Semigroup Pγ(Λ×Λ) with the Unique Left Unit

Let $\Lambda$ be an arbitrary nonempty set, and $\Gamma$ be a semilattice on the set $\Lambda$, ${\cal P}_{\Gamma}(\Lambda\times\Lambda)$ is a semigroup of binary relations determined by the semilattice $\Gamma$ on the set $\Lambda$. In the semigroup ${\cal P}_{\Gamma}(\Lambda\times\Lambda)$ , by using the existing conclusions of left units, the greatest left unit is obtained. By employing constructive method of the left units, we find the conditions that this left unit must satisfy for the given senmigroup ${\cal P}_{\Gamma}(\Lambda\times\Lambda)$ to have a unique right unit.