Abstract
For n ∈ 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) = {α ∈ On: (∀ k ∈A) kα ≤ 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-1𝒟*-classes. Moreover, On(A) is idempotent-generated and its idempotent rank is 2n-2- |A\ {n}|. 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.
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