Invariant homomorphisms of nonstandard extensions of Boolean algebras and vector lattices
Let \(B\) be a Boolean algebra and let \({}^{\ast}B\) be its nonstandard extension in a polysaturated Robinson superstructure. A mapping \(h:{}^{\ast}B\rightarrow B\) is called a \(\ast\)-invariant Boolean homomorphism (briefly, \(\ast\)-IBH) if \(h\) is a Boolean homomorphism and \(h(b)=b\) for all \(b\in B\). The main theorem asserts that a \(\ast\)-IBH exists if and only if \(B\) is complete; a \(\ast\)-IBH is uniquely defined if and only if \(B\) is atomic. The method of Boolean-valued models is used in the proof. Further, this result is extended to \(\ast\)-invariant linear homomorphisms of vector lattices. Finally, the author considers a Boolean algebra \(B\) with positive \(\sigma\)-additive measure \(\mu : B\rightarrow \mathbf{R}_+\). He proves that if \((B, \mu)\) is a nonatomic complete normed Boolean algebra and \(h: {}^{\ast}B\rightarrow B\) is a \(\ast\)-IBH then, for every standard \(\varepsilon >0\), there exists a \(\chi_{\varepsilon}\in {}^{\ast}B\) such that \({}^{\ast}\mu (\chi_{\varepsilon})<\varepsilon\) and \(h(b\land\chi_{\varepsilon})=h(b)\) for all \(b\in {}^{\ast}B\). This implies that the measure \(\mu\circ h\) is singular with respect to the Loeb measure \(\text{st}\circ\mu\).
scientific article; zbMATH DE number 1037968
scientific article