Classification of finitely generated lattice-ordered abelian groups with order-unit

TL;DR

This paper provides a classification of finitely generated unital lattice-ordered abelian groups using sequences of weighted simplicial complexes, linking algebraic properties to combinatorial structures.

Contribution

It introduces a novel classification method connecting unital $ extit{ ext{l}}$-groups with sequences of weighted simplicial complexes, enabling property analysis.

Findings

Classification via sequences of weighted simplicial complexes

Recognition criterion for isomorphic unital $ extit{ ext{l}}$-groups

Properties like total order and finitely presented can be read from sequences

Abstract

A unital $\ell$-group $(G,u)$ is an abelian group $G$ equipped with a translation-invariant lattice-order and a distinguished element $u$, called order-unit, whose positive integer multiples eventually dominate each element of $G$. We classify finitely generated unital $\ell$-groups by sequences $\mathcal W = (W_{0},W_{1},...)$ of weighted abstract simplicial complexes, where $W_{t+1}$ is obtained from $W_{t}$ either by the classical Alexander binary stellar operation, or by deleting a maximal simplex of $W_{t}$. A simple criterion is given to recognize when two such sequences classify isomorphic unital $\ell$-groups. Many properties of the unital $\ell$-group $(G,u)$ can be directly read off from its associated sequence: for instance, the properties of being totally ordered, archimedean, finitely presented, simplicial, free.