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On the order dimension of locally countable partial orderings


Authors: Kojiro Higuchi, Steffen Lempp, Dilip Raghavan and Frank Stephan
Journal: Proc. Amer. Math. Soc. 148 (2020), 2823-2833
MSC (2010): Primary 06A06, 03E04; Secondary 03D28
DOI: https://doi.org/10.1090/proc/14946
Published electronically: February 26, 2020
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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
Email: higuchi.koujirou@nihon-u.ac.jp

Steffen Lempp
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1325
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
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
Email: fstephan@comp.nus.edu.sg

DOI: https://doi.org/10.1090/proc/14946
Keywords: Partial ordering, dimension, Turing degrees
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
Article copyright: © Copyright 2020 American Mathematical Society