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.