Editing Lattice (order) (section)

=== Modularity ===
{{main|Modular lattice}}
For some applications the distributivity condition is too strong, and the following weaker property is often useful. A lattice <math>(L, \vee, \wedge)</math> is {{dfni|modular}} if, for all elements <math>a, b, c \in L,</math> the following identity holds:
<math>(a \wedge c) \vee (b \wedge c) = ((a \wedge c) \vee b) \wedge c.</math> ({{dfn|Modular identity}})<br>
This condition is equivalent to the following axiom:
<math>a \leq c</math> implies <math>a \vee (b \wedge c) = (a \vee b) \wedge c.</math> ({{dfn|Modular law}})<br>
A lattice is modular if and only if it does not have a [[sublattice]] isomorphic to N<sub>5</sub> (shown in Pic.&nbsp;11).<ref name="Davey.Priestley.2002.10.6"/>
Besides distributive lattices, examples of modular lattices are the lattice of submodules of a [[Module (mathematics)|module]] (hence ''modular''), the lattice of [[two-sided ideal]]s of a [[Ring (mathematics)|ring]], and the lattice of [[normal subgroup]]s of a [[Group (mathematics)|group]]. The [[Subsumption lattice|set of first-order terms]] with the ordering "is more specific than" is a non-modular lattice used in [[automated reasoning]].