On the order dimension of locally countable partial orderings

Higuchi, Kojiro; Lempp, Steffen; Raghavan, Dilip; Stephan, Frank

doi:10.1090/proc/14946

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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