Pruning of intervals

(you need flash player for Fig. 2)

We call a lattice L isoform, if for any congruence relation θ of L, all congruence classes of θ are isomorphic sublattices.
We have proved that for every finite distributive lattice D, there exists a finite, isoform lattice L such that the congruence
lattice of L is isomorphic to D  ( G. Grätzer and E. T. SchmidtFinite lattices with isoform congruences   (pdf) ).

New result (2003):  G. Grätzer , R. W. Quackenbush and E. T. SchmidtCongruence- preserving extensions
of finite lattices to isoform lattices.
  (pdf) .We have proved:

Theorem. Every finite lattice K has a congruence-preserving extension to a finite isoform lattice L.

Here is an example. Take  the lattice K given on Figure 1:







We have an embedding into a cubic extension (this is a direct product) and prune (delete) an edge to get the lattice L,

see on Figure 2  the blinking edge: