Non-well-founded sets modeled as ideal fixed points

Abstract

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 conisder 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 HF₁, the non-well founded analog of HF.