On rank properties of endomorphisms of finite circular orders

A mapping α from X = {1, 2, . . . , n} to X is orientation-preserving if the sequence (1α, 2α, . . . , nα) 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 ofX have the same type if they have identical sizes and numbers of classes. Let τ 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 τ . 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 ∈ S such that s = s), and show that the minimum number of idempotent elements required to generate S is ` n r ́ .