The essence of ideal completion in quantitative form

Abstract

If a poset lacks joins of directed subsets, one can pass to its ideal completion. But doing this means also changing the setting: The universal property of ideal completion of posets suggests that it should be regarded as a functor from the category of posets with monotone maps to the category of dcpos with Scott-continuous functions as morphisms. The same applies for the quantitative version of ideal completion suggested in the literature. As in the case of posets, it seems advantageous to consider a different topology with the completed spaces. We introduce topological V - continuity spaces and their Smyth completion and show that this is an adequate setting to consider ideal completion of quantitative domains: Performing the Smyth completion of a V -continuity space regarded as topological V -continuity space gives the ideal completion of the original space together with its Scott topology.