Existence of right and left identity in minimalistic algebraic structure Let $(A,\cdot)$ be some algebraic structure in which there exists elements $e_r,e_l$ such that
$$e_l\cdot x = x, \forall x\in A$$
$$x\cdot e_r = x, \forall x\in A$$ 
By definition, if $(A,\cdot)$ is a monoid or a group then we must have $e_r = e_l$. 
But how about the case when $(A,\cdot)$ is a strict magma or a strict semigroup? Can we have $e_l \neq e_r$ ? 
Would this lead to a contradiction?

Note that 
$$ e_r = e_l\cdot e_r = e_l$$
