  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 $k$ generators, for every $k \geq 3$. As a special case of our
construction, we obtain a finitely generated positive cone for $F_2 \times
\mathbb{Z}$.
