On finite and totally finite elements in vector lattices
Let \(X\) be an Archimedean vector lattice. An element \(\phi\) in \(X\) is called finite if there exists a \(z\in X\) such that for every \(x\in X\) there exists a number \(C_x\) such that \(|x|\wedge n|\phi|\leq C_xz\) for all \(n\in\mathbb{N}\). The element \(z\) is then called a majorant for \(\phi\). The set of all finite elements is an ideal in \(X\) and is denoted by \(\Phi_1(X)\). An element of \(X\) which has a majorant which itself belongs to \(\Phi_1(X)\), is called totally finite. The set of all totally finite elements is an ideal in \(X\) and is denoted by \(\Phi_2(X)\). It is clear that \(\{0\}\subset\Phi_1(X)\subset \Phi_2(X)\subset X\). If, in the latter chain of inclusions, one replaces all of the inclusion signs by either equality or strict inclusion signs, the question arises whether each of the eight ensuing possibilities is realized by some vector lattice \(X\). That question is fully answered in the present paper. For instance, the situation \(\{0\}= \Phi_1(X)\neq \Phi_2(X)\neq X\) turns out to be impossible. Furthermore, a certain topological condition in the set of all maximal ideals of \(X\) equipped with the hull-kernel topology is introduced, under which \(\Phi_1(X)= \Phi_2(X)\).
scientific article; zbMATH DE number 817242