Finitely generic abelian lattice-ordered groups

The authors characterize the finitely generic abelian lattice-ordered groups and make application of this characterization to specific examples. A key goal in Abraham Robinson's development of model-theoretic forcing was to explicate the notion of algebraically closed, even when the appropriate classes may not be first-order axiomatizable. Interesting links sometimes appear between purely algebraic properties and model-theoretic properties such as existentially closed (e.c.) and finitely generic. In this spirit we consider the characterization of finitely generic abelian i-groups, as well as the model-theoretic properties of certain e.c. abelian i-groups. The model theory of abelian lattice-ordered (1-) groups was developed by Glass and Pierce in (G-P) and (G-P2). They showed that every finitely generic structure is hyperarchimedean, and that the group C(X, R) is existentially closed. They also stated several problems, including: (i) Distinguish the finitely generic models among the hyperarchimedean e.c. ones (G-P,p. 263).