A Characterization of Permutation Models in Terms of Forcing

Abstract

We show that if N and M are transitive models of ZFA such that N $\subseteq$ M, N and M have the same kernel and same set of atoms, and M $\models$ AC, then N is a Fraenkel-Mostowski-Specker (FMS) submodel of M if and only if M is a generic extension of N by some almost homogeneous notion of forcing. We also develop a slightly modified notion of FMS submodels to characterize the case where M is a generic extension of N not necessarily by an almost homogeneous notion of forcing.