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Hasse Diagram


A Hasse diagram is a graphical rendering of a partially ordered set displayed via the cover relation of the partially ordered set with an implied upward orientation. A point is drawn for each element of the poset, and line segments are drawn between these points according to the following two rules:

1. If x<y in the poset, then the point corresponding to x appears lower in the drawing than the point corresponding to y.

2. The line segment between the points corresponding to any two elements x and y of the poset is included in the drawing iff x covers y or y covers x.

Hasse diagrams are also called upward drawings.

Hasse diagrams for a graph g are implemented as HasseDiagram[g] in the Wolfram Language package Combinatorica` , where g is a directed acyclic Combinatorica graph object. They may be implemented in a future version of the Wolfram Language as HasseGraph.

HasseDiagramBooleanAlgebras

The above figures show the Hasse diagrams for Boolean algebras of orders n=2, 3, 4, and 5. In particular, these figures illustrate the partition between left and right halves of the lattice, each of which is the Boolean algebra on n-1 elements (Skiena 1990, pp. 169-170). These correspond precisely to the hypercube graphs Q_n.


See also

Between, Cover Relation, Hypercube Graph, Partially Ordered Set

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References

Skiena, S. "Hasse Diagrams." §5.4.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 163, 169-170, and 206-208, 1990.

Referenced on Wolfram|Alpha

Hasse Diagram

Cite this as:

Weisstein, Eric W. "Hasse Diagram." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HasseDiagram.html

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