Studies On Maximum Of "D-->D"(μ)and Continuity Under Linear Order

Domain theory plays a fiIndaxnental role in denotational semanics ofProgranuning langUages, Which is familiar to us. The concepts ofWximation and convergence etc. are imPortan. wnle we have no ideaabout the degree of aPproxhaion. Mar. 2 000,Keye Mch first discus sedthis problem in his doctor's degree thesis, namely ((A Fotmdation forComPutaton). Based on these ideas, he established a comParativelycomPlete analysis system, and brough forward new concePts such asindue ed tOPOlogy, meot on dOmain, p tOPOlogy Whch areseeIning1y sultable fOr establislunent of comPutatonal models. Of courseMer research is still necessary. ln ths paPer, we discussed fo1lowingproblems:l. Measurement is raised fOr the characteriZation of the degree oft aPProximation. We discussed the measuremeni on a to-continuous domain.- In this way, we are enabled to have a further tmderstanding ofmeasuremellt itself2. We gave the charaCeriZation theorem of maximum of [D--+D](U),thus comPldely solved the problem about the U-Q continuous maPs withmonotone measures raised by Keye Matin in Mar.2000.3. Metric space as a classic structure is known very well. But, no order exists on a metric space. Keye Martin raised an order on a metric space, so-called linear order. In this paper, we raised linear approximation on a metric space. Further, we solved the problem, which is under what kind of conditions a metric space with a linear order forms a continuous partial order set (i.e. poset) or a domain? And the results concerned on the quotient space also had been discussed.