Math 2.0

Order, lattices, ordered algebraic structures Definition

Preorder

Builds on Reflexive relationTransitive relation

Definition

Let AA be a set. A preorder on AA is a binary relation ≤\le on AA that is reflexive and transitive. For all a,b,c∈Aa, b, c \in A:

  1. a≤aa \le a;
  2. a≤ba \le b and b≤cb \le c imply a≤ca \le c.

Remarks

Divisibility on ℤ\mathbb{Z} is a preorder. It is not antisymmetric, because 2∣−22 \mid -2 and −2∣2-2 \mid 2. An antisymmetric preorder is a partial order. For a preorder, a≤ba \le b and b≤ab \le a together define an equivalence relation. The preorder gives a partial order on the quotient set.

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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 relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionPreorder, definition, no evidence levelPreorderdefinition
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. Universal quantifier definition
  8. Cartesian product definition
  9. Subset definition
  10. Binary relation definition
  11. Reflexive relation definition
  12. Transitive relation definition
  13. Preorder definition

Where this leads

122 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Preorder, lead section
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WritingW1 stated
MSC06A06
DefinesPreorder
UsesReflexive relation, Transitive relation

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