Math 2.0

Order, lattices, ordered algebraic structures Definition

Order dimension

Builds on Partial orderLinear extensionSzpilrajn extension theorem

Definition

Let (P,≤)(P, \le) be a partially ordered set. A realizer of PP is a set ℛ\mathcal{R} of linear extensions of ≤\le with the following property. For all x,y∈Px, y \in P, x≤yx \le y holds if and only if x⪯yx \preceq y for every ⪯\preceq in ℛ\mathcal{R}. The order dimension dim⁡P\dim P is the least cardinality of a realizer of PP.

Remarks

The set of all linear extensions is a realizer, by the Szpilrajn extension theorem, so the dimension is defined. Example: an antichain {a,b,c}\set{a, b, c} has dimension 22. The orders a≺b≺ca \prec b \prec c and c≺b≺ac \prec b \prec a form a realizer, since each pair of distinct elements appears in both directions, and one linear order alone would make two elements comparable. Non-example of a realizer: for the same antichain, the set {a≺b≺c, b≺a≺c}\set{a \prec b \prec c,\ b \prec a \prec c} is not a realizer, since aa precedes cc in both orders but a≰ca \not\le c. A poset has dimension 11 exactly when it is a total order. Equivalently, dim⁡P\dim P is the least dd such that PP embeds in a product of dd chains.

Reading path

To understand this page from the ground up, read these 34 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionLinear extension, concept, no evidence levelLinear extensionconceptPartial order, concept, no evidence levelPartial orderconceptProposition, concept, no evidence levelPropositionconceptTotal order, concept, no evidence levelTotal orderconceptSet, definition, no evidence levelSetdefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionFunction, definition, no evidence levelFunctiondefinitionPreorder, definition, no evidence levelPreorderdefinitionChoice function, definition, no evidence levelChoice functiondefinitionPartial order, definition, no evidence levelPartial orderdefinitionAxiom of choice, statement, no evidence levelAxiom of choicestatementMaximal element, definition, no evidence levelMaximal elementdefinitionTotal order, definition, no evidence levelTotal orderdefinitionUpper bound, definition, no evidence levelUpper bounddefinitionChain, definition, no evidence levelChaindefinitionZorn's lemma, statement, no evidence levelZorn's lemmastatementSzpilrajn extension theorem, statement, no evidence levelSzpilrajn extension t…statementOrder dimension, definition, no evidence levelOrder dimensiondefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Linear extension concept
  3. Partial order concept
  4. Proposition concept
  5. Total order concept
  6. Set definition
  7. Disjunction definition
  8. Implication definition
  9. Predicate definition
  10. Family of sets definition
  11. Ordered pair definition
  12. Union definition
  13. Existential quantifier definition
  14. Universal quantifier definition
  15. Cartesian product definition
  16. Subset definition
  17. Union of a family of sets definition
  18. Uniqueness quantifier definition
  19. Binary relation definition
  20. Antisymmetric relation definition
  21. Reflexive relation definition
  22. Transitive relation definition
  23. Function definition
  24. Preorder definition
  25. Choice function definition
  26. Partial order definition
  27. Axiom of choice statement
  28. Maximal element definition
  29. Total order definition
  30. Upper bound definition
  31. Chain definition
  32. Zorn's lemma statement
  33. Szpilrajn extension theorem statement
  34. Order dimension definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Order_dimension
About this page
TagAPD4NYG9TG
URIm2://APD4NYG9TG
Kinddefinition
WritingW0 stub
MSC06A07
DefinesOrder dimension
UsesPartial order, Linear extension, Szpilrajn extension theorem
Changed2 November 2026

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