Modular lattice
This phrasing emphasizes an interpretation in terms of projection onto the sublattice [a, b], a fact known as the diamond isomorphism theorem.
[1] An alternative but equivalent condition stated as an equation (see below) emphasizes that modular lattices form a variety in the sense of universal algebra.
For example, the subspaces of a vector space (and more generally the submodules of a module over a ring) form a modular lattice.
In other words, no lattice with more than one element satisfies the unrestricted consequent of the modular law.
Since a ≤ b implies a = a ∧ b and since a ∧ b ≤ b, replace a with a ∧ b in the defining equation of the modular law to obtain: This shows that, using terminology from universal algebra, the modular lattices form a subvariety of the variety of lattices.
As a special case, the lattice of subgroups of an abelian group is modular.
For an example, the lattice of subgroups of the dihedral group of order 8 is not modular.
Then using absorption and modular identity: For the other direction, let the implication of the theorem hold in G. Let a,b,c be any elements in G, such that c ≤ a.
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.
A lattice is modular if and only if the diamond isomorphism theorem holds for every pair of elements.
The definition of modularity is due to Richard Dedekind, who published most of the relevant papers after his retirement.
In a paper published in 1894[citation needed] he studied lattices, which he called dual groups (German: Dualgruppen) as part of his "algebra of modules" and observed that ideals satisfy what we now call the modular law.
[10] In a digression he introduced and studied lattices formally in a general context.
He called such lattices dual groups of module type (Dualgruppen vom Modultypus).
[10]: 13 In the same paper, Dedekind also investigated the following stronger form[10]: 14 of the modular identity, which is also self-dual:[10]: 9 He called lattices that satisfy this identity dual groups of ideal type (Dualgruppen vom Idealtypus).
