Modular Lattice - Diamond Isomorphism Theorem

Diamond Isomorphism Theorem

For any two elements a,b of a modular lattice, one can consider the intervals and . They are connected by order-preserving maps

φ: → and

ψ: →

that are defined by φ(x) = x ∨ a and ψ(x) = x ∧ b.

• In a modular lattice, the maps φ and ψ indicated by the arrows are mutually inverse isomorphisms.

• Failure of the diamond isomorphism theorem in a non-modular lattice.

The composition ψφ is an order-preserving map from the interval to itself which also satisfies the inequality ψ(φ(x)) = (x ∨ a) ∧ b ≥ x. The example shows that this inequality can be strict in general. In a modular lattice, however, equality holds. Since the dual of a modular lattice is again modular, φψ is also the identity on, and therefore the two maps φ and ψ are isomorphisms between these two intervals. This result is sometimes called the diamond isomorphism theorem for modular lattices. A lattice is modular if and only if the diamond isomorphism theorem holds for every pair of elements.

The diamond isomorphism theorem for modular lattices is analogous to the third isomorphism theorem in algebra, and it is a generalization of the lattice theorem.