Math 2.0

Order, lattices, ordered algebraic structures Definition

Duality in order theory

Builds on Partial order

Definition

Let (P,≤)(P, \le) be a partially ordered set. The dual of PP is the poset Pop=(P,≥)P^{\mathrm{op}} = (P, \ge) on the same set, where x≥yx \ge y means y≤xy \le x. The dual of a statement about partial orders is the statement obtained by replacing ≤\le with ≥\ge throughout. Then each defined notion is replaced by its dual notion, such as upper bound by lower bound and greatest element by least element.

Remarks

The relation ≥\ge is again a partial order, since reflexivity, antisymmetry and transitivity are symmetric in the two sides. For example, the dual of (ℕ,≤)(\N, \le) is ℕ\N with the reverse order, in which 00 is the greatest element and there is no least element. The dual of the dual of PP is PP itself. Duality principle: if a statement holds in every poset, then its dual statement holds in every poset. The reason is that the dual statement for PP is the original statement for PopP^{\mathrm{op}}. The dual of PP is also written P∂P^{\partial} or P∗P^{*}.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionPreorder, definition, no evidence levelPreorderdefinitionPartial order, definition, no evidence levelPartial orderdefinitionDuality in order theory, definition, no evidence levelDuality in order theo…definition
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  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Implication definition
  5. Predicate definition
  6. Ordered pair definition
  7. Universal quantifier definition
  8. Cartesian product definition
  9. Subset definition
  10. Binary relation definition
  11. Antisymmetric relation definition
  12. Reflexive relation definition
  13. Transitive relation definition
  14. Preorder definition
  15. Partial order definition
  16. Duality in order theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Duality_(order_theory)
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DefinesDuality in order theory
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