# Hasse Diagram/Examples

## Examples of Hasse Diagrams

### 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.

## Informal Examples

### British Monarchs

Recall the ordering on the British monarchy:

Let $K$ denote the set of British monarchs.

Let $\MM$ denote the relation on $K$ defined as:

- $a \mathrel \MM b$ if and only if $a$ was monarch after or at the same time as $b$.

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

- $a \mathrel {\MM^{-1} } b$ if and only if $a$ was monarch before or at the same time as $b$.

Then $\MM$ and $\MM^{-1}$ are orderings on $K$.

This **Hasse diagram** illustrates the restriction of $\MM$ to all $x$ of $K$ such that $x \mathrel \MM \text {Victoria}$ and $\text {Elizabeth II} \mathrel \MM x$.

### British Monarchs Line of Descent

Recall the partial ordering on the set of people:

Let $P$ denote the set of all people who have ever lived.

Let $\DD$ denote the relation on $P$ defined as:

- $a \mathrel \DD b$ if and only if $a$ is a descendant of or the same person as $b$.

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

- $a \mathrel {\DD^{-1} } b$ if and only if $a$ is an ancestor of or the same person as $b$.

Then $\DD$ and $\DD^{-1}$ are partial orderings on $P$.

This **Hasse diagram** illustrates the restriction of $\DD$ to the set of British monarchs such that $x \mathrel \DD \text {Victoria}$.

### Genealogy from Terah to Joseph

Recall the partial ordering on the set of people:

Let $P$ denote the set of all people who have ever lived.

Let $\DD$ denote the relation on $P$ defined as:

- $a \mathrel \DD b$ if and only if $a$ is a descendant of or the same person as $b$.

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

- $a \mathrel {\DD^{-1} } b$ if and only if $a$ is an ancestor of or the same person as $b$.

Then $\DD$ and $\DD^{-1}$ are partial orderings on $P$.

This **Hasse diagram** illustrates the restriction of $\DD$ to all $x$ of $P$ such that $x \mathrel \DD \text {Terah}$ and $\text {Joseph} \mathrel \DD x$, according to the Book of Genesis in the Bible.

## Source of Name

This entry was named for Helmut Hasse.

## Sources

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- 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): $\S 1.2$ - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 0.2$. Sets - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.5$: Ordered Sets