Invertible Element

An invertible element of a Monoid $M$ is an $x:M$ such that there exists a $x^{-1}:M$ with

• $x\star x^{-1}=\varepsilon=x^{-1}\star x$

As a decategorification of isomorphisms

If we view a monoid as a single-object Category, then an invertible element is a morphism $f$ that is an Isomorphism.