Explicit left orders on free groups extending the lexicographic order on free monoids

TL;DR

This paper constructs explicit left orders on finitely generated free groups that extend the lexicographic order on free monoids, using piecewise linear homeomorphisms, with efficient membership decision and a context-free positive cone.

Contribution

It introduces a novel explicit left order on free groups extending monoid orders, defined via free actions on the line, with decidable membership and a context-free positive cone.

Findings

Order extension from monoids to free groups

Linear-time membership decision procedure

Positive cone is a context-free language

Abstract

For every finitely generated free group we construct an explicit left order extending the lexicographic order on the free monoid generated by the positive letters. The order is defined by a left, free action on the orbit of 0 of a free group of piecewise linear homeomorphisms of the line. The membership in the positive cone is decidable in linear time in the length of the input word. The positive cone forms a context-free language closed under word reversal.