# Definition:Hasse Diagram

## Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

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

• 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:

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 $\N_{>0}$ which divide $24$.

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

## Also known as

Some sources refer to this as a nodal diagram.

## Source of Name

This entry was named for Helmut Hasse.

## Sources

The diagram is not named, and it is applied only to the subset relation
The diagram is not named