Editing Modular lattice (section)

== Diamond isomorphism theorem ==

For any two elements ''a'',''b'' of a modular lattice, one can consider the intervals [''a'' ∧ ''b'', ''b''] and [''a'', ''a'' ∨ ''b'']. They are connected by order-preserving maps
::φ: [''a'' ∧ ''b'', ''b''] → [''a'', ''a'' ∨ ''b''] and
::ψ: [''a'', ''a'' ∨ ''b''] → [''a'' ∧ ''b'', ''b'']
that are defined by φ(''x'') = ''x'' ∨ ''a'' and ψ(''y'') = ''y'' ∧ ''b''.

<gallery heights="180px" widths="225px">
Image:Modular_pair.svg|In a modular lattice, the maps φ and ψ indicated by the arrows are mutually inverse isomorphisms.
Image:Not a modular pair.svg|Failure of the diamond isomorphism theorem in a non-modular lattice.
</gallery>

The composition ψφ is an order-preserving map from the interval [''a'' ∧ ''b'', ''b''] 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 [''a'', ''a'' ∨ ''b''], 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 second [[isomorphism theorem]] in algebra, and it is a generalization of the [[lattice theorem]].