Math 2.0

Order, lattices, ordered algebraic structures Definition

Minimal 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 minimal element of SS when no s∈Ss \in S satisfies s<ms < m. Equivalently, for each s∈Ss \in S, s≤ms \le m implies s=ms = m.

Remarks

It is the maximal element for the reversed order ≥\ge. In {2,3,4,6,9}\{2, 3, 4, 6, 9\} under divisibility, the minimal elements are 22 and 33. A well-founded relation is one in which each nonempty subset has a minimal element.

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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 orderdefinitionMinimal element, definition, no evidence levelMinimal elementdefinition
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  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. Minimal element definition

Where this leads

18 pages build on this one.

Further reading

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