Some comments and examples on generation of (hyper-)Archimedean \(\ell\)-groups and \(f\)-rings
It is important to know the theory of subcategories with the objects that are obtained by adjoining a special element to the original objects. For instance, Mundici's equivalence also holds between the categories of abelian \(\ell\)-groups and perfect MV-algebras, which is due to the fact that for an abelian \(\ell\)-group \(G\), the lexicographical product of \(G\) and \(\mathbb Z\) admits a strong order unit. The paper under review discusses the process of adjoining a weak order unit to an \(\ell\)-group, or an identity to an \(f\)-ring, and finds significant contrasts between these cases. In \(\ell\)-groups, hyperarchimedeanness and similar properties fail to pass from generating structures to the structures they generate, as illustrated by a basic example of Conrad and Martinez. For reduced \(f\)-rings, on the other hand, these properties do inherit upwards.
scientific article; zbMATH DE number 5799082
scientific article