Funayama's theorem revisited

Obviously each lattice that satisfies either (JID) or (MID) is distributive. A classic result in lattice theory is Funayama’s theorem [5] stating that there is an embedding e of L into a complete Boolean algebra B that preserves all existing joins and meets in L iff L satisfies both (JID) and (MID). Funayama’s original proof was quite involved. For complete L, Grätzer [6, Sec. II.4] gave a more accessible proof of Funayama’s theorem. The key ingredient of Grätzer’s proof is to show that if L satisfies both (JID) and (MID), then the embedding of L into its free Boolean extension B(L) is a complete lattice embedding. Then taking the MacNeille completion B(L) of B(L) produces a complete Boolean algebra and the embedding B(L) ↪→ B(L) preserves all existing joins and meets in B(L). Thus, the composition L ↪→ B(L) ↪→ B(L) is a complete lattice embedding. For complete L, Johnstone [8, Sec. II.2] gave a different proof of Funayama’s theorem. Let L be a complete lattice satisfying (JID). Then L is a frame. Therefore, the poset N(L) of all nuclei on L is also a frame, and the embedding L ↪→ N(L) is a frame homomorphism. Let N(L)¬¬ be the Booleanization of N(L); that is, the Boolean frame of regular nuclei on L. Thus, N(L)¬¬ is a complete Boolean algebra and the composition L ↪→ N(L) N(L)¬¬ is a frame embedding. If in addition L satisfies (MID), then the embedding L ↪→ N(L)¬¬ is a complete lattice embedding. Our aim is to show that Grätzer’s proof has an obvious generalization to the case when L is not necessarily complete, thus providing an accessible proof of Funayama’s theorem in its full generality. If L is complete, we show that the complete Boolean algebras B(L) and N(L)¬¬