Let $S$ be a regular semigroup. The Green's equivalence relation $\mathcal{L}$ is defined on $S$ as follows. $$s\mathcal{L}t \mbox{ if and only if } Ss=St.$$ Similarly using cyclic right ideals of $S$, the Green's relation $\mathcal{R}$ is defined. Finally note that $\mathcal{H}=\mathcal{L}\cap \mathcal{R}$. It is a known fact that $\mathcal{L}$ is a right congruence on $S$ and $\mathcal{R}$ is a left congruence. However for some classes of semigroups (e.g. commutative semigroups) we know that $\mathcal{L}=\mathcal{R}=\mathcal{H}$ and they are congruence on $S$.
My question is the following.
Is there a regular semigroup $S$ such that the Green relation $\mathcal{L}$ on $S$ is a congruence but $\mathcal{H}$ is not a congruence?
Sorry! I correct a mistake about completely simple semigroups in my question.