# Definition:Hasse Diagram

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## 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 "Divisor" 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)... (previous)... (next): $\S 1.2$ - Seth Warner:
*Modern Algebra*(1965)... (previous)... (next): $\S 14$ - Richard A. Dean:
*Elements of Abstract Algebra*(1966)... (previous)... (next): $\S 0.2$

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

- T.S. Blyth:
*Set Theory and Abstract Algebra*(1975)... (previous)... (next): $\S 7$, Examples $7.3, \ 7.4$ - P.M. Cohn:
*Algebra Volume 1*(2nd ed., 1982)... (previous)... (next): $\S 1.5$: Ordered Sets

*The diagram is not named*