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  in the poset, then the point corresponding to  appears lower in the drawing than the point corresponding to .

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

Hasse diagrams are also called upward drawings. Hasse diagrams for a graph  are implemented as HasseDiagram[g] in the Wolfram Language package Combinatorica` , where  is a directed acyclic Combinatorica graph object.

The above figures show the Hasse diagrams for Boolean algebras of orders , 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  elements (Skiena 1990, pp. 169-170). These correspond precisely to the hypercube graphs .

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.