Injective positively ordered monoids. II
This is a continuation of work on positively ordered monoids (POM) began by the author in the paper reviewed above, where the author gave arithmetical characterizations of injective POM's. Here, criteria are found for embedding an arbitrary POM into an injective POM, in which case the POM is called regular. Unlike the case in Abelian groups or Boolean algebras, many POM's are not regular. The most fundamental regular POM is the extended real line \(P\), and injective POM's are exactly the retracts of powers of \(P\). A class of POM's larger than the injective POM's is refinement POM's. Statements proved by \textit{A. Tarski} [Cardinal algebras (1949; Zbl 0041.345)] for cardinal algebras are shown to be ``approximately true'' in all refinement POM's. A functor from the category of refinement POM's to itself allows characterization of the evaluation map of a refinement POM with applications to equidecomposability type POM's, a regularity criterion, and the existence of Banach limits.
scientific article; zbMATH DE number 166153
scientific article