Representing Homomorphisms of Congruence Lattices as Restrictions of Congruences of Isoform Lattices

Let L1 be a finite lattice with an ideal L2. Then the restriction map is a {0, 1}-homomorphism from ConL1 into ConL2. In 1986, the present authors published the converse. If D1 and D2 are finite distributive lattices, and φ : D1 → D2 is a {0, 1}-homomorphism, then there are finite lattices L1 and L2 with an embedding η of L2 as an ideal of L1, and there are isomorphisms ε1 : ConL1 → D1 and ε2 : ConL2 → D2 such that φ is represented as the restriction map of congruences from L1 to L2, up to the two isomorphisms. Let us call a lattice isoform, if for any congruence, all congruence classes are isomorphic lattices. In 2003, G. Grätzer and E. T. Schmidt proved that every finite distributive lattice can be represented as the congruence lattice of an isoform lattice. In this paper we combine the two results, reproving the 1986 result with isoform lattices.