## Characterize the invertible elements in a semigroup

Let $S$ be a semigroup with identity $e$ and let $a \in S$. Show that $a$ is invertible if and only if $Sa = aS = S$.

Suppose $a$ is invertible; then there exists an element $b$ such that $ab = ba = e$. Now let $s \in S$. We have $s = se = sba \in Sa$ and $s = es = abs \in aS$. So $aS = Sa = S$.

Conversely, suppose $aS = Sa = S$. Now $e \in S$, so there exist elements $b,c \in S$ such that $ab = ca = e$. Now $b = eb = cab = ce = c$, so that in fact $a$ is invertible.