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Boolean families of valuation rings. (English. Russian original) Zbl 0791.12004

Algebra Logic 31, No. 3, 170-181 (1992); translation from Algebra Logika 31, No. 3, 276-296 (1992).
Every Boolean space is homeomorphic to the space of maximal ideals of a certain Boolean algebra. So, it is quite natural to call a compact totally disconnected space a Boolean space. A family \(W\) of valuation rings of a field \(F\) is called weakly Boolean if the collection of subsets of \(W\) of the type \(V_ A^ F=\{R_ v \mid R_ v \in W,\;A \subseteq R_ v \}\), where \(A\) is a finite subset of \(F\), forms a closed-open basis of a Boolean topology on \(W\).
Let \(W\) be a weakly Boolean family of valuation rings of a field \(F\), and \(F_ 0\) an algebraic extension of \(F\). It is proved that the set \(W_ 0\) of all valuation rings \(R_{v_ 0}\) of \(F_ 0\) such that \(R_{v_ 0} \cap F \in W\) is a weakly Boolean family of valuation rings of \(F_ 0\). So it is possible to “lift” a weakly Boolean family of \(F\) to a weakly Boolean family of its algebraic extension \(F_ 0\).
A weakly Boolean family of valuations \(W\) is called Boolean if (1) for any \(a,b \in F\) there exists \(c \in F\) such that \(V_ a^ F \cap V^ F_ b=V_ c^ F\), (2) for any \(a \in F\) there exists \(a^*\) such that \(W \backslash V^ F_ a=V_{a^*}^ F\). A theorem on “lifting” Boolean families to algebraic extensions is proved.
Reviewer: G.Pestov (Tomsk)

MSC:

12J10 Valued fields
54G99 Peculiar topological spaces
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References:

[1] Yu. L. Ershov,Decidability Problems and Constructive Models [in Russian], Nauka, Moscow (1980).
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