Math 2.0

Order, lattices, ordered algebraic structures Definition

Total order

Builds on Partial order

Definition

Let AA be a set. A total order on AA is a partial order ≤\le on AA such that, for all a,b∈Aa, b \in A, a≤ba \le b or b≤ab \le a.

Remarks

Other names are linear order and simple order. The usual orders on ℕ\mathbb{N}, ℤ\mathbb{Z}, ℚ\mathbb{Q} and ℝ\mathbb{R} are total orders. The order ⊆\subseteq on the power set of {1,2}\{1, 2\} is 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 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 orderdefinitionTotal order, definition, no evidence levelTotal orderdefinition
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. 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. Total order definition

Where this leads

69 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Total_order, lead section
About this page
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WritingW1 stated
MSC06A05
DefinesTotal order
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  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

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