Magnifying elements of some semigroups of partial transformations

Abstract

Let X be a nonempty set and let P (X) denote the semigroup (under the composition) of partial transformations from a subset of X to X and E(X) denote the subsemigroup of P (X) containing surjective partial transformations on X. For a fixed nonempty subset Y of X, let P T (X, Y ) = {α ∈ P (X) | (dom α ∩ Y )α ⊆ Y } and P T(X,Y) = {α ∈ P (X) | (dom α ∩ Y )α = Y }. We give necessary and sufficient conditions for elements in semigroups P T (X, Y ) and P T(X,Y) to be left or right magnifying.