Math 2.0

Order, lattices, ordered algebraic structures Definition

Lexicographic order

Builds on Partial orderCartesian product

Definition

Let (A,≤A)(A, \le_A) and (B,≤B)(B, \le_B) be partially ordered sets. The lexicographic order on the Cartesian product A×BA \times B is the relation ≤\le with (a,b)≤(a′,b′)(a, b) \le (a', b') when a<Aa′or(a=a′ and b≤Bb′).a <_A a' \quad \text{or} \quad (a = a' \text{ and } b \le_B b').

Remarks

It is a partial order. When ≤A\le_A and ≤B\le_B are total orders, it is a total order. Example: in ℕ×ℕ\mathbb{N} \times \mathbb{N}, (1,5)<(2,0)(1, 5) < (2, 0) and (2,0)<(2,3)(2, 0) < (2, 3). A dictionary orders words in the same way, letter by letter.

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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 orderdefinitionLexicographic order, definition, no evidence levelLexicographic 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. Lexicographic order definition

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7 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Lexicographic_order, Cartesian products
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Tag0G77XFBV00
URIm2://0G77XFBV00
Kinddefinition
WritingW1 stated
MSC06A05
DefinesLexicographic order
UsesPartial order, Cartesian product

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