Congruences on Nil-extension of a Clifford Semigroup

The concept of congruence pair on S is introduced,where S is a nil-extension of a Clifford semigroup K.It is shown that every congruence σ on such a semigroup S can be uniquely represented by a congruence pair on S.It is also proved that there is an order-preserving bijection Γ∶σ→(σQ,σk) from set of all congruences on S onto the set of all congruence pairs on S.