Fans and generators of free abelian l-groups

TL;DR

This paper investigates the decidability of isomorphism problems for free abelian lattice-ordered groups generated by specific elements, revealing both decidable and undecidable cases, and explores related geometric structures using fans.

Contribution

It establishes decidability results for recognizing free generators and isomorphisms in free abelian -groups, and demonstrates undecidability in more general cases, advancing understanding of their structure.

Findings

Decidability of -isomorphism to \u00a7_n is proven.

Undecidability of -isomorphism to _l for arbitrary l.

Decidability of recognizing free generating sets when m=n.

Abstract

Let $t_1,\ldots,t_n$ be $\ell$-group terms in the variables $X_1,\ldots,X_m$. Let $\hat t_1,\ldots,\hat t_n$ be their associated piecewise homogeneous linear functions. Let $G $ be the $\ell$-group generated by $\hat t_1, \ldots,\hat t_n$ in the free $m$-generator $\ell$-group $\mathcal A_m.$ We prove: (i) the problem whether $G$ is $\ell$-isomorphic to $\mathcal A_n$ is decidable; (ii) the problem whether $G$ is $\ell$-isomorphic to $\mathcal A_l$ ($l$ arbitrary) is undecidable; (iii) for $m=n$, the problem whether $\{\hat t_1,\ldots,\hat t_n\}$ is a {\it free} generating set is decidable. In view of the Baker-Beynon duality, these theorems yield recognizability and unrecognizability results for the rational polyhedron associated to the $\ell$-group $G$. We make pervasive use of fans and their stellar subdivisions.