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Local properties of complete Boolean products. (English) Zbl 0584.06006
From the author’s introduction: ”Products of complete Boolean algebras find applications in the theory of models of axiomatic set theory. The iteration of Boolean-valued models is closely connected with products of the underlying Boolean algebras. [...] In this paper we investigate the structure and properties, especially the local ones, of complete Boolean products. As consequences we answer some questions concerning (m,0)- products. Other consequences are connected to models of set theory. [...] The first section is devoted to local properties. It is shown there that any complete product can be decomposed into its locally independent and locally nowhere independent parts. The main result says that for independent products local independence and minimality are equivalent. The final part of the section discusses the implication between the concepts of independence, disjointness, local independence and local disjointness. In the second section examples of complete Boolean products are given for two special algebras: the Cantor algebra \({\mathcal C}\) and the random algebra \({\mathcal R}\). [...] The main results of the section state that there is a product of the algebras \({\mathcal R}\) and \({\mathcal R}\), the so-called quadratic product, which is incomparable with the minimal product. The quadratic product of \({\mathcal R}\) and \({\mathcal R}\) is not locally independent, but is locally disjoint. The third section shows the connection of localized properties of complete products with properties of pairs of Cohen or random numbers.”
Reviewer: S.Rudeanu
MSC:
06E10 Chain conditions, complete algebras
06E05 Structure theory of Boolean algebras
03C90 Nonclassical models (Boolean-valued, sheaf, etc.)
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