Rooted wreath products

We introduce “rooted valuation products” and use them to construct universal Abelian lattice-ordered groups (with prescribed set of components) from the more classical theory of H. Hahn. The wreath product construction of W.C. Holland and S.H. McCleary generalised the Abelian (lattice-ordered) permutation group ideas to give universals for transitive (ℓ-)permutation groups with prescribed set of primitive components. In the case of (not necessarily transitive) sublattice subgroups of order-preserving permutations of totally ordered sets, the set of natural congruences forms a root system. We generalise the rooted valuation product construction to the permutation case when all natural primitive components are regularly obtained; we analogously obtain universals for these permutation groups (for a prescribed set of natural primitive components) which we call “rooted wreath products.” We identify the rooted valuation product with an appropriate subgroup of the corresponding rooted wreath product. The maximal Abelian group actions on the ordered real line were characterised in by R. Winkler, and their digital representations were consequently obtained. We use the rooted wreath product construction to get a more general result, and deduce Winklerʹs characterisation as a consequence.