Form equivalence class Howard-Rubin Number: 14
Statement: Robinson's Valuation Lemma: Let \(\{\phi_{x}:x\in X\}\) be a family of partial valuations on a set \(S\). (\(\phi\) is apartial valuation on \(S\) if the domain of \(\phi\subseteq S\) and therange of \(\phi \subseteq \{ 0,1\}\).) Suppose \(\frak B\) is a filterbase on \(X\) such that for every finite \(F\subseteq S\) and forevery \(B\in\frak B\) there is \(x\in B\) such that \(F\subseteq\) dom\(\phi_{x}\). Then there is a total valuation \(f\) on \(S\) (that is, dom \(f= S\)) such that for all finite \(F\subseteq S\) and all \(B\in\frak B\),\(\exists x\in B\) such that \(F\subseteq\) dom \(\phi_{x}\) and \(f|F = \phi_x|F\). (\(\frak B\) is a filter base means \(\forall B_{1},\ B_{2}\in\frak B\),\(\exists B_{3}\in\frak B\) such that \(B_{3}\subseteq B_{1}\cap B_{2}.)\)Rav [1977] and Cowen [1973]. Compare with [14 I].
Howard-Rubin number: 14 AJ
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