Embedding theorems and generalized discrete ordered abelian groups

Abstract:

Let G be a totally ordered commutative group. For each nonzero element $g \in G$, let $L(g)$ denote the largest convex subgroup of G not containing g. Denote by $U(g)$ the smallest convex subgroup of G that contains g. The group G is said to be generalized discrete if $U(g)/L(g)$ is order isomorphic to the additive group of integers for all $g \ne 0$ in G. This paper is principally concerned with the structure of generalized discrete groups. In particular, the problem of embedding a generalized discrete group in the lexicographic product of its components, $U(g)/L(g)$, is studied. We prove that such an embedding is not always possible (contrary to statements in the literature). However, we do establish the validity of this embedding when G is countable. In case F is o-separable as well as countable, the structure of G is completely determined.