Ranks of Semigroups Generated by Order-Preserving Transformations with a Fixed Partition Type
GEORGE R. BARNES, Inessa Levi Β· Communications in Algebra Β· 2003
The rank of a finite semigroup is the minimal size of its generating set. Let T n be the semigroup of all total transformations of the set {1, 2,β¦, n}, and let πͺ n be the subsemigroup of T n of all the order-preserving transformations whose images consist of at most n β 1 elements. We study the subsemigroups S πͺ(Ο) of πͺ n generated by the order-preserving transformations whose kernels are partitions of X n of a given partition type Ο. We characterize idempotent-generated semigroups S πͺ(Ο), and show that S πͺ(Ο) is idempotent-generated precisely when it is regular, and that occurs if and only if the weight r of Ο is n β 1 or 1. For arbitrary partition type Ο of weight r we determine the Green's relations on S πͺ(Ο), and we show that its rank equals to .