Locally pure topological abelian groups: elementary invariants

Abstract

Traditional model theory deals with first-order theories of algebraic systems. A basic result in the model theory of abelian groups, obtained by Szmielew [13] in 1955, is the decidability of the full theory of abelian groups. Szmielew uses the method of elimination of quantifiers, which typically produces the sharpest results. More abstract model theoretic methods can be used to obtain Szmielew’s results. In the process the results lose some of their effectivity, but gain in algebraic content. Eklof and Fisher [2] reworked Szmielew’s results in terms of a detailed analysis of saturated abelian groups. They were able to give a complete classification of somewhat (i.e. wl-) saturated abelian groups. As it turned out, the algebraic tools needed for this are all to be found in Kaplansky’s monograph [7].