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Metric properties of positively ordered monoids. (English) Zbl 0769.06008

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.

MSC:

06F05 Ordered semigroups and monoids
06F30 Ordered topological structures
28B10 Group- or semigroup-valued set functions, measures and integrals
54E35 Metric spaces, metrizability
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