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.
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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.
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Presented by W. Wm. McGovern.
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Jordan, F. Connected topological lattice-ordered groups. Algebra Univers. 84, 4 (2023). https://doi.org/10.1007/s00012-022-00800-6
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DOI: https://doi.org/10.1007/s00012-022-00800-6