Decomposition of Garside groups and self-similar L-algebras

Picantin's iterated crossed product representation of Garside monoids is extended and reproved for a wide class of not necessarily noetherian partially ordered groups with a right invariant lattice structure. It is shown that the tree-like structure of such an iterated crossed product is equivalent to a partial cycle set, closely related to a class of set-theoretic solutions of the quantum Yang-Baxter equation. The decomposition of finite square-free solutions is related to the crossed product representation of the corresponding structure group.