Abstract
We call a linear order L strongly surjective if whenever K is a suborder of L then there is a surjective f : L → K so that x ≤ y implies f(x) ≤ f(y). We prove various results on the existence and non-existence of uncountable strongly surjective linear orders answering questions of R. Camerlo, R. Carroy and A. Marcone. In particular, ♢+ implies the existence of a lexicographically ordered Suslin tree which is strongly surjective and minimal; every strongly surjective linear order must be an Aronszajn type under \(2^{\aleph _{0}}<2^{\aleph _{1}}\) or in the Cohen and other canonical models (where \(2^{\aleph _{0}}= 2^{\aleph _{1}}\)); finally, we prove that it is consistent with CH that there are no uncountable strongly surjective linear orders at all.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abraham, U., Shelah, A.: Martin’s axiom does not imply that every two ℵ 1-dense sets of reals are isomorphic. Israel J. Math. 38, 161–176 (1981)
Abraham, U., Matatyahu, R., Saharon, S.: On the consistency of some partition theorems for continuous colorings, and the structure of ℵ 1-dense real order types. Ann. Pure Appl. Logic 29(2), 123–206 (1985)
Baumgartner, J.E.: All ℵ 1-dense sets of reals can be isomorphic. Fundam. Math. 79(2), 101–106 (1973)
Baumgartner, J.E.: Order Types of Real Numbers and Other Uncountable Orderings. Ordered Sets, pp 239–277. Springer, Netherlands (1982)
Camerlo, R, Carroy, R, Marcone, A: Epimorphisms between linear orders. Order 32, 387–400 (2015). Erratum, Order 33 (2016), 187
Camerlo, R., Carroy, R., Marcone, A.: Linear orders: when embeddability and epimorphism agree. preprint, arXiv:1701.02020
Devlin, K.J., Saharon, S.: A weak version of ♢ which follows from \(2^{\aleph _{0}}<2^{\aleph _{1}}\). Israel J. Math. 29(2–3), 239–247 (1978)
Guzman, O., Hrušák, M.: Parametrized ♢-principles and canonical models. Slides from Retrospective workshop on Forcing and its applications. Fields Institute (2015)
Hajnal, A., Nagy, Zs., Soukup, L.: On the number of certain subgraphs of graphs without large cliques and independent subsets. In: A Tribute to Paul Erdos, p. 223 (1990)
Jech, T: Set theory. The third millennium edition, revised and expanded. Springer Monographs in Mathematics. Springer, Berlin (2003). xiv+ 769 pp. ISBN: 3-540-44085-2
Moore, J.T.: A five element basis for the uncountable linear orders. Ann. of Math. (2) 163(2), 669–688 (2006)
Moore, J.T.: ω 1 and − ω 1 may be the only minimal uncountable linear orders. Mich. Math. J. 55(2), 437–457 (2007)
Moore, J., Hrušák, M., Džamonja, M.: Parametrized ♢ principles. Trans. Am. Math. Soc. 356(6), 2281–2306 (2004)
Todorcevic, S.: Trees and linearly ordered sets. In: Handbook of Set-Theoretic Topology, pp 235–293 (1984)
Todorcevic, S.: Walks on ordinals and their characteristics, vol. 263. Springer Science & Business Media (2007)
Acknowledgements
We thank S. Friedman, M. Hrušák, J. Moore and A. Rinot for helpful discussions and remarks. We are especially grateful to R. Carroy for stimulating conversations throughout this project. Finally, we are grateful for the anonymous referee’s careful reading and many useful comments.
The author was supported in part by the FWF Grant I1921 and OTKA grant no.113047.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Soukup, D.T. Uncountable Strongly Surjective Linear Orders. Order 36, 43–64 (2019). https://doi.org/10.1007/s11083-018-9454-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-018-9454-7