Extension from substitution-defined trees to general prefix-closed right-extendable trees

Determine whether the methods for characterizing positionality of Dumont–Thomas complement numeration systems can be adapted to abstract numeration systems defined by prefix-closed and right-extendable trees that do not arise from an underlying substitution, with the aim of characterizing positionality for general abstract numeration systems.

Background

The paper’s methods apply to abstract numeration systems generated by trees arising from substitutions. The authors observe that abstract numeration systems can also be defined by trees without an underlying substitution, provided that the associated language is prefix-closed and right-extendable. They explicitly ask whether their techniques extend to this broader setting, potentially advancing the general characterization of positional abstract numeration systems.