Cartesian closed categories of FƵ-domains

Abstract

A subset system Z assigns to each partially ordered set P a certain collection Z(P) of subsets. In this paper, a new kind of subset systems called directable subset systems is introduced. For a directable subset system Ƶ, the concepts of FƵ-way-below relation and FƵ-domain are introduced. The well-known Scott topology is naturally generalized to the Ƶ-level and the resulting topology is called FƵ-Scott topology, and the continuous functions with respect to this topology are characterized by preserving the suprema of directed Ƶ-sets. Then, we mainly consider a generalization of the cartesian closedness of the categories DCPO of directed complete posets, BF of bifinite domains and FS of FS-domains to the Ƶ-level. Corresponding to them, it is proved that, for a suitable subset system Ƶ, the categories F Ƶ CPO of Ƶ-complete posets, FSF Ƶ of finitely separated FƵ-domains and BFF Ƶ of bifinite FƵ-domains are all cartesian closed. Some examples of these categories are given.