology with toric variety theory.

A new description of these groups is provided, connecting lattice-ordered group theory with modern algebraic geometry.

Abstract

An l-group G is an abelian group equipped with a translation invariant lattice order. Baker and Beynon proved that G is finitely generated projective iff it is finitely presented. A unital l-group is an l-group G with a distinguished order unit, i.e., an element u in G whose positive integer multiples eventually dominate every element of G. While every finitely generated projective unital l-group is finitely presented, the converse does not hold in general. Classical algebraic topology (a la Whitehead) will be combined in this paper with the W{\l}odarczyk-Morelli solution of the weak Oda conjecture for toric varieties, to describe finitely generated projective unital l-groups.