On the semigroups of order-preserving transformations generated by idempotents of rank n-1

Let O-n be the semigroup of all singular order-preserving mappings on the finite set X-n = {1, 2,..., n}. It is known that O-n is generated by its set of idempotents of rank n-1, and its rank and idempotent rank are n and 2n-2, respectively. In this paper, we study the structure of the semigroup generated by any nonempty subset of idempotents of rank n - 1 in O-n. We also calculate its rank and idempotent rank.