= Group axioms from two-sided local inverse identities
Suppose a nonempty <semigroup> has, for each $a$, elements $l_a,r_a$ satisfying $bl_aa=b$ and $ar_ab=b$ for every $b$. Then it is a <group>. For any fixed $a$, the element $e_R=l_aa$ is a right <identity element>, and $e_L=ar_a$ is a left <identity element>. Since $e_Le_R=e_L=e_R$, they coincide in a two-sided identity $e$. For each $a$, putting $b=e$ in the two assumed identities gives $l_aa=e$ and $ar_a=e$. <Associativity> then yields $l_a=l_ae=l_a(ar_a)=(l_aa)r_a=er_a=r_a$, a two-sided <inverse element>. Thus the usual <group axioms> follow without an assumed identity or cancellation law.
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