Metric properties of positively ordered monoids
Résumé
We introduce here an intrinsic (quasi-) metric on each positively ordered monoid (POM), which is defined in terms of the evaluation map from the given POM to its bidual and for which POM homomorphisms are continuous. Moreover, we find a class of refinement POMs which, equipped with the canonical metric, are complete metric spaces; this class includes the class of weak cardinal algebras, but also most cases of completions of a certain kind (we will call it 'strongly reduced products') of POMs, and of which a prototype has been used in a previous paper for the description of the evaluation map from a given refinement POM into its bidual. This result can also be viewed as a wide generalization to the non-linearly ordered case (for example weak cardinal algebras) of the (Cauchy-) completeness of the real line.