摘要
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.
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.
