Math 2.0

Order, lattices, ordered algebraic structures Definition

Maximal element

Builds on Partial order

Definition

Let (P,≤)(P, \le) be a partially ordered set and let SS be a subset of PP. An element m∈Sm \in S is a maximal element of SS when no s∈Ss \in S satisfies m<sm < s. Equivalently, for each s∈Ss \in S, m≤sm \le s implies s=ms = m.

Remarks

A maximal element need not be the greatest element, and a set can have several maximal elements. In {{1},{2},{1,2,3},{4}}\{\{1\}, \{2\}, \{1, 2, 3\}, \{4\}\} under ⊆\subseteq, the maximal elements are {1,2,3}\{1, 2, 3\} and {4}\{4\}. Zorn's lemma gives maximal elements in many infinite posets.

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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 orderdefinitionMaximal element, definition, no evidence levelMaximal elementdefinition
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. Maximal element definition

Where this leads

15 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Maximal_and_minimal_elements, lead section
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