Editing Duality (order theory) (section)

== Examples ==
[[File:Duale Verbaende.svg|thumb|A bounded distributive lattice, and its dual]]
Naturally, there are a great number of examples for concepts that are dual:
* [[Greatest element|Greatest elements and least elements]]
* [[Maximal element|Maximal elements and minimal elements]]
* [[Least upper bound]]s (suprema, ∨) and [[greatest lower bound]]s (infima, ∧)
* [[Upper set|Upper sets and lower sets]]
* [[ideal (order theory)|Ideals]] and [[filter (mathematics)|filters]]
* [[Closure operator]]s and [[kernel operator]]s.

Examples of notions which are self-dual include:
* Being a ([[complete lattice|complete]]) [[lattice (order)|lattice]]
* [[monotonic function|Monotonicity]] of functions
* [[distributive lattice|Distributivity of lattices]], i.e. the lattices for which ∀''x'',''y'',''z'': ''x'' ∧ (''y'' ∨ ''z'') = (''x'' ∧ ''y'') ∨ (''x'' ∧ ''z'') holds are exactly those for which the dual statement ∀''x'',''y'',''z'': ''x'' ∨ (''y'' ∧ ''z'') = (''x'' ∨ ''y'') ∧ (''x'' ∨ ''z'') holds<ref>The quantifiers are essential: for individual elements ''x'', ''y'', ''z'', e.g. the first equation may be violated, but the second may hold; see the [[modular lattice|N<sub>5</sub> lattice]] for an example.</ref>
* Being a [[Boolean algebra (structure)|Boolean algebra]]
* Being an [[order isomorphism]].

Since partial orders are [[Antisymmetric relation|antisymmetric]], the only ones that are self-dual are the [[equivalence relations]] (but the notion of partial order '''is''' self-dual).