Math 2.0

Order, lattices, ordered algebraic structures Definition

Partial order

Builds on PreorderAntisymmetric relation

Definition

Let AA be a set. A partial order on AA is a preorder ≤\le on AA that is antisymmetric. For all a,b,c∈Aa, b, c \in A:

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

The pair (A,≤)(A, \le) is a partially ordered set.

Remarks

Examples are ≤\le on ℝ\mathbb{R}, divisibility on ℕ\mathbb{N}, and ⊆\subseteq on the power set of a set. "Partial" means that two elements need not be comparable: neither {1}⊆{2}\{1\} \subseteq \{2\} nor {2}⊆{1}\{2\} \subseteq \{1\}. The strict order a<ba < b means a≤ba \le b and a≠ba \neq b. A finite poset can be drawn as a Hasse diagram.

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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 orderdefinition
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  • 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

Where this leads

110 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Partially_ordered_set, lead section
About this page
Tag011C29Q1S1
URIm2://011C29Q1S1
Kinddefinition
WritingW1 stated
MSC06A06
DefinesPartial order
UsesPreorder, Antisymmetric relation

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