modular lattice
A modular lattice obeys a weaker, conditional form of the distributive law — one that still holds in the rich examples from algebra where full distributivity fails. The motivating example is the lattice of subgroups of a group, or of submodules of a module: these are generally not distributive, yet they satisfy a useful ‘modular’ identity that underpins the Jordan-Holder and Schreier refinement theorems.
Formally, a lattice is modular if it satisfies the modular law: a ≤ c implies a ∨ (b ∧ c) = (a ∨ b) ∧ c, for all a, b, c. Equivalently (and as an unconditional identity) (a ∧ c) ∨ (b ∧ c) = ((a ∧ c) ∨ b) ∧ c. Every distributive lattice is modular, but not conversely. The clean forbidden-sublattice criterion is: a lattice is modular if and only if it does not contain the pentagon N_5 as a sublattice; the diamond M_3 is modular but not distributive, marking the gap between the two classes.
Modularity is exactly the structural feature that gives lattices of subobjects their good behaviour. The submodule lattice of any module over any ring is modular, as is the normal-subgroup lattice of a group; the isomorphism theorems are essentially statements that certain ‘parallelogram’ intervals in these modular lattices are isomorphic. Dedekind isolated the modular law for precisely this reason, so modular lattices are sometimes called Dedekind lattices.
The lattice of subspaces of a vector space (e.g. the planes, lines and {0} in R^3) is modular: it satisfies the modular law but is not distributive once the dimension is at least 2.
Subspace lattices are modular but not distributive — the canonical separating example.
The full subgroup lattice of a non-abelian group can fail to be modular, e.g. that of S_4 contains N_5; modularity is guaranteed only for normal subgroups or for submodules.