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 digraph, 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 symmetries 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 symmetries 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