An analysis of the logic of Riesz Spaces with strong unit

Lukasiewicz逻辑与Riesz空间

Lukasiewicz logic and Riesz spaces

本文首先对具有标量乘法的MV-代数{ \em Riesz MV-代数}进行了深入的研究，它是由$[0,1]$得到的.推广了MV-代数与$\ell$ -群之间的Mundici等价性，证明了在具有强单位的Riesz空间中，Riesz MV-代数与单位区间范畴等价.此外，范数完备Riesz MV-代数的子类等价于交换单位C $^*$ -代数类.以Riesz MV-代数为模型的命题演算${\mathbb R}{\cal L}$是Rumukasiewicz $\infty$ -值命题演算的保守推广，并且它在标准模型$[0,1]$中的求值是完备的.证明了该逻辑的一个范式定理，推广了McNaughton定理.我们定义了${\mathbb R}{\cal L}$中公式的拟线性组合和拟线性跨度的概念，并将它们与${\mathbb R}{\cal L}$的de Finetti相干准则的类似物联系起来。

半单MV-代数的Riesz壳

The Riesz hull of a semisimple MV-algebra

MV-代数和Riesz MV-代数分别与具有强单位的交换格-序群和具有强单位的Riesz空间（向量格）范畴等价.格序群的一个标准构造是阿基米德格序群的向量格壳。遵循类似的方法，本文定义了半单MV-代数的Riesz壳。

Uniform logical proofs for Riesz representation theorem, Daniell-Stone theorem and Stone's representation theorem for probability algebras

Riesz representation theorem, Daniell-Stone theorem for Daniell integrals and Stone's representation theorem for probability and measure algebras are three important classical results in analysis concerning existence of measures with certain properties. Many proofs of these theorems can be found in the literature of analysis, from elementary ones which use ordinary techniques from measure theory, to more sophisticated ones, such as those employing techniques from nonstandard analysis, in particular for Riesz representation theorem. In this paper, as the first goal, we give new proofs for all t…

Dedekind sigma-complete l-groups and Riesz spaces as varieties

We prove that the category of Dedekind $σ$-complete Riesz spaces is an infinitary variety, and we provide an explicit equational axiomatization. In fact, we show that finitely many axioms suffice over the usual equational axiomatization of Riesz spaces. Our main result is that $\mathbb{R}$, regarded as a Dedekind $σ$-complete Riesz space, generates this category as a quasi-variety, and therefore as a variety. Analogous results are established for the categories of (i) Dedekind $σ$-complete Riesz spaces with a weak order unit, (ii) Dedekind $σ$-complete lattice-ordered groups, and (iii) Dedekin…

Operations that preserve integrability, and truncated Riesz spaces

For any real number $p\in [1,+\infty)$, we characterise the operations $\mathbb{R}^I \to \mathbb{R}$ that preserve $p$-integrability, i.e., the operations under which, for every measure $μ$, the set $\mathcal{L}^p(μ)$ is closed. We investigate the infinitary variety of algebras whose operations are exactly such functions. It turns out that this variety coincides with the category of Dedekind $σ$-complete truncated Riesz spaces, where truncation is meant in the sense of R. N. Ball. We also prove that $\mathbb{R}$ generates this variety. From this, we exhibit a concrete model of the free Dedekin…

Infinitary logic and basically disconnected compact Hausdorff spaces

We extend Łukasiewicz logic obtaining the infinitary logic $\mathcal{IR}Ł$ whose models are algebras $C(X,[0,1])$, where $X$ is a basically disconnected compact Hausdorff space. Equivalently, our models are unit intervals in $σ$-complete Riesz spaces with strong unit. The Lindenbaum-Tarski algebra of $\mathcal{IR}Ł$ is, up to isomorphism, an algebra of $[0,1]$-valued Borel functions. Finally, our system enjoys standard completeness with respect to the real interval $[0,1]$.