Abstract
Let G and H be totally ordered Abelian groups such that, for some integer k, the lexicographic powers G kand H kare isomorphic (as ordered groups). It was proved by F. Oger that G and H need not be isomorphic. We show here that they are whenever G is either divisible or ω1 -saturated (and in a few more cases). Our proof relies on a general technique which we also use to prove that G and H must be elementary equivalent as ordered groups (a fact also proved by F. Delon and F. Lucas) and isomorphic as chains. The same technique applies to the question of whether G and H should be isomorphic as groups, but, in this last case, no hint about a possible negative answer seems available.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. C. Eklof and E. R. Fisher (1972) The elementary theory of Abelian groups, Ann. Math. Logic 4, 115–171.
C. C. Chang and H. J. Keisler (1973) Model Theory, Studies in Logic No. 73. North-Holland, Amsterdam.
A. L. S. Corner (1963) Every countable torsion-free ring is an endomorphism ring, Proc. London Math. Soc. (3) 13, 687–710.
F. Delon and F. Lucas; Inclusions et produits de groupes abéliens ordonnés étudiés au premier ordre, J. Symbolic Logic, to appear.
S. Feferman and R. L. Vaught (1959) The first order properties of products of algebraic systems, Fund. Math. 47, 57–103.
F. Oger (1987) Produits lexicographiques de groupes ordonnés: Isomorphisme et equivalence élémentaire, J. Algebra 109, 452–467.
F. Oger, An example of two nonisomorphic countable ordered Abelian groups with isomorphic lexicographical squares, Bull. London Math. Soc. To appear.
J. G. Rosenstein (1982) Linear Orderings, Academic Press, New York.
P. H. Schmidt (1982) Model theory of ordered Abeling groups, Habilitationsschrift, Heidelberg.
W. Sierpinski (1965) Cardinal and Ordinal Numbers, PWN Warsaw.
Author information
Authors and Affiliations
Additional information
Communicated by A. M. W. Glass
Rights and permissions
About this article
Cite this article
Giraudet, M. Cancellation and absorption of lexicographic powers of totally ordered Abelian groups. Order 5, 275–287 (1988). https://doi.org/10.1007/BF00354895
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00354895