Editing Modular lattice (section)

==Introduction==
The modular law can be seen as a restricted [[associativity|associative law]] that connects the two lattice operations similarly to the way in which the associative law λ(μ''x'') = (λμ)''x'' for vector spaces connects multiplication in the field and [[scalar multiplication]].

The restriction {{Math|''a'' ≤ ''b''}} is clearly necessary, since it follows from {{Math|1=''a'' ∨ (''x'' ∧ ''b'') = (''a'' ∨ ''x'') ∧ ''b''}}.  In other words, no lattice with more than one element satisfies the unrestricted consequent of the modular law.

It is easy to see<ref>The following is true for any lattice: {{Math|''a'' ∨ (''x'' ∧ ''b'') ≤ (''a'' ∨ ''x'') ∧ (''a'' ∨ ''b'')}}. Also, whenever {{Math|''a'' ≤ ''b''}}, then {{Math|''a'' ∨ ''b'' {{=}} ''b''}}.</ref> that {{Math|''a'' ≤ ''b''}} implies {{Math|1=''a'' ∨ (''x'' ∧ ''b'') ≤ (''a'' ∨ ''x'') ∧ ''b''}} in every lattice. Therefore, the modular law can also be stated as

;'''Modular law (variant)''': {{Math|''a'' ≤ ''b''}} implies {{Math|(''a'' ∨ ''x'') ∧ ''b'' ≤ ''a'' ∨ (''x'' ∧ ''b'')}}.

The modular law can be expressed as an equation that is required to hold unconditionally. Since  {{Math|''a'' ≤ ''b''}} implies {{Math|''a'' {{=}} ''a'' ∧ ''b''}} and since {{Math|''a'' ∧ ''b'' ≤ ''b''}}, replace {{Math|''a''}} with {{Math|''a'' ∧ ''b''}} in the defining equation of the modular law to obtain:
;'''Modular identity''': {{Math|1=(''a'' ∧ ''b'') ∨ (''x'' ∧ ''b'') = ((''a'' ∧ ''b'') ∨ ''x'') ∧ ''b''}}.
This shows that, using terminology from [[universal algebra]], the modular lattices form a subvariety of the [[variety (universal algebra)|variety]] of lattices. Therefore, all homomorphic images, [[sublattice]]s and direct products of modular lattices are again modular.