Magnifiers in semigroups of transformations whose restrictions belong to a given semigroup

Abstract

Let [Formula: see text] be the full transformation semigroup on a set [Formula: see text]. For a subset [Formula: see text] of [Formula: see text] and a submonoid [Formula: see text] of [Formula: see text], denote by [Formula: see text] the semigroup under composition consisting of all transformations [Formula: see text] such that the restriction [Formula: see text] of [Formula: see text] to [Formula: see text] belongs to [Formula: see text]. We give necessary and sufficient conditions for an element in [Formula: see text] to be left or right magnifier. We apply these descriptions to obtain more concrete results for the semigroups [Formula: see text] and [Formula: see text], where [Formula: see text] (respectively, [Formula: see text]) is the specific submonoid of [Formula: see text] consisting of all injective (respectively, surjective) transformations. The paper also identifies some results on [Formula: see text] that have appeared in the literature.