On the ranks of certain monoids of transformations that preserve a uniform partition - Fernandes, Vítor H. (2015)
The rank of a semigroup, an important and relevant concept in Semigroup Theory, is the cardinality of a least-size generating set. Semigroups of transformations that preserve or reverse the order or the orientation as well as semigroups of transformations preserving an equivalence relation have been widely studied over the past decades by many authors. The purpose of this article is to compute the ranks of the monoid OR mxn of all orientation-preserving or orientation-reversing full transformations on a chain with mn elements that preserve a uniform m-partition and of its submonoids OP mxn of all orientation-preserving transformations and OD mxn of all order-preserving or order-reversing full transformations. These three monoids are natural extensions of O mxn, the monoid of all order-preserving full transformations on a chain with mnelements that preserve a uniform m-partition.