Cayley Table/Examples/Self-Maps on Doubleton

Example of Cayley Table

Let $S$ be the set of self-maps on the doubleton $D = \set {a, b}$.

Let these be enumerated:

$\qquad \epsilon := \begin{pmatrix} a & b \\ a & b \end{pmatrix} \quad \alpha := \begin{pmatrix} a & b \\ b & a \end{pmatrix} \quad \beta := \begin{pmatrix} a & b \\ a & a \end{pmatrix} \quad \gamma := \begin{pmatrix} a & b \\ b & b \end{pmatrix}$

Let $\struct {S, \circ}$ be the semigroup of self-maps under composition of mappings.

The Cayley table of $\struct {S, \circ}$ can be written:

$\qquad \begin{array}{c|cccc} \circ & \epsilon & \alpha & \beta & \gamma \\ \hline \epsilon & \epsilon & \alpha & \beta & \gamma \\ \alpha & \alpha & \epsilon & \gamma & \beta \\ \beta & \beta & \beta & \beta & \beta \\ \gamma & \gamma & \gamma & \gamma & \gamma \\ \end{array}$

Sources

• 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Chapter $\text{I}$: Semi-Groups and Groups: $1$: Definition and examples of semigroups