Natural non-dcpo domains and f-spaces

As Dag Normann has recently shown, the fully abstract model for PCF of hereditarily-sequential functionals is not ω-complete (in contrast to the old fully abstract continuous dcpo model of Milner). This is also applicable to a potentially wider class of models such as the recently constructed by the author fully abstract (universal) model for PCF = PCF + pif (parallel if). Here we will present an outline of a general approach to this kind of ‘natural’ domains which, although being non-dcpos, allow considering ‘naturally’ continuous functions (with respect to existing directed ‘pointwise’, or ‘natural’ least upper bounds). There is also an appropriate version of ‘naturally’ algebraic and ‘naturally’ bounded complete ‘natural’ domains which serves as the non-dcpo analogue of the well-known concept of Scott domains, or equivalently, the complete f-spaces of Ershov. It is shown that this special version of ‘natural’ domains, if considered under ‘natural’ Scott topology, exactly corresponds to the class of f-spaces, not necessarily complete.