My group theory is very rusty. If I want to just start with left inverses and left identities, must I link the axioms, or can I leave them independent?
e.g., is it enough to say "there exists at least one $e \in G$ s.t. $ea=a$ for all $a \in G$" and "for each $a$ in $G$, there exists $a^{-1} \in G$ s.t. $a^{-1}a$ is an identity", or must I say "there exists at least one $e \in G$ s.t. ($ea=a$ for all $a \in G$ AND for each $a \in G$ there exists $a^{-1} \in G$ s.t. $a^{-1}a=e$)".
This seems like it must be a FAQ, but I just can't find it. If I don't assume the linkage, I run into problems where I show (for example) that if $a^{-1}a=e_1$, $b=aa^{-1}$, $b^{-1}b=e_2$, then $b=e_2$. That will get me things like $ae_1=a$, but not the general case.