On the Ranks of Certain Monoids of Transformations that Preserve a Uniform Partition

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 O{script}ℛ_m×n 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 O{script}P{script}_m×n of all orientation-preserving transformations and O{script}D{script}_m×n of all order-preserving or order-reversing full transformations. These three monoids are natural extensions of O{script}_m×n, the monoid of all order-preserving full transformations on a chain with mn elements that preserve a uniform m-partition. Given m, n ≥ 2, we show that the rank of O{script}P{script}_m×n is equal to (Formula presented.), for m > 2, and equal to (Formula presented.), for m = 2, the rank of O{script}D{script}_m×n is equal to (Formula presented.) and the rank of O{script}ℛ_m×n is equal to (Formula presented.), for m > 2, and equal to (Formula presented.), for m = 2. These results have quite nontrivial proofs and were intuited by performing massive calculations with GAP [24], a system for computational discrete algebra. Moreover, in this article, we also determine the ranks of certain semigroups of orientation-preserving full transformations with restricted ranges.