The infinitesimal approach to the representation of vector lattices by spaces of continuous functions on a compactum (Q1363240)

We suggest an infinitesimal method of realization of Archimedean vector lattices with a unit in the form of lattices of extended continuous functions on a compact topological space. By this method we obtain nonstandard proofs of Vulikh's theorem on the functional representation of \(K\)-spaces with a unit and of the theorem on the conditional completeness of an arbitrary \((r)\)-complete vector lattice with projection onto strips. In this work we primarily use the embedding of the superstructure \(M\), which contains the objects in question, into the nonstandard Robinson extension \({}^*M\). We assume everywhere that \(E\) is a vector lattice with a unit \(e\). We assume without stipulation that the embedding \(M \subseteq {}^*M\) is \(|E|_+\)-saturated, where \(|E|\) is the cardinality of the lattice \(E\).