Math 2.0

Order, lattices, ordered algebraic structures Definition

Antichain

Builds on Partial order

Definition

Let (P,≤)(P, \le) be a partially ordered set. A subset AA of PP is an antichain when, for all a,b∈Aa, b \in A with a≠ba \neq b, neither a≤ba \le b nor b≤ab \le a.

Remarks

In the power set of {1,2,3}\{1, 2, 3\}, the set {{1},{2},{3}}\{\{1\}, \{2\}, \{3\}\} is an antichain. A chain and an antichain have at most one common element. The largest size of an antichain is the width of the poset.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

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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 orderdefinitionAntichain, definition, no evidence levelAntichaindefinition
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. Antichain definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Antichain, lead section
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WritingW1 stated
MSC06A07
DefinesAntichain
UsesPartial order

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  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

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