Initial self-embeddings of models of set theory

By a classical theorem of Harvey Friedman (1973), every countable nonstandard model \(\mathcal{M}\) of a sufficiently strong fragment of ZF has a proper rank-initial self-embedding \(j\), i.e., \(j\) is a self-embedding of \(\mathcal{M}\) such that \(j[\mathcal{M}]\subsetneq\mathcal{M}\), and the ordinal rank of each member of \(j[\mathcal{M}]\) is less than the ordinal rank of each element of \(\mathcal{M}\setminus j[\mathcal{M}]\). Here we investigate the larger family of proper initial-embeddings \(j\) of models \(\mathcal{M}\) of fragments of set theory, where the image of \(j\) is a transitive submodel of \(\mathcal{M}\).