Separating minimal valuations, point-continuous valuations and continuous valuations

TL;DR

This paper provides concrete examples of continuous valuations on dcpos to distinguish minimal, point-continuous, and continuous valuations, highlighting their differences through specific constructions involving Johnstone's dcpo and the Sorgenfrey line.

Contribution

The paper introduces explicit examples of continuous valuations that separate minimal, point-continuous, and continuous valuations on dcpos, aiding in understanding their distinctions.

Findings

Example of a point-continuous valuation on Johnstone's dcpo that is not minimal.

Construction of a continuous valuation on the Sorgenfrey line that is not point-continuous.

Potential use of the constructed valuation as a counterexample for Fubini-type equations on dcpos.

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 $\mu$ be the continuous valuation on $\mathcal J$ with $\mu(U) =1$ for nonempty Scott opens $U$ and $\mu(U) = 0$ for $U=\emptyset$. Then $\mu$ is a point-continuous valuation on $\mathcal J$ that is not minimal. (2) Lebesgue measure extends to a measure on the Sorgenfrey line $\mathbb R_{l}$. Its restriction to the open subsets of $\mathbb R_{l}$ is a continuous valuation $\lambda$. Then its image valuation $\overline\lambda$ through the embedding of $\mathbb R_{l}$ into its Smyth powerdomain $\mathcal Q\mathbb R_{l}$ 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.