represent decompositions of isometries of ℝ
into symmetries and rotations. Some convergent (that is noetherian and confluent) rewrite system for this structure was introduced by the first author. One of the rules is similar to Yang-Baxter equation. It involves a map
In order to obtain the algebraic properties of
, we study the confluence of critical peaks (or critical pairs) for our rewrite system. For that purpose, we introduce
describing the calculation of angles of rotations generated by rewriting. In particular, one of those properties is related to the
(also called Zamolodchikov equation).