  We prove that if $S$ is a $le$-semigroup in which left ideal elements commute
(condition which is called $\mathbf{\Lambda}$), then any $\mathcal{J}$-class
satisfying the Green condition is a subsemigroup of $S$. As a corollary of this
we show that semisimple $le$-semigroups satisfying $\mathbf{\Lambda}$ are
precisely those that decompose as a semilattice of left simple $\vee
e$-semigroups which are in addition semisimple, intra-regular and satisfy
$\mathbf{\Lambda}$.
