Math 2.0

Order, lattices, ordered algebraic structures Definition

Least 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 l∈Sl \in S is the least element of SS when l≤sl \le s for each s∈Ss \in S.

Remarks

Another name is minimum. By antisymmetry, a set has at most one least element. Each nonempty subset of ℕ\mathbb{N} has a least element. This is the well-ordering principle of ℕ\mathbb{N}. See well-order. The interval (0,1](0, 1] in ℝ\mathbb{R} has no least element.

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 orderdefinitionLeast element, definition, no evidence levelLeast 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. Least element definition

Where this leads

44 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Greatest_element_and_least_element, lead section
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Tag0T2NTJCV83
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WritingW0 stub
MSC06A06
DefinesLeast element
UsesPartial order

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