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

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.

Continuous valuation and logic

We consider the continuous valuation of logic, where the certainty of a statement is measured with a number in the closed unit interval I =[0, 1].The idea originates in continuous logics, which have been investigated from various standpoints, in [1] ~ [4] and [8] ~ [10], for example. More comprehensive information can be seen in [4] and others.

Continuous Valuation

SynopsisBy expressing the values of both liabilities and assets as it is possible to obtain the valuation surplus and its analysis not only at the end of the year but also at any time during the year. Practical forms of gi(t) are discussed and formulae for a continuous analysis of surplus are given.