Math 2.0

Order, lattices, ordered algebraic structures Definition

Order type

Builds on Total orderOrder isomorphism

Definition

Let (A,≤A)(A, \le_A) and (B,≤B)(B, \le_B) be totally ordered sets. They have the same order type when there is an order isomorphism from AA onto BB. The order type of (A,≤A)(A, \le_A), written type⁡(A)\operatorname{type}(A), is the class of all totally ordered sets that have the same order type as AA.

Remarks

The sets ℕ\N and 2ℕ={0,2,4,… }2\N = \set{0, 2, 4, \dots} with their usual orders have the same order type, ω\omega, by the isomorphism n↦2nn \mapsto 2n. The sets ℕ\N and ℤ\Z do not: ℕ\N has a least element, ℤ\Z has none, and an order isomorphism maps a least element to a least element. Having the same order type is an equivalence relation on totally ordered sets. Each well-ordered set is order isomorphic to a unique ordinal number, which is used to represent its order type. Common notation: ω\omega for ℕ\N, ω∗\omega^* for the reverse of ℕ\N, ζ\zeta for ℤ\Z, η\eta for ℚ\Q and λ\lambda for ℝ\R. The same definition makes sense for partially ordered sets.

Reading path

To understand this page from the ground up, read these 25 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 pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness 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 levelPreorderdefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionPartial order, definition, no evidence levelPartial orderdefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionTotal order, definition, no evidence levelTotal orderdefinitionBijective function, definition, no evidence levelBijective functiondefinitionOrder isomorphism, definition, no evidence levelOrder isomorphismdefinitionOrder type, definition, no evidence levelOrder typedefinition
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. Proposition concept
  3. Set definition
  4. Implication definition
  5. Predicate definition
  6. Ordered pair definition
  7. Existential quantifier definition
  8. Universal quantifier definition
  9. Cartesian product definition
  10. Subset definition
  11. Uniqueness quantifier definition
  12. Binary relation definition
  13. Antisymmetric relation definition
  14. Reflexive relation definition
  15. Transitive relation definition
  16. Function definition
  17. Preorder definition
  18. Codomain definition
  19. Injective function definition
  20. Partial order definition
  21. Surjective function definition
  22. Total order definition
  23. Bijective function definition
  24. Order isomorphism definition
  25. Order type definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Order_type
About this page
Tag28PE4GRFD7
URIm2://28PE4GRFD7
Kinddefinition
WritingW0 stub
MSC06A05, 03E10
DefinesOrder type
UsesTotal order, Order isomorphism

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