A remarkable family of left-ordered groups: Central extensions of Hecke groups

We provide an infinite family of left-ordered groups that have a positive cone that is generated by two elements as a semigroup. This family corresponds to that of certain central extensions of Hecke groups, and includes the Klein bottle group and the braid group B3. Using the classical convex extension (flipping) procedure, on each of the groups of this family we define an ordering sharing many properties with Dehornoy's. Several related questions and problems are addressed.