On the monoids of transformations that preserve the order and a uniform partition

Communications in Algebra

In this paper we consider the monoid O mxn of all order-preserving full transformations on a chain with mn elements that preserve a uniform m-partition and its submonoids O+ mxn and O− mxn of all extensive transformations and of all co-extensive transformations, respectively. We give formulas for the number of elements of these monoids and determine their ranks. Moreover, we construct a bilateral semidirect product decomposition of Omxn in terms of O− mxn and O+ mxn.