For a diagonalizable group scheme D(M) S acting on an algebraic space X over a scheme S, we introduce for any submonoid N of M an attractor space X^N. We then extend and refine the study of G m-actions on algebraic spaces to actions of diagonalizable group schemes and general monoids. We expect that this formalism will shed light on some primordial objects, as an evidence we observe that root groups are natural from the view point of attractors associated to monoids.