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:

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

• W.E. Deskins: Abstract Algebra (1964): $\S 1.2$: Definition $1.7 \ \text{(b)}$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 14$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.2$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 7$, Examples $7.3, \ 7.4$