To appear in Communications in Algebra (DOI:10.1080/00927872.2012.720322)

The purpose of this paper is to compute the ranks of the monoid
${\cal{OR}}_{m \times n}$ 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
${\cal{OP}}_{m \times n}$ of all orientation-preserving transformations and
${\cal{OD}}_{m \times n}$ of all order-preserving or order-reversing full transformations.
These three monoids are natural extensions of ${\cal{O}}_{m \times n}$,
the monoid of all order-preserving full transformations on a chain
with $mn$ elements that preserve a uniform $m$-partition. Moreover, we also determine the ranks of certain semigroups of orientation-preserving full transformations with restricted ranges.