Let Cn be the semigroup of all order-preserving and order-decreasing transformations on an n-element chain. Let ζ be an idempotent in Cn. The set Cn(ζ) of all α ∈ Cn such that the image of ζ is the set of all fixed points in α is a maximal nilpotent subsemigroup of Cn. For this semigroup, the cardinality and the rank have already determined by Yağci and Kormaz. In the present paper, we characterize all ideals and the maximal as well as minimal congruences on Cn(ζ). Moreover, we consider a class of congruences different from the Rees congruences.