Elementary properties of ordered abelian groups

Abstract

Introduction. A complete classification of abelian groups by their elementary properties (i.e. properties that can be formalized in the lower predicate calculus) was given by Szmielew [9]. No such attempt, however, has so far been made with respect to ordered groups. In the present paper the classification by elementary properties will be carried out for all archimedean ordered abelian groups and, moreover, for a certain more general class of groups which we shall call regularly ordered. Simultaneously, necessary and sufficient conditions will be established for two such groups to be elementarily equivalent, i.e. to have all their elementary properties in common (see Theorem 4.7). By a complete classification of a class of groups we mean its partition into disjoint subclasses in such a way that two groups belong to one subclass if, and only if, they are elementarily equivalent. This goal will be attained by setting up a series of (finite or infinite) complete systems of axioms, each system defining a certain subclass of ordered abelian groups. The notation and terminology of [8] will be used throughout. In particular, the concept of model-completeness introduced in [8] and [7] will be applied. A novelty feature of the present paper is that the method based on model-completeness will be combined with what we shall call adjunction of new relations. To illustrate the usefulness of the method, we shall also apply it to give new simplified proofs of some theorems by Langford and Tarski on the completeness of certain systems of axioms referring to ordered sets. This will constitute an additional result of the paper (see ?2). 1. Preliminaries, terminology and notation. The concept of an ordered set can be formalized in the lower predicate calculus by means of the following system of axioms based on a binary relation of equivalence, E(x, y), (read: x is equivalent to and a binary relation of order, Q(x, y) (read: x is less than, or equivalent to, y):