Expanded Theory of Ordered Abelian Groups

The first-order theory of ordered abelian groups was analyzed in 3 (opens in new tab). However, algebraic results on ordered abelian groups in the literature usually cannot be stated in first-order logic. Typically they involve so-called convex subgroups. Here we introduce an expanded theory of ordered abelian groups that allows quantification over convex subgroups and expresses almost all relevant algebra. We classify ordered abelian groups by the properties expressible in the expanded theory, and we prove that the expanded theory of ordered abelian groups is decidable. Curiously, the decidability proof is simpler than that in 3 (opens in new tab). Furthermore, the decision algorithm is primitive recursive.

This is a journal version of 19 (opens in new tab) which could not be published in USSR for non-scientific reasons.