We characterize the regular, both regular and intra-regular, and the regular and left duo $le$-semigroups in terms of left ideal elements, ideal elements, bi- and quasi-ideal elements. The regular and left duo $poe$-semigroups are left regular and left duo. The left regular and left duo $le$-semigroups are exactly the semilattices of left simple semigroups. On the other hand, a $poe$-semigroup is a semilattice of left simple semigroups if and only if it is a semilattice of left simple $poe$-semigroups.