Volume 35 • Number 4 • 2012   • The Cardinal of Various Monoids of Transformations That Preserve a Uniform Partition Vítor H. Fernandes and Teresa M. Quinteiro Abstract. In this paper we give formulas for the number of elements of the monoids ${\mathcal{OR}}_{m \times n}$ of all full transformations on a finite chain with $mn$ elements that preserve a uniform $m$-partition and preserve or reverse the orientation and for its submonoids ${\mathcal{OD}}_{m \times n}$ of all order-preserving or order-reversing elements, ${\mathcal{OP}}_{m\times n}$ of all orientation-preserving elements, ${\mathcal{O}}_{m \times n}$ of all order-preserving elements, ${\mathcal{O}}_{m \times n}^+$ of all extensive order-preserving elements and ${\mathcal{O}}_{m \times n}^-$ of all co-extensive order-preserving elements. 2010 Mathematics Subject Classification: 20M20, 05A10. Full text: PDF