Representations of Join-homomorphisms of Distributive Lattices with Doubly 2-distributive Lattices

In the early eighties, A. Huhn proved that if D, E are finite distributive lattices and ψ : D → E is a {0}-preserving join-embedding, then there are finite lattices K, L and there is a lattice homomorphism φ : K → L such that ConK (the congruence lattice of K) is isomorphic to D, ConL (the congruence lattice of L) is isomorphic to E, and the natural induced mapping extφ : ConK → ConL represents ψ. The present authors with H. Lakser generalized this result to an arbitrary {0}-preserving join-homomorphism ψ. It was also A. Huhn who introduced the 2-distributive identity: x ∧ (y1 ∨ y2 ∨ y3) = (x ∧ (y1 ∨ y2)) ∨ (x ∧ (y1 ∨ y3)) ∨ (x ∧ (y2 ∨ y3)). We shall call a lattice doubly 2-distributive, if it satisfies the 2-distributive identity and its dual. In this note, we prove that the lattices K and L in the above result can be constructed as doubly 2-distributive lattices.