\(\alpha\)-Dedekind complete archimedean vector lattices versus \(\alpha\)- quasi-\(F\) spaces (Q1203839)

Let \(\alpha\) denote an uncountable cardinal number, \(\mathcal W\) the category of Archimedean vector lattices with a distinguished weak order unit and vector lattice homomorphisms preserving this weak order unit and \({\mathcal W}(\alpha)\) the category of \(\mathcal W\) objects with \(\alpha\)-complete morphisms. In this paper it is shown, among others, that in \({\mathcal W}(\alpha)\) the full subcategory of \(\alpha\)-Dedekind complete objects is epireflective. It is also explained how the Yosida functor connects the algebraic notions of \(\alpha\)-Dedekind complete, \(\alpha\)-dense and \(\alpha\)-jam-dense with the topological notions of \(\alpha\)-quasi-\(F\), \(\alpha\)-irreducible and \(\alpha\)-SpFi morphism. Moreover, the \(\alpha\)- quasi-\(F\) cover of a compact space \(X\) is the Yosida space of the \(\alpha\)-Dedekind complete epireflection of \(C(X)\) in \({\mathcal W}(\alpha)\).