Duality (order theory)

In themathematicalarea oforder theory,everypartially ordered setPgives rise to adual(oropposite) partially ordered set which is often denoted byP^oporP^d.This dual orderP^opis defined to be the same set, but with theinverse order,i.e.x≤yholds inP^opif and only ify≤xholds inP.It is easy to see that this construction, which can be depicted by flipping theHasse diagramforPupside down, will indeed yield a partially ordered set. In a broader sense, two partially ordered sets are also said to be duals if they aredually isomorphic,i.e. if one poset isorder isomorphicto 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 theDuality Principlefor 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 beself-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[edit]

Naturally, there are a great number of examples for concepts that are dual:

Greatest elements and least elements
Maximal elements and minimal elements
Least upper bounds(suprema, ∨) andgreatest lower bounds(infima, ∧)
Upper sets and lower sets
Idealsandfilters
Closure operatorsandkernel operators.

Examples of notions which are self-dual include:

Being a (complete)lattice
Monotonicityof functions
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^[1]
Being aBoolean algebra
Being anorder isomorphism.

Since partial orders areantisymmetric,the only ones that are self-dual are theequivalence relations(but the notion of partial orderisself-dual).

References[edit]

^The quantifiers are essential: for individual elementsx,y,z,e.g. the first equation may be violated, but the second may hold; see theN₅latticefor an example.

Davey, B.A.; Priestley, H. A. (2002),Introduction to Lattices and Order(2nd ed.),Cambridge University Press,ISBN 978-0-521-78451-1

[1] The quantifiers are essential: for individual elementsx,y,z,e.g. the first equation may be violated, but the second may hold; see theN₅latticefor an example.

[1]

v t e Order theory
Topics Glossary Category
Key concepts	Binary relation Boolean algebra Cyclic order Lattice Partial order Preorder Total order Weak ordering
Results	Boolean prime ideal theorem Cantor–Bernstein theorem Cantor's isomorphism theorem Dilworth's theorem Dushnik–Miller theorem Hausdorff maximal principle Knaster–Tarski theorem Kruskal's tree theorem Laver's theorem Mirsky's theorem Szpilrajn extension theorem Zorn's lemma
Properties & Types (list)	Antisymmetric Asymmetric Boolean algebra topics Completeness Connected Covering Dense Directed (Partial)Equivalence Foundational Heyting algebra Homogeneous Idempotent Lattice Bounded Complemented Complete Distributive Join and meet Reflexive Partial order Chain-complete Graded Eulerian Strict Prefix order Preorder Total Semilattice Semiorder Symmetric Total Tolerance Transitive Well-founded Well-quasi-ordering(Better) (Pre)Well-order
Constructions	Composition Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure
Topology& Orders	Alexandrov topology&Specialization preorder Ordered topological vector space Normal cone Order topology Order topology Topological vector lattice Banach Fréchet Locally convex Normed
Related	Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding Isomorphism Order type Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz space Partially ordered group Positive cone of a partially ordered group Upper set Young's lattice

Examples[edit]

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