THE THEORY OF ORDERED ABELIAN GROUPS DOES NOT HAVE THE INDEPENDENCE PROPERTY

ABSTRACT. We prove that no complete theory of ordered abelian groups hasthe independence property, thus answering a question by B. Poizat. The maintool is a result contained in the doctoral dissertation of Yuri Gurevich and alsoin P. H. Schmitt's Elementary properties of ordered abelian groups, which basicallytransforms statements on ordered abelian groups into statements on colouredchains. We also prove that every ra-type in the theory of coloured chains hasat most 2" coheirs, thereby strengthening a result by B. Poizat. 0. Background and introduction. The independence property is one of theproperties introduced by S. Shelah in [16].DEFINITION. A complete theory T is said to have the independence property, if there is a formula 0: T\j(3xs\ f\ v(xs,cl)k A -.<p(f,c') : s C fc is consistent. To convey an idea of the relevance of this definition we remember two resultsfrom (logical) stability theory.