Abstract
We answer two open problems about lattice-ordered groups that admit a connected lattice-ordered group topology. We show that, in the general case, admitting a connected lattice-ordered group topology does not effect the algebraic structure of the lattice-ordered group. For example, admitting a connected lattice-ordered group topology does not imply that the lattice-ordered group is Archimedean or even representable. On the other hand, if one assumes that the lattice-ordered group has a basis, then admitting a lattice-ordered group topology implies that the lattice-ordered group is a subdirect product of copies of the real numbers.
Similar content being viewed by others
Data Availability Statement
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Anderson, M., Feil, T.: Lattice Ordered Groups (An Introduction). D. Redel Publishing Company, The Netherlands (1988)
Ball, R.N.: Topological lattice-ordered groups. Pac. J. Math. 83(1), 1–26 (1979)
Bludov, V.V., Glass, A.M.W., Kopytov, V.M., Medvedev, N.Ya.: Unsolved problems on ordered and orderable groups. ar**v:0906.2621 [math.GR]
Darnel, M.R.: Theory of Lattice-Ordered Groups. Marcel Dekker, New York (1995)
Husain, T.: Introduction to Topological Group. Saunders, Philadelphia (1996)
Kopytov, V.M., Medvedev, N.Ya.: Ordered groups, Selected Questions in Algebra, Collection of Works in Memory of N.Ya. Medvedev, Barnaul, Altaisky State University, pp. 15–112 (2007) (Russian)
Nadler, S.B.: Continuum Theory (An Introduction). Marcel Dekker, New York (1992)
Scrimger, E.B.: A large class of small varieties of lattice-ordered groups. Proc. Am. Math. Soc. 51(2), 301–306 (1975)
Smarda, B.: Topologies in \(\ell \)-groups. Arch. Math. (Brno) T. 3(2), 60–81 (1967)
Acknowledgements
The author would like to thank Homeira Pajoohesh for making him aware of these problems, discussing the problems at length, and suggesting ideas without which this paper could not have been written.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by W. Wm. McGovern.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Jordan, F. Connected topological lattice-ordered groups. Algebra Univers. 84, 4 (2023). https://doi.org/10.1007/s00012-022-00800-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00012-022-00800-6