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Separating minimal valuations, point-continuous valuations, and continuous valuations

Published online by Cambridge University Press:  07 December 2021

Jean Goubault-Larrecq Open the ORCID record for Jean Goubault-Larrecq [Opens in a new window]  and
Xiaodong Jia Open the ORCID record for Xiaodong Jia [Opens in a new window]
Jean Goubault-Larrecq
Affiliation:
Université Paris-Saclay, CNRS, ENS Paris-Saclay, Laboratoire Méthodes Formelles, 91190, Gif-sur-Yvette, France
Xiaodong Jia*
Affiliation:
School of Mathematics, Hunan University, Changsha, Hunan, 410082, China
*
*Corresponding author. Email: jia.xiaodong@yahoo.com
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Abstract

We give two concrete examples of continuous valuations on dcpo’s to separate minimal valuations, point-continuous valuations, and continuous valuations:

  1. (1) Let ${\mathcal J}$ be the Johnstone’s non-sober dcpo, and μ be the continuous valuation on ${\mathcal J}$ with μ(U)=1 for nonempty Scott opens U and μ(U)=0 for $U=\emptyset$. Then, μ is a point-continuous valuation on ${\mathcal J}$ that is not minimal.

  2. (2) Lebesgue measure extends to a measure on the Sorgenfrey line $\mathbb{R}_\ell$. Its restriction to the open subsets of $\mathbb{R}_\ell$ is a continuous valuation λ. Then, its image valuation $\overline\lambda$ through the embedding of $\mathbb{R}_\ell$ into its Smyth powerdomain $\mathcal{Q}\mathbb{R}_\ell$ in the Scott topology is a continuous valuation that is not point-continuous.

We believe that our construction $\overline\lambda$ might be useful in giving counterexamples displaying the failure of the general Fubini-type equations on dcpo’s.

Keywords

Minimal valuationspoint-continuous valuationscontinuous valuationsFubini equations

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Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Issue 6 , June 2021 , pp. 614 - 632
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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