Hasse diagram (original) (raw)

If (A,≤) is a finite poset, then it can be represented by a Hasse diagram, which is a graph whose vertices are elements of A and the edges correspond to the covering relation. More precisely an edge from x∈A to y∈A is present if

• •
  x<y.
• •
  There is no z∈A such that x<z and z<y. (There are no in-between elements.)

If x<y, then in y is drawn higher than x. Because of that, the direction of the edges is never indicated in a Hasse diagram.

Example: If A=𝒫⁢({1,2,3}), the power set of {1,2,3}, and ≤is the subset relation ⊆, then Hasse diagram is

Even though {3}<{1,2,3} (since {3}⊂{1,2,3}), there is no edge directly between them because there are inbetween elements:{2,3} and {1,3}. However, there still remains an indirect path from {3} to {1,2,3}.