Non-well-founded sets obtained from ideal fixed points. (English) Zbl 0717.03023
Logic in computer science, Proc. 4th Annual Symp., Pacific Grove/CA (USA) 1989, 263-272 (1989).
Summary: [For the entire collection see Zbl 0713.00018.]
Motivated by ideas from the study of abstract data types, we show how to interpret non-well-founded sets as fixed points of continuous transformations of an initial continuous algebra. We consider a preordered structure closely related to the set HF of well-founded, hereditarily finite sets. By taking its ideal completion, we obtain an initial continuous algebra in which we are able to solve all of the usual systems of equations that characterize hereditarily finite, non-well- founded sets. In this way, we are able to obtain a structure which is isomorphic to \(HF_ 1\), the non-well-founded analog of HF.
The complete version of this paper is published in Inf. Comput. 93, No.1, 16-54 (1991).
Motivated by ideas from the study of abstract data types, we show how to interpret non-well-founded sets as fixed points of continuous transformations of an initial continuous algebra. We consider a preordered structure closely related to the set HF of well-founded, hereditarily finite sets. By taking its ideal completion, we obtain an initial continuous algebra in which we are able to solve all of the usual systems of equations that characterize hereditarily finite, non-well- founded sets. In this way, we are able to obtain a structure which is isomorphic to \(HF_ 1\), the non-well-founded analog of HF.
The complete version of this paper is published in Inf. Comput. 93, No.1, 16-54 (1991).
MSC:
| 03E65 | Other set-theoretic hypotheses and axioms |
| 68Q65 | Abstract data types; algebraic specification |
| 06B35 | Continuous lattices and posets, applications |
