Monoids of intervals of ordered abelian groups
Résumé
For any partially ordered abelian group G, we relate the structure of the ordered monoid ?(G) of intervals of G (i.e., nonempty, upward directed lower subsets of G), to various properties of G, as for example interpolation properties, or topological properties of the state space when G has an order-unit. This allows us to solve a problem by K.R. Goodearl by proving that even in most natural cases, multiplier groups of dimension groups often fail to be interpolation groups. Furthermore, the study of monoids of intervals in the totally ordered case yields a characterization of Hahn powers of the real line by a ?rst-order sentence on the positive interval monoid.
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