Definition:Hasse Diagram

Definition

Let $\struct {S, \preceq}$ be an ordered set.

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

$(1):\quad$ The vertices of $G$ represent the elements of $S$

$(2):\quad$ The edges of $G$ represent the elements of $\preceq$

$(3):\quad$ 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 $y$

$(4):\quad$ 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.

Also known as

Some sources refer to this as a nodal diagram.

Also presented as

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

Some sources do not label the vertices, on the grounds that the structure of the Hasse diagram is the important part.

Examples

Divisors of $12$

This Hasse diagram illustrates the "Divisor" ordering on the set $S = \set {1, 2, 3, 4, 6, 12}$, where $S$ is the set of all elements of $\N_{>0}$ which divide $12$.

Divisors of $24$

This Hasse diagram illustrates the "Divisor" ordering on the set $S = \set {1, 2, 3, 4, 6, 8, 12, 24}$, where $S$ is the set of all elements of $\N_{>0}$ which divide $24$.

Divisors of $30$

This Hasse diagram illustrates the "Divisor" ordering on the set $S = \set {1, 2, 3, 5, 6, 10, 30}$.

That is, $S$ is the set of all elements of $\N_{>0}$ which divide $30$ except for $15$, which for the purposes of this example has been deliberately excluded.

Subsets of $\set {1, 2, 3}$

This Hasse diagram illustrates the "Subset" relation on the power set $\powerset S$ where $S = \set {1, 2, 3}$.

Subsets of $\set {1, 2, 3, 4}$

This Hasse diagram illustrates the "Subset" relation on the power set $\powerset S$ where $S = \set {1, 2, 3, 4}$.

Subgroups of Symmetry Group of Rectangle

Consider the symmetry group of the rectangle:

Let $\RR = ABCD$ be a (non-square) rectangle.

The various symmetry mappings of $\RR$ are:

The identity mapping $e$

The rotation $r$ (in either direction) of $180^\circ$

The reflections $h$ and $v$ in the indicated axes.

The symmetries of $\RR$ form the dihedral group $D_2$.

This Hasse diagram illustrates the subgroup relation on $\map D 2$.

Subgroups of Symmetry Group of Square

Consider the symmetry group of the square:

Let $\SS = ABCD$ be a square.

The various symmetry mappings of $\SS$ are:

the identity mapping $e$

the rotations $r, r^2, r^3$ of $90^\circ, 180^\circ, 270^\circ$ around the center of $\SS$ anticlockwise respectively

the reflections $t_x$ and $t_y$ are reflections in the $x$ and $y$ axis respectively

the reflection $t_{AC}$ in the diagonal through vertices $A$ and $C$

the reflection $t_{BD}$ in the diagonal through vertices $B$ and $D$.

This group is known as the symmetry group of the square, and can be denoted $D_4$.

This Hasse diagram illustrates the subgroup relation on $\map D 4$.

Parallel Lines

Recall this partial ordering on the set of lines:

Let $S$ denote the set of all infinite straight lines embedded in a cartesian plane.

Let $\LL$ denote the relation on $S$ defined as:

$a \mathrel \LL b$ if and only if:

$a$ is parallel $b$

if $a$ is not parallel to the $y$-axis, then coincides with or lies below $b$

but if $b$ is parallel to the $y$-axis, then $a$ coincides with or lies to the right of $b$

Its dual $\LL^{-1}$ is defined as:

$a \mathrel {\LL^{-1} } b$ if and only if:

$a$ is parallel $b$

if $a$ is not parallel to the $y$-axis, then coincides with or lies above $b$

but if $b$ is parallel to the $y$-axis, then $a$ coincides with or lies to the left of $b$.

Then $\LL$ and $\LL^{-1}$ are partial orderings on $S$.

This Hasse diagram illustrates the restriction of $\LL$ to the set of all infinite straight lines in the cartesian plane which are parallel to and one unit away from either the $x$-axis or the $y$-axis.

Also see

• Results about Hasse diagrams can be found here.

Source of Name

This entry was named for Helmut Hasse.

Sources

• 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.2$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings

The diagram is not named

• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.2$. Sets

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

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.5$: Ordered Sets

The diagram is not named