Semigroup of Partition-Preserving Transformations Whose Images Intersect Each Block

For a partition [Formula: see text] of a set [Formula: see text], we say that a transformation [Formula: see text] of [Formula: see text] preserves [Formula: see text] if for every [Formula: see text] there exists [Formula: see text] such that [Formula: see text]. Let [Formula: see text] denote the semigroup of all transformations [Formula: see text] of [Formula: see text] such that [Formula: see text] preserves [Formula: see text] and its image [Formula: see text] intersects each block of [Formula: see text]. We describe Green's relations on [Formula: see text] and prove that Green's relations [Formula: see text] and [Formula: see text] on [Formula: see text] are equal if and only if [Formula: see text] is finite. Moreover, we characterize ideals and the kernel of [Formula: see text] and determine when the semigroup [Formula: see text] is unit-regular, orthodox, inverse, and completely regular separately.