Orthodox congruences on an eventually regular semigroup

It is shown that each regular congruence on an eventually regular semigroup is uniquely determined by its kernel and hyper-trace. Furthermore, the orthodox congruences (resp., the regular congruences) on an eventually regular (resp., orthodox) semigroup S are described by means of certain congruence pair (xi, K), where xi is a certain normal congruence on the subsemigroup (E(S)) generated by E(S) and K is a certain normal subsemigroup of S.