Magnifiers in various semigroups of transformations that preserve a partition

Abstract

Let [Formula: see text] be an arbitrary infinite set, and let [Formula: see text] be a partition of [Formula: see text] indexed by [Formula: see text]. Denote by [Formula: see text] the full transformation semigroup on [Formula: see text], and let [Formula: see text]. The character map of an element [Formula: see text] of [Formula: see text] is the self-map [Formula: see text] defined as [Formula: see text] whenever [Formula: see text]. In this paper, we describe magnifiers in [Formula: see text] in terms of their character and restriction maps. We further describe magnifiers in the subsemigroups [Formula: see text], [Formula: see text], and [Formula: see text] of [Formula: see text] that consist of all transformations whose character maps are injective, surjective, and bijective, respectively.