First-order theories of subgroups of divisible Hahn products

Abstract(All the ℓ-groups we consider are Abelian.) Some first-order theories of divisible ℓ-groups are well known, for example the theory of the totally ordered ones and the theories of the projectable ones (in: A.M.W. Glass, W.C. Holland (Eds.), Lattice-ordered Groups, Kluwer Academic Press, Dordrecht, 1989, pp. 41–79). In this paper we study some theories of nonprojectable divisible ℓ-groups, the simplest example of which is R×→(R×R) (the lexicographic product of R by the direct product R×R). We introduce a generalization of the projectability property (r-projectability). We prove that the class of r-projectable special-valued divisible ℓ-groups is an elementary class and give a classification of its completions