Definition:Invertible Operation

Definition

Let $\struct {S, \circ}$ be an algebraic structure.

The operation $\circ$ is invertible if and only if:

$\forall a, b \in S: \exists r, s \in S: a \circ r = b = s \circ a$

Example

An example of a $4$-element algebraic structure whose operation is invertible is given by the following Cayley table:

$\begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & d & b & c \\ b & c & b & d & a \\ c & d & a & c & b \\ d & b & c & a & d \\ \end{array}$

The invertible nature of $\circ$ can readily be determined by inspection:

$a \circ c = b = d \circ a$

$a \circ d = c = b \circ a$

$a \circ b = d = c \circ a$

etc.

Sources

• 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): Exercise $1.4: \ 14$