We propose a criterion for preserving the regularity of a formal language representation when passing from groups to subgroups. We use this criterion to show that the regularity of a positive cone language in a left-orderable group passes to its finite index subgroups, and to show that there exists no left order on a finitely generated acylindrically hyperbolic group such that the corresponding positive cone is represented by a quasi-geodesic regular language. We also answer one of Navas’ questions by giving an example of an infinite family of groups which admit a positive cone that is generated by exactly [Formula: see text] generators, for every [Formula: see text]. As a special case of our construction, we obtain a finitely generated positive cone for [Formula: see text].

A regular left-order on a finitely generated group G is a total, left-multiplication invariant order on G whose corresponding positive cone is the image of a regular language over the generating set of the group under the evaluation map. We show that admitting regular leftorders is stable under extensions and wreath products and we give a classification of the groups whose left-orders are all regular left-orders. In addition, we prove that a solvable Baumslag-Solitar group B.1; n/ admits a regular left-order if and only if n 1. Finally, Hermiller and Sunić showed that no free product admits a regular left-order. We show that if A and B are groups with regular left-orders, then .A B/ Z admits a regular left-order.

A regular left-order on finitely generated group a group G is a total, left-multiplication invariant order on G whose corresponding positive cone is the image of a regular language over the generating set of the group under the evaluation map. We show that admitting regular left-orders is stable under extensions and wreath products and give a classification of the groups all whose left-orders are regular left-orders. In addition, we prove that solvable Baumslag-Solitar groups B(1, n) admits a regular left-order if and only if n ≥ −1. Finally, Hermiller and Sunić showed that no free product admits a regular left-order, however we show that if A and B are groups with regular left-orders, then (A * B) × Z admits a regular left-order.