Definition:Hasse Diagram

Definition
Let $$S$$ be a set, and let $$\mathcal R \subseteq S \times S$$ be an ordering on $$S$$.

In particular, let $$\mathcal R$$ be a partial ordering on $$S$$.

A Hasse diagram is a method of representing the ordered set $$\left({S; \preceq}\right)$$ as a graph $$G$$, in which:


 * The vertices of $$G$$ represent the elements of $$S$$;


 * The edges of $$G$$ represent the elements of $$\preceq$$;


 * If $$x, y \in S: x \preceq y$$ then the edge representing $$x \preceq y$$ is drawn so that $$x$$ is lower down the page than $$y$$, that is, the edge ascends (usually obliquely) from $$x$$ to $$y$$;


 * If $$x \preceq y$$ and $$y \preceq z$$, then as an ordering is transitive it follows that $$x \preceq z$$. But in a Hasse diagram, the relation $$x \preceq z$$ is not shown. Transitivity is implicitly expressed by the fact that $$z$$ is higher up than $$x$$, and can be reached by tracing a path from $$x$$ to $$z$$ completely through ascending edges.

Some sources draw arrows on their edges, so as to make $$G$$ a directed graph, but this is usually considered unnecessary.

These are examples of Hasse diagrams:


 * Hasse-Diagram.png

The diagram on the left illustrates the "Divides" ordering on the set $$S = \left\{{1, 2, 3, 4, 6, 8, 12, 24}\right\}$$ where $$S$$ is the set of all elements of $$\N^*$$ which divide $$24$$.

The diagram on the right illustrates the "Subset" relation on the power set $$\mathcal P \left({S}\right)$$ where $$S = \left\{{1, 2, 3}\right\}$$.