On transformation semigroup induced by symmetric group characters

Let [Formula: see text] be a partition of a nonempty set [Formula: see text]. A transformation [Formula: see text] preserves [Formula: see text] if for every [Formula: see text], there exists [Formula: see text] such that [Formula: see text]. We consider the semigroup [Formula: see text], consisting of all such transformations where the induced mapping [Formula: see text], defined by [Formula: see text] whenever [Formula: see text], is bijective. This study explores the structure of [Formula: see text] by establishing conditions for the semigroup [Formula: see text] to be orthodox, inverse, and completely regular. Furthermore, we also characterize its ideals and find the kernel.