On Natural Non-dcpo Domains

in his 87th year whose influence on me and help cannot be overstated. model for PCF of hereditarily sequential functionals is not ω-complete and therefore not continuous in the traditional terminology (in contrast to the old fully abstract continuous dcpo model of Milner). This is also applicable to a wider class of models such as the recently constructed by the author fully abstract (universal) model for PCF + = PCF + 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) and also have appropriate version of “naturally ” algebraic and “naturally ” bounded complete “natural ” domains. This is the non-dcpo analogue of the wellknown concept of Scott domains, or equivalently, the complete f-spaces of Ershov. In fact, the latter version of natural domains, if considered under “natural ” Scott topology, exactly corresponds to the class of f-spaces, not necessarily complete.