For n E N, let On be the semigroup of all singular order-preserving mappings on [n] = (1, 2,..., n}. For each nonempty subset A of [n], let On (A) = (a ∈ On: (A k ∈ A) ka ≤ k} be the semigroup of all order-preserving and A-decreasing mappings on [n]. In this paper it is shown that On(A)is an abundant semigroup with n - 1 *-classes. Moreover, On(A) is idempotent-generated and its idempotent rank is 2n - 2 - IA/(n}l. Further, it is shown that the rank of On(A) is equal to n - 1 if 1 ∈ A, and it is equal to n otherwise.