Wehrung, Friedrich Metric properties of positively ordered monoids. (English) Zbl 0769.06008 Forum Math. 5, No. 2, 183-201 (1993). Summary: We introduce here an intrinsic (quasi-) metric on each positively ordered monoid (P.O.M.), which is defined in terms of the evaluation map from the given P.O.M. to its bidual and for which P.O.M. homomorphisms are continuous. Moreover, we find a class of refinement P.O.M.’s 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 P.O.M.’s, and of which a prototype has been used in a previous paper for the description of the evaluation map of a given refinement P.O.M. 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. Cited in 3 Documents MSC: 06F05 Ordered semigroups and monoids 06F30 Ordered topological structures 28B10 Group- or semigroup-valued set functions, measures and integrals 54E35 Metric spaces, metrizability Keywords:quasimetric; strongly reduced products; positively ordered monoid; complete metric spaces; weak cardinal algebras PDFBibTeX XMLCite \textit{F. Wehrung}, Forum Math. 5, No. 2, 183--201 (1993; Zbl 0769.06008) Full Text: DOI EuDML HAL