Math 2.0

Order, lattices, ordered algebraic structures Definition

Chain

Builds on Partial orderTotal order

Definition

Let (P,≤)(P, \le) be a partially ordered set. A subset CC of PP is a chain when, for all a,b∈Ca, b \in C, a≤ba \le b or b≤ab \le a.

Remarks

So a chain is a subset on which ≤\le is a total order. In the power set of ℕ\mathbb{N}, the sets {0}⊆{0,1}⊆{0,1,2}⊆…\{0\} \subseteq \{0, 1\} \subseteq \{0, 1, 2\} \subseteq \dots form a chain. Chains are the key notion in Zorn's lemma and in the Hausdorff maximal principle.

Reading path

To understand this page from the ground up, read these 17 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 orderdefinitionTotal order, definition, no evidence levelTotal orderdefinitionChain, definition, no evidence levelChaindefinition
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

Where this leads

16 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Total_order, Chains
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Tag0GP9ASVQ1X
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Kinddefinition
WritingW0 stub
MSC06A06
DefinesChain
UsesPartial order, Total order

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