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  • 1 University of Szeged Bolyai Institute Aradi vértanúk tere 1 H-6720 Szeged Hungary
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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.