Monoids of Intervals of Ordered Abelian Groups Original Research Article

For any partially ordered abelian groupG, we relate the structure of the ordered monoid Λ(G) ofintervalsofG(i.e., nonempty, upward directed lower subsets ofG), to various properties ofG, as for example interpolation properties, or topological properties of the state space whenGhas 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 afirst-order sentenceon the positive interval monoid.