A category of compositional domain-models for separable Stone spaces

✍ Scribed by Fabio Alessi; Paolo Baldan; Furio Honsell

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 463 KB
• Volume: 290
• Category: Article
• ISSN: 0304-3975
• DOI: 10.1016/s0304-3975(02)00037-3

✦ Synopsis

In this paper we introduce SFP M , a category of SFP domains which provides very satisfactory domain-models, i.e. "partializations", of separable Stone spaces (2-Stone spaces). More speciÿcally, SFP M is a subcategory of SFP ep , 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). SFP M is "structurally well behaved", in the sense that the functor MAX, which associates to each object of SFP M the Stone space of its maximal elements, is compositional with respect to the constructors above, and !-continuous. A correspondence can be established between these constructors over SFP M and appropriate constructors on Stone spaces, whereby SFP M domain-models of Stone spaces deÿned as solutions of a vast class of recursive equations in SFP M , can be obtained simply by solving the corresponding equations in SFP M . Moreover any continuous function between two 2-Stone spaces can be extended to a continuous function between any two SFP M domain-models of the original spaces. The category SFP M 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 SFP M objects. Then the results proved for SFP M easily extends to the wider category having CSFP's as objects.

Using SFP M we can provide a plethora of "partializations" of the space of ÿnitary hypersets (the hyperuniverse N! (Ann. New York Acad. Sci. 806 (1996) 140). These includes the classical ones proposed in Abramsky (A Cook's tour of the ÿnitary non-well-founded sets unpublished manuscript, 1988; Inform. Comput. 92(2) (1991) 161) and Mislove et al. (Inform. Comput. Partially supported by Progetto MURST Coÿn'99 TOSCA.