Ordering groups and validity in lattice-ordered groups

TL;DR

This paper characterizes subsets of groups that extend to positive cones of orders, linking equation validity in lattice-ordered groups to properties of free groups, and provides new proofs for key decidability results.

Contribution

It introduces a new characterization of subsets extending to positive cones, connecting validity in lattice-ordered groups with properties of free groups and automorphisms.

Findings

Decidability of the word problem for free l-groups

Generation of the variety of l-groups by automorphisms of the real line

New proofs of validity criteria in lattice-ordered groups

Abstract

A characterization is given of the subsets of a group that extend to the positive cone of a right order on the group and used to relate validity of equations in lattice-ordered groups (l-groups) to subsets of free groups that extend to positive cones of right orders. This correspondence is used to obtain new proofs of the decidability of the word problem for free l-groups and generation of the variety of l-groups by the l-group of automorphisms of the real number line. A characterization of the subsets of a group that extend to the positive cone of an order on the group is also used to establish a correspondence between the validity of equations in varieties of representable l-groups (equivalently, classes of ordered groups) and subsets of relatively free groups that extend to positive cones of orders.