论文标题
连续的R价态
Continuous R-valuations
论文作者
论文摘要
我们通过将持续估值的价值从真实的价值扩展到所谓的Abelian D-Rags $ R $来引入定向完整的POSET（简称DCPO）的连续$ r $ $ $ valuations（简称DCPO）。 就像琼斯和Plotkin引入的估值Monad $ \ Mathbf {V} $一样，我们表明，在DCPOS和Scott-Continous Map的类别上，连续的$ r $ valuations的构建扩展到了强大的Monad $ \ Mathbf {V} r $。此外，就像两位作者和C.théron的最新作品一样，第二作者B. Lindenhovius，M。Mislove和V. Zamdzhiev，我们表明我们可以提取一个交换性的Monad $ \ mathbf {v}^r_m $，其中我们称之为Minimal $ r $ $ $ -Valuations。 We also show that continuous $R$-valuations have close connections to measures when $R$ is taken to be $\mathbf{I}\mathbb{R}^\star_+$, the interval domain of the extended nonnegative reals: (1) On every coherent topological space, every non-zero, bounded $τ$-smooth measure $μ$ (defined on the Borel $σ$ -Algebra），在规范上确定连续的$ \ mathbf {i} \ Mathbb {r}^\ star _+$ - 估值; and (2) such a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation is the most precise (in a certain sense) continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation that approximates $μ$, when the support of $μ$ is a compact Hausdorff subspace of a第二个可稳定的紧凑拓扑空间。这特别适用于单位间隔的Lebesgue度量。结果，可以将Lebesgue度量标识为连续$ \ Mathbf {i} \ Mathbb {r}^\ star _+$ - 估值。此外，我们表明后者很小。
We introduce continuous $R$-valuations on directed-complete posets (dcpos, for short), as a generalization of continuous valuations in domain theory, by extending values of continuous valuations from reals to so-called Abelian d-rags $R$. Like the valuation monad $\mathbf{V}$ introduced by Jones and Plotkin, we show that the construction of continuous $R$-valuations extends to a strong monad $\mathbf{V}^R$ on the category of dcpos and Scott-continuous maps. Additionally, and as in recent work by the two authors and C. Théron, and by the second author, B. Lindenhovius, M. Mislove and V. Zamdzhiev, we show that we can extract a commutative monad $\mathbf{V}^R_m$ out of it, whose elements we call minimal $R$-valuations. We also show that continuous $R$-valuations have close connections to measures when $R$ is taken to be $\mathbf{I}\mathbb{R}^\star_+$, the interval domain of the extended nonnegative reals: (1) On every coherent topological space, every non-zero, bounded $τ$-smooth measure $μ$ (defined on the Borel $σ$-algebra), canonically determines a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation; and (2) such a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation is the most precise (in a certain sense) continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation that approximates $μ$, when the support of $μ$ is a compact Hausdorff subspace of a second-countable stably compact topological space. This in particular applies to Lebesgue measure on the unit interval. As a result, the Lebesgue measure can be identified as a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation. Additionally, we show that the latter is minimal.
