 Group axioms from two-sided local inverse identities

Suppose a nonempty semigroup has, for each $a$, elements $l_a,r_a$ satisfying $b{l}_{a}a=b$ and $a{r}_{a}b=b$ for every $b$. Then it is a group. For any fixed $a$, the element $e_R=l_{a}a$ is a right identity element, and $e_L=a{r}_a$ is a left identity element. Since $e_L{e}_R=e_L=e_R$, they coincide in a two-sided identity $e$. For each $a$, putting $b=e$ in the two assumed identities gives $l_{a}a=e$ and $a{r}_a=e$. Associativity then yields $l_a=l_{a}e=l_a(a{r}_a)=(l_{a}a)r_a=e{r}_a=r_a$, a two-sided inverse element. Thus the usual group axioms follow without an assumed identity or cancellation law.