2. Modularity for arbitrary lattices

Theorem. The following conditions on a lattice $L$ are equivalent.

(1) $L$ obeys the modular condition $x \leq z \Rightarrow x \vee (y \wedge z) = (x \vee y) \wedge z$.

(2) $L$ obeys the law $(x \vee y) \wedge (x \vee z) = x \vee (y \wedge (x \vee z))$.

(3) $L$ has no sublattice isomorphic to $N _ 5$.

(4) Transposed intervals of $L$ are isomorphic under the obvious maps up and down.

(5) Any three elements of $L$ generate a distributive sublattice provided two of them are comparable.