Editing Modular lattice (section)

==Examples==
[[File:Smallest nonmodular lattice 2.svg|thumb|''N''<sub>5</sub>, the smallest non-modular lattice: {{nowrap|''x''∨(''a''∧''b'')}} = ''x''∨0 = ''x'' ≠ ''b'' = 1∧''b'' ={{nowrap|(''x''∨''a'')∧''b''}}.]]
The lattice of submodules of a [[module over a ring]] is modular.  As a special case, the lattice of subgroups of an [[abelian group]] is modular.

The lattice of [[normal subgroup]]s of a [[group (mathematics)|group]] is modular.  But in general the [[lattice of subgroups|lattice of all subgroups]] of a group is not modular.  For an example, the lattice of subgroups of the [[dihedral group]] of order 8 is not modular.

The smallest non-modular lattice is the "pentagon" lattice ''N''<sub>5</sub> consisting of five elements 0, 1, ''x'', ''a'', ''b'' such that 0 < ''x'' < ''b'' < 1, 0 < ''a'' < 1, and ''a'' is not comparable to ''x'' or to ''b''. For this lattice,
:''x'' ∨ (''a'' ∧ ''b'') = ''x'' ∨ 0 = ''x'' < ''b'' = 1 ∧ ''b'' = (''x'' ∨ ''a'') ∧ ''b''
holds, contradicting the modular law. Every non-modular lattice contains a copy of ''N''<sub>5</sub> as a sublattice.<ref name=":0">{{Cite book |chapter-url=https://link.springer.com/book/10.1007%2Fb139095 |title=Lattices and Ordered Algebraic Structures |last=Blyth |first=T. S. |publisher=Springer |year=2005 |isbn=978-1-85233-905-0 |series=Universitext |location=London |at=Theorem 4.4 |chapter=Modular lattices |doi=10.1007/1-84628-127-X_4}}</ref>