On Kierstead’s conjecture
HTML articles powered by AMS MathViewer
- by Keng Meng Ng and Maxim Zubkov PDF
- Trans. Amer. Math. Soc. 372 (2019), 3713-3753 Request permission
Abstract:
We settle the long-standing Kierstead conjecture in the negative. We do this by constructing a computable linear order with no rational subintervals, where every block has order type finite or $\zeta$, and where every computable copy has a strongly nontrivial $\Pi ^0_1$ automorphism. We also construct a strongly $\eta$-like linear order where every block has size at most $4$ with no rational subinterval such that every $\Delta ^0_2$ isomorphic computable copy has a nontrivial $\Pi ^0_1$ automorphism.References
- R. G. Downey, Computability theory and linear orderings, Handbook of recursive mathematics, Vol. 2, Stud. Logic Found. Math., vol. 139, North-Holland, Amsterdam, 1998, pp. 823–976. MR 1673590, DOI 10.1016/S0049-237X(98)80047-5
- Rodney G. Downey, Asher M. Kach, and Daniel Turetsky, Limitwise monotonic functions and their applications, Proceedings of the 11th Asian Logic Conference, World Sci. Publ., Hackensack, NJ, 2012, pp. 59–85. MR 2868506, DOI 10.1142/9789814360548_{0}004
- Rodney G. Downey and Michael F. Moses, On choice sets and strongly nontrivial self-embeddings of recursive linear orders, Z. Math. Logik Grundlag. Math. 35 (1989), no. 3, 237–246. MR 1000966, DOI 10.1002/malq.19890350307
- Ben Dushnik and E. W. Miller, Partially ordered sets, Amer. J. Math. 63 (1941), 600–610. MR 4862, DOI 10.2307/2371374
- Andrey N. Frolov and Maxim V. Zubkov, Increasing $\eta$-representable degrees, MLQ Math. Log. Q. 55 (2009), no. 6, 633–636. MR 2582163, DOI 10.1002/malq.200810031
- A. N. Frolov and M. V. Zubkov, Limitwise monotonic functions relative to the Kleene’s Ordinal Notation System, Lobachevskii J. Math. 35 (2014), no. 4, 295–301. MR 3284616, DOI 10.1134/S1995080214040167
- Kenneth Harris, $\eta$-representation of sets and degrees, J. Symbolic Logic 73 (2008), no. 4, 1097–1121. MR 2467206, DOI 10.2178/jsl/1230396908
- Charles M. Harris, Kyung Il Lee, and S. Barry Cooper, Automorphisms of $\eta$-like computable linear orderings and Kierstead’s conjecture, MLQ Math. Log. Q. 62 (2016), no. 6, 481–506. MR 3601091, DOI 10.1002/malq.201400109
- Asher M. Kach and Daniel Turetsky, Limitwise monotonic functions, sets, and degrees on computable domains, J. Symbolic Logic 75 (2010), no. 1, 131–154. MR 2605885, DOI 10.2178/jsl/1264433912
- Henry A. Kierstead, On $\Pi _1$-automorphisms of recursive linear orders, J. Symbolic Logic 52 (1987), no. 3, 681–688. MR 902983, DOI 10.2307/2274356
- Joseph G. Rosenstein, Linear orderings, Pure and Applied Mathematics, vol. 98, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. MR 662564
- Steven Thomas Schwarz, QUOTIENT LATTICES, INDEX SETS, AND RECURSIVE LINEAR ORDERINGS, ProQuest LLC, Ann Arbor, MI, 1982. Thesis (Ph.D.)–The University of Chicago. MR 2611822
- Steven Schwarz, Recursive automorphisms of recursive linear orderings, Ann. Pure Appl. Logic 26 (1984), no. 1, 69–73. MR 739913, DOI 10.1016/0168-0072(84)90041-1
- Guohua Wu and Maxim Zubkov, The Kierstead’s conjecture and limitwise monotonic functions, Ann. Pure Appl. Logic 169 (2018), no. 6, 467–486. MR 3780011, DOI 10.1016/j.apal.2018.01.003
- M. V. Zubkov, Sufficient conditions for the existence of ${\bf 0}’$-limitwise monotone functions for computable $\eta$-like linear orders, Sibirsk. Mat. Zh. 58 (2017), no. 1, 107–121 (Russian, with Russian summary); English transl., Sib. Math. J. 58 (2017), no. 1, 80–90. MR 3686945, DOI 10.1134/s0037446617010128
Additional Information
- Keng Meng Ng
- Affiliation: Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371
- MR Author ID: 833062
- Maxim Zubkov
- Affiliation: N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal University, Kremlevskaya 18, Kazan 420008, Russia
- Received by editor(s): August 15, 2018
- Received by editor(s) in revised form: March 30, 2019
- Published electronically: May 9, 2019
- Additional Notes: The first author was partially supported by grants MOE-RG131/17 and MOE2015-T2-2-055
The second author was supported by an RSF grant (project 18-11-00028) - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 3713-3753
- MSC (2010): Primary 03C57, 03D45
- DOI: https://doi.org/10.1090/tran/7858
- MathSciNet review: 3988623