Math 2.0

Order, lattices, ordered algebraic structures Definition

Order embedding

Builds on Partial orderMonotone function

Definition

Let (P,≤P)(P, \le_P) and (Q,≤Q)(Q, \le_Q) be partially ordered sets. A map f ⁣:P→Qf \colon P \to Q is an order embedding when, for all x,y∈Px, y \in P, x≤Py  ⟺  f(x)≤Qf(y).x \le_P y \iff f(x) \le_Q f(y).

Remarks

The inclusion ℚ→ℝ\Q \to \R is an order embedding, since the order of ℚ\Q is the restriction of the order of ℝ\R. Let P={a,b}P = \set{a, b} with aa and bb incomparable, and let Q={0,1}Q = \set{0, 1} with 0<10 < 1. The map f(a)=0f(a) = 0, f(b)=1f(b) = 1 is monotone, but it is not an order embedding: f(a)≤f(b)f(a) \le f(b) while a≰ba \not\le b. An order embedding is injective: f(x)=f(y)f(x) = f(y) gives x≤yx \le y and y≤xy \le x, so x=yx = y by antisymmetry. So ff is an order isomorphism from PP onto its image f(P)f(P) with the induced order. For total orders, a strictly increasing map is the same as an order embedding.

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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 levelPreorderdefinitionPartial order, definition, no evidence levelPartial orderdefinitionMonotone function, definition, no evidence levelMonotone functiondefinitionOrder embedding, definition, no evidence levelOrder embeddingdefinition
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.
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  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. Partial order definition
  19. Monotone function definition
  20. Order embedding definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Order_embedding
About this page
TagAZES1C6VMT
URIm2://AZES1C6VMT
Kinddefinition
WritingW0 stub
MSC06A06
DefinesOrder embedding
UsesPartial order, Monotone function

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