Resumen de Maximal abelian group actions on the ordered real line and their digital representations

Reinhard Winkler

• Investigations on commuting functions, done by many authors into diverse directions, lead to the search for maximal abelian groups of maps respecting a certain structure. Here we investigate the example of automorphisms of the linear ordering $ (\mathbb{R}, \leq) $. Nontrivial phenomena occur which might be typical for much more general situations. In our example one can use results from the theory of lattice ordered groups (in particular the Conrad-Harvey-Holland Theorem) to get representations by real valued functions. This leads to a set of invariants which, modulo a notion of equivalence which is rather well understood, classifies all maximal abelian subgroups of the automorphism group of $ (\mathbb{R}, \leq) $ up to conjugation.