Let $S$ be a hypothetical algebraic set. It is a True Group, a Monoid where the elements may be inversed (not necessarily an Abelian group).
$S$ is of infinite size but only three of its members concern us: $a$ is a left identity element, $b$ is a right identity element, $c$ is a dual-identity element (both left and right identity). For argument's sake, $a$ ≠ $b$ ≠ $c$ ≠ $a$
Does this mean $S$ has different inverse functions, one for each identity element? Or can a True Group only have a single identity element?
Thanks in advance
(I'm only an armchair mathematician looking at the properties of algebraic operations so please be gentle! :)