Non-well-founded sets obtained from ideal fixed points

Motivated by ideas from the study of abstract data types, the authors show how to interpret non-well-founded sets as fixed points of continuous transformations of an initial continuous algebra. They consider a preordered structure closely related to the set HF of well-founded, hereditarily finite sets. By taking its ideal completion, the authors obtain an initial continuous algebra in which they are able to solve all of the usual systems of equations that characterize hereditarily finite, non-well-founded sets. In this way, they are able to obtain a structure which is isomorphic to HF₁, the non-well-founded analog to HF