Math 2.0

Order, lattices, ordered algebraic structures Definition

Upper set

Builds on Partial orderSubset

Definition

Let (P,≤)(P, \le) be a partially ordered set and let UU be a subset of PP. The set UU is an upper set when, for all x,y∈Px, y \in P, if x∈Ux \in U and x≤yx \le y, then y∈Uy \in U. Dually, a subset LL of PP is a lower set when, for all x,y∈Px, y \in P, if y∈Ly \in L and x≤yx \le y, then x∈Lx \in L.

Remarks

In (ℕ,≤)(\N, \le), the set {n∈ℕ:n≥5}\set{n \in \N : n \ge 5} is an upper set: if n≥5n \ge 5 and n≤mn \le m, then m≥5m \ge 5. The set of even numbers is not an upper set of (ℕ,≤)(\N, \le), since 22 is in it, 2≤32 \le 3, and 33 is not. A subset is an upper set if and only if its complement is a lower set. Unions and intersections of any family of upper sets are upper sets. Other names are up-set and upward closed set for upper sets, and down-set, decreasing set and order ideal for lower sets. The smallest lower set containing a subset SS is its down-closure.

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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 orderdefinitionUpper set, definition, no evidence levelUpper setdefinition
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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. Upper set definition

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2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Upper_and_lower_sets
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Kinddefinition
WritingW0 stub
MSC06A06
DefinesUpper set
UsesPartial order, Subset

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