Editing Duality (order theory)

{{Short description|Term in the mathematical area of order theory}}
{{Other uses|Order dual (disambiguation)}}
In the [[mathematics|mathematical]] area of [[order theory]], every [[partially ordered set]] ''P'' gives rise to a '''dual''' (or '''opposite''') partially ordered set which is often denoted by ''P''<sup>op</sup> or ''P''<sup>''d''</sup>. This dual order ''P''<sup>op</sup> is defined to be the same set, but with the '''inverse order''', i.e. ''x'' ≤ ''y'' holds in ''P''<sup>op</sup> [[if and only if]] ''y'' ≤ ''x'' holds in ''P''. It is easy to see that this construction, which can be depicted by flipping the [[Hasse diagram]] for ''P'' upside down, will indeed yield a partially ordered set. In a broader sense, two partially ordered sets are also said to be duals if they are '''dually isomorphic''', i.e. if one poset is [[order isomorphism|order isomorphic]] to the dual of the other.

The importance of this simple definition stems from the fact that every definition and theorem of order theory can readily be transferred to the dual order. Formally, this is captured by the '''Duality Principle''' for ordered sets:

: If a given statement is valid for all partially ordered sets, then its dual statement, obtained by inverting the direction of all order relations and by dualizing all order theoretic definitions involved, is also valid for all partially ordered sets.

If a statement or definition is equivalent to its dual then it is said to be '''self-dual'''. Note that the consideration of dual orders is so fundamental that it often occurs implicitly when writing ≥ for the dual order of ≤ without giving any prior definition of this "new" symbol.

== 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).

==See also==
* [[Converse relation]]
* [[List of Boolean algebra topics]]
* [[Transpose graph]]
*[[Dual (category theory)|Duality in category theory]], of which duality in order theory is a special case

==References==

{{reflist}}

* {{Citation | last1=Davey | first1=B.A. | last2=Priestley | first2=H. A. | title=Introduction to Lattices and Order|title-link= Introduction to Lattices and Order | edition=2nd | publisher=[[Cambridge University Press]] | isbn=978-0-521-78451-1 | year=2002}}

{{Order theory}}

[[Category:Order theory]]
[[Category:Duality theories|Order theory]]