Math 2.0

Order, lattices, ordered algebraic structures Definition

Well-order

Builds on Total orderLeast element

Definition

Let AA be a set. A well-order on AA is a total order ≤\le on AA such that each nonempty subset of AA has a least element. The pair (A,≤)(A, \le) is a well-ordered set.

Remarks

The usual order on ℕ\mathbb{N} is a well-order. The usual orders on ℤ\mathbb{Z} and on [0,1][0, 1] are not well-orders: ℤ\mathbb{Z} and (0,1](0, 1] have no least element. A total order is a well-order exactly when its strict order << is well-founded. The well-ordering theorem says that each set has a well-order.

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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 orderdefinitionLeast element, definition, no evidence levelLeast elementdefinitionTotal order, definition, no evidence levelTotal orderdefinitionWell-order, definition, no evidence levelWell-orderdefinition
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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. Least element definition
  17. Total order definition
  18. Well-order definition

Where this leads

42 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Well-order, lead section
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WritingW1 stated
MSC06A05
DefinesWell-order
UsesTotal order, Least element

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