Existence of right and left identity in minimalistic algebraic structure

Question

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?

Answer 1

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

Answer 2

By the defining property of $e_l$, we have

$$e_l\cdot e_r = e_r.$$

And by the defining property of $e_r$, we have

$$e_l\cdot e_r = e_l.$$