Definition:Invertible Element

Let $$\left({S, \circ}\right)$$ be a monoid whose identity is $$e_S$$.

If $$x \in S$$ has an inverse, then $$x$$ is said to be invertible for $$\circ$$.

That is, $$x$$ is invertible iff:

$$\exists y \in S: x \circ y = e_S = y \circ x$$