The cardinal of various monoids of transformations that preserve a uniform partition

Abstract

In this paper we give formulas for the number of elements of the monoids OR mxn of all full transformations on a nite chain with mn elements that preserve a uniform m-partition and preserve or reverse the orientation and for its submonoids OD mxn of all order-preserving or order-reversing elements, OP mxn of all orientation- preserving elements, O mxn of all order-preserving elements, O+ mxn of all extensive order-preserving elements and O- mxn of all co-extensive order-preserving elements.