No positive cone in a free product is regular

TL;DR

The paper proves that positive cones in free products of certain groups cannot be represented by regular languages, establishing a limit on their language complexity within the Chomsky hierarchy.

Contribution

It demonstrates that no regular language can represent positive cones in free products of nontrivial, finitely generated, left-orderable groups, refining previous complexity bounds.

Findings

No regular positive cone exists in free products of the specified groups.

Positive cones in free groups of rank ≥ 2 can be context-free but not regular.

The result strengthens previous semigroup generation limitations for positive cones.

Abstract

We show that there exists no left order on the free product of two nontrivial, finitely generated, left-orderable groups such that the corresponding positive cone is represented by a regular language. Since there are orders on free groups of rank at least two with positive cone languages that are context-free (in fact, 1-counter languages), our result provides a bound on the language complexity of positive cones in free products that is the best possible within the Chomsky hierarchy. It also provides a strengthening of a result by Cristobal Rivas stating that the positive cone in a free product of nontrivial, finitely generated, left-orderable groups cannot be finitely generated as a semigroup.