Let $(M, \cdot)$ be a finite semigroup such that
$$ x,y\in M\wedge \exists a,b\in M:x=a⋅y\wedge y=b⋅x\Rightarrow x=y. $$
Show that M contains at least one right absorbant element(or right zero).
Using that there exists an $x^n=x$ for $x\in M$ because it is a finite semigroup I got that $(M, \cdot)$ is a idempotent semigroup but I'm not sure it's correct.I don't know how to continue.
Your condition means that $M$ is $\mathcal{L}$-trivial, i.e. the Green's relation $\mathcal{L}$ is the equality. Since $M$ is finite, it has a minimum ideal $I$, which is a completely simple semigroup. Moreover, since $M$ is $\mathcal{L}$-trivial, $I$ is actually a right zero band, and all its elements are right zeroes.