Math 2.0

Order, lattices, ordered algebraic structures Definition

Order isomorphism

Builds on Bijective functionPartial order

Definition

Let (P,≤P)(P, \le_P) and (Q,≤Q)(Q, \le_Q) be partially ordered sets. An order isomorphism from PP to QQ is a bijective function f ⁣:P→Qf \colon P \to Q such that, for all a,b∈Pa, b \in P, a≤Pb  ⟺  f(a)≤Qf(b).a \le_P b \iff f(a) \le_Q f(b).

Remarks

The inverse of an order isomorphism is an order isomorphism. The map x↦2xx \mapsto 2x is an order isomorphism from ℤ\mathbb{Z} to the even integers. The ordered sets ℕ\mathbb{N} and ℤ\mathbb{Z} are not order isomorphic, because ℕ\mathbb{N} has a least element and ℤ\mathbb{Z} has none. A monotone bijection need not be an order isomorphism, when the orders are not total.

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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 functiondefinitionBijective function, definition, no evidence levelBijective functiondefinitionOrder isomorphism, definition, no evidence levelOrder isomorphismdefinition
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. Bijective function definition
  23. Order isomorphism definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Order_isomorphism, lead section
About this page
Tag0Y74KN99A5
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Kinddefinition
WritingW0 stub
MSC06A06
DefinesOrder isomorphism
UsesBijective function, Partial order

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