A category of compositional domain-models for separable Stone spaces

In this paper we introduce SFPM, a category of SFP domains which provides very satisfactory domain-models, i.e. "partializations", of separable Stone spaces (2-Stone spaces). More specifically, SFPM is a subcategory of SFPep, closed under direct limits as well as many constructors, such as lifting, sum, product and Plotkin powerdomain (with the notable exception of the function space constructor). SFPM is "structurally well behaved", in the sense that the functor MAX, which associates to each object of SFPM the Stone space of its maximal elements, is compositional with respect to the constructors above, and ω-continuous. A correspondence can be established between these constructors over SFPM and appropriate constructors on Stone spaces, whereby SFPM domain-models of Stone spaces defined as solutions of a vast class of recursive equations in SFPM, can be obtained simply by solving the corresponding equations in SFPM. Moreover any continuous function between two 2-Stone spaces can be extended to a continuous function between any two SFPM domain-models of the original spaces. The category SFPM does not include all the SFP's with a 2-Stone space of maximal elements (CSFP's). We show that the CSFP's can be characterized precisely as suitable retracts of SFPM objects. Then the results proved for SFPM easily extends to the wider category having CSFP's as objects. Using SFPM we can provide a plethora of "partializations" of the space of finitary hypersets (the hyperuniverse Nω (Ann. New York Acad. Sci. 806 (1996) 140). These includes the classical ones proposed in Abramsky (A Cook's tour of the finitary non-well-founded sets unpublished manuscript, 1988; Inform. Comput. 92(2) (1991) 161) and Mislove et al. (Inform. Comput. 93(1) (1991) 16), which are also shown to be non-isomorphic, thus providing a negative answer to a problem raised in Mislove et al.