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

Abstract : Abstract We give two concrete examples of continuous valuations on dcpo’s to separate minimal valuations, point-continuous valuations, and continuous valuations: (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) 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.
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https://hal.inria.fr/hal-03469452
Contributor : Jean Goubault-Larrecq Connect in order to contact the contributor
Submitted on : Tuesday, December 7, 2021 - 4:51:18 PM
Last modification on : Friday, January 7, 2022 - 3:37:50 AM

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Jean Goubault-Larrecq, Xiaodong Jia. Separating minimal valuations, point-continuous valuations, and continuous valuations. Mathematical Structures in Computer Science, Cambridge University Press (CUP), 2021, pp.1-19. ⟨10.1017/S0960129521000384⟩. ⟨hal-03469452⟩

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