Groups with right-invariant multiorders

TL;DR

This paper characterizes when free abelian groups of rank m can have generic n-tuples of linear orders as Cayley objects, linking group structure to order properties using diophantine approximation.

Contribution

It establishes a necessary and sufficient condition for free abelian groups to admit certain Cayley objects with multiple linear orders.

Findings

A free abelian group of rank m admits the generic n-tuple of linear orders as a Cayley object if and only if m > n.

The proof employs Kronecker's Theorem on diophantine approximation.

Provides background and context for the main theorem.

Abstract

A Cayley object for a group G is a structure on which G acts regularly as a group of automorphisms. The main theorem asserts that a necessary and sufficient condition for the free abelian group G of rank m to have the generic n-tuple of linear orders as a Cayley object is that m>n. The background to this theorem is discussed. The proof uses Kronecker's Theorem on diophantine approximation.