Math 2.0

Order, lattices, ordered algebraic structures Definition

Dense order

Builds on Partial order

Definition

Let (P,≤)(P, \le) be a partially ordered set, and write x<yx < y for x≤yx \le y and x≠yx \ne y. The order is dense when, for all x,y∈Px, y \in P with x<yx < y, there is z∈Pz \in P with x<z<yx < z < y.

Remarks

The order of ℚ\Q is dense: for rationals x<yx < y, the rational z=(x+y)/2z = (x + y)/2 satisfies x<z<yx < z < y. The order of ℤ\Z is not dense: for x=0x = 0 and y=1y = 1 there is no integer strictly between. A dense order has no covering pairs. A poset with at most one element is dense, since there is no pair x<yx < y to check. The term is used mostly for total orders, where it is a basic property in the study of order types. The order topology of a dense total order with more than one element has no isolated points.

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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 orderdefinitionDense order, definition, no evidence levelDense orderdefinition
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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. Dense order definition

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2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Dense_order
About this page
TagJB9HJ9CPEN
URIm2://JB9HJ9CPEN
Kinddefinition
WritingW0 stub
MSC06A05, 06A06
DefinesDense order
UsesPartial order
Changed2 November 2026

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