Math 2.0

Order, lattices, ordered algebraic structures Binary relations and basic order notions Concept

Hasse diagram

Builds on Partial order

Summary

A Hasse diagram is a drawing of a finite partially ordered set. Each element is a point, and there is a line upward from aa to bb when bb covers aa. This means that a<ba < b and no cc satisfies a<c<ba < c < b. Then a≤ba \le b exactly when one can go from aa to bb by a path that goes only upward.

Remarks

The diagram does not draw the loops of reflexivity or the lines that transitivity implies. The Hasse diagram of the power set of {1,2,3}\{1, 2, 3\} under ⊆\subseteq is the picture of a cube.

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 orderdefinitionHasse diagram, concept, no evidence levelHasse diagramconcept
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. Hasse diagram concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Hasse_diagram, lead section
About this page
Tag00EYM7MDKS
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Aliasesorder-diagram
Kindconcept
WritingW1 stated
MSC06A06
UsesPartial order
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