On rank properties of endomorphisms of finite circular orders

A mapping alpha from X = {1, 2,..., n} to X is orientation-preserving if the sequence {1 alpha,2 alpha,..., n alpha} is a cyclic permutation of a nondecreasing sequence (with respect to some total order on X). Orientation-preserving mappings can be thought of as preserving a circular order on X. Two partitions of X have the same type if they have identical sizes and numbers of classes. Let tau be a partition with r classes, and let S be the semigroup generated by the set of orientation-preserving mappings from X to X with kernels of same type as tau. We show that the minimum cardinality of a generating set for S is ((n)(r)). Moreover, we characterize all such S generated by their idempotent elements (i.e., s is an element of S such that s(2) = s), and show that the minimum number of idempotent elements required to generate S is ((n)(r)).