Almost distributive sublattices and congruence heredity (Q995375)

Let \(L\) be the congruence lattice of a finite algebra \(A\). If every 0-1 sublattice of \(L\) is also the congruence lattice of an algebra with the same universe as \(A\), then \(L\) is called a hereditary congruence lattice. If every 0-1 sublattice of \(L^n\) is the congruence lattice of an algebra on \(A^n\) for every \(n\geq 1\), then \(L\) is named a power-hereditary congruence lattice. It is known that every congruence lattice representation of \(N_5\) is power-hereditary. Now the author gives an extension of the former results by proving that any congruence lattice representation of a finite lattice obtained by doubling a convex subset of a distributive lattice is hereditary. A number of open problems are posed.