The combinatorial simple principal ideal semigroups generated by two elements were described by L. Megyesi and G. Poll�k.
The ‘most general’ among them is called the R�dei semigroup. The ‘most special’ combinatorial simple principal ideal semigroup
generated by two elements is the bicyclic semigroup. D. B. McAlister determined the compatible semilattice orders on the bicyclic
semigroup. Our aim is to study the compatible semilattice orders on the homomorphic images of the R�dei semigroup. We prove
that there are four compatible total orders on these semigroups. We show that on the R�dei semigroup, the total orders are
the only compatible semilattice orders. Moreover, on each proper homomorphic image of the R�dei semigroup, we give a compatible
semilattice order which is not a total order.