Math 2.0

Order, lattices, ordered algebraic structures Definition

Maximal chain

Builds on Partial orderChain

Definition

Let (P,≤)(P, \le) be a partially ordered set. A chain C⊆PC \subseteq P is maximal when there is no chain D⊆PD \subseteq P with C⊊DC \subsetneq D.

Remarks

In the subsets of {1,2,3}\set{1, 2, 3} ordered by inclusion, {∅,{1},{1,2},{1,2,3}}\set{\emptyset, \set{1}, \set{1, 2}, \set{1, 2, 3}} is a maximal chain. Every other subset is incomparable with one of its members, so no subset can be added. The chain {∅,{1},{1,2,3}}\set{\emptyset, \set{1}, \set{1, 2, 3}} is not maximal, since adding {1,2}\set{1, 2} gives a larger chain. The subsets of {1,…,n}\set{1, \dots, n} have exactly n!n! maximal chains, one for each order in which the elements are added. A maximal chain is a maximal element of the set of chains ordered by inclusion. The statement that every chain of a poset lies in a maximal chain is the Hausdorff maximal principle, which is equivalent to Zorn's lemma.

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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 orderdefinitionChain, definition, no evidence levelChaindefinitionMaximal chain, definition, no evidence levelMaximal chaindefinition
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
  17. Chain definition
  18. Maximal chain definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Total_order, Chains
About this page
TagWZKPDWBW2Z
URIm2://WZKPDWBW2Z
Kinddefinition
WritingW0 stub
MSC06A06, 06A07
DefinesMaximal chain
UsesPartial order, Chain
Changed2 November 2026

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