Semantic Domain Based On The R-posets Preliminary Study

In the paper, our research object are posets with families of approximating partial orders (seen in [1])( called R-poset).We devote to explore if the mathematical structure R-poset can be a better mathematical framework for research in semantic domains. We define the Scott topology on R-poset, which is the foundation of establishing the notions of approximation and continuity of functions, and makes preparations for exploring the mathematical characters on R-poset. The theory of semantic domains based on metric spaces has been developed a lot . [2] recreates part of the theory on sets with families of equivalences(sfe) . R-poset are more generalized structures than sfe. We imitate part of the results obtained on sfe in [2] and the proof of Tarski fixed-point theorem for dcpo . We recreate a fixed-point theorem for approximating maps on R-poset and Tarski fixed-point theorem on R-poset. And we construct a new category R-POSET, build an adjunction between R-POSET and GUMS ,which supplies a train of thoughts for researching the R-POSET from the standpoint of metric space .