Simplicial geometry of unital lattice-ordered abelian groups

TL;DR

This paper explores the geometric structure of finitely presented unital lattice-ordered abelian groups using duality with rational polyhedra, providing new constructions, a classification theorem, and geometric characterizations.

Contribution

It introduces a duality-based framework to analyze finitely presented unital $ extit{ ext{l}}$-groups, including constructions of limits, a classification theorem, and geometric characterizations.

Findings

Constructed finite limits and co-limits in the category of finitely presented unital $ extit{ ext{l}}$-groups.

Proved a Cantor-Bernstein-Schr"oder theorem for these groups.

Provided a geometric characterization of finitely generated subalgebras of free objects.

Abstract

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 $\FP$.