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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the order dimension of locally countable partial orderings
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by Kojiro Higuchi, Steffen Lempp, Dilip Raghavan and Frank Stephan PDF
Proc. Amer. Math. Soc. 148 (2020), 2823-2833 Request permission

Abstract:

We show that the order dimension of any locally countable partial ordering $(P, <)$ of size $\kappa ^+$, for any $\kappa$ of uncountable cofinality, is at most $\kappa$. In particular, this implies that it is consistent with ZFC that the dimension of the Turing degrees under partial ordering can be strictly less than the continuum.
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Additional Information
  • Kojiro Higuchi
  • Affiliation: College of Engineering, Nihon University, 1 Nakagawara, Tokusada, Tamuramachi, Koriyama, Fukushima Prefecture 963-8642, Japan
  • MR Author ID: 943964
  • Email: higuchi.koujirou@nihon-u.ac.jp
  • Steffen Lempp
  • Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1325
  • MR Author ID: 247988
  • Email: lempp@math.wisc.edu
  • Dilip Raghavan
  • Affiliation: Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076, Republic of Singapore
  • MR Author ID: 870765
  • Email: matrd@nus.edu.sg
  • Frank Stephan
  • Affiliation: Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076, Republic of Singapore
  • MR Author ID: 335879
  • ORCID: 0000-0001-9152-1706
  • Email: fstephan@comp.nus.edu.sg
  • Received by editor(s): February 15, 2019
  • Received by editor(s) in revised form: March 29, 2019, October 9, 2019, and November 10, 2019
  • Published electronically: February 26, 2020
  • Additional Notes: The first author was partially supported by Grant-in-Aid for JSPS Fellows.
    The second author was partially supported by NSF grant DMS-160022
    The third author was partially supported by the Singapore Ministry of Education’s research grant number MOE2017-T2-2-125.
    The fourth author was partially supported by the Singapore Ministry of Education Academic Research Fund Tier 2 grant MOE2016-T2-1-019 / R146-000-234-112.
  • Communicated by: Heike Mildenberger
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 2823-2833
  • MSC (2010): Primary 06A06, 03E04; Secondary 03D28
  • DOI: https://doi.org/10.1090/proc/14946
  • MathSciNet review: 4099772