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

摘要

In this article we consider the monoid O_(m×n) of all order-preserving full transformations on a chain with mn elements that preserve a uniform m-partition and its submonoids O~+_(m×n) and O~-_(m×n) of all extensive transformations and of allo-extensive transformations, respectively. We determine their ranks and construct a bilateral semidirect product decomposition of O_(m×n) in terms of O~(-m×n) and O~+(m×n).