Definition:Invertible Operation

Definition
Let $$\left({S, \circ}\right)$$ be an algebraic structure.

Then $$\circ$$ is invertible iff:
 * $$\forall a, b \in S: \exists r, s \in S: a \circ r = b = s \circ a$$

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.