On cartesian closed extensions of non-pointed domains

Abstract

Let P denote a cartesian closed full subcategory of the category POSET of posets and Scott continuous functions. We define C-P to be the full subcategory of objects from P whose D-completion is isomorphic to an object from C, where C is a subcategory of the category CONT of domains. The category C-P is always a subcategory of the category CONTP of continuous posets and Scott continuous functions. We prove that if C is a cartesian closed full subcategory of F-L, U-L, F-RB or U-RB, then the category C-P is also cartesian closed. It is known that the category CDCPO of consistent directed complete posets and Scott continuous functions is cartesian closed. In particular, we have the following cartesian closed categories: F-L-CDCPO, U-L-CDCPO, F-RB-CDCPO, U-RB-CDCPO, F-aL-CDCPO, U-aL-CDCPO, F-B-CDCPO, U-B-CDCPO, etc. If the categories FS and RB coincide, then it leads directly to the most potential of this uniform way of finding new cartesian closed categories of continuous posets: for every cartesian closed full subcategory C of CONT, the category C-P is also cartesian closed.