On diagonal acts of monoids

It is proved that the monoid R-N of all partial recursive functions of one variable is finitely generated, and that R-N x R-N is a cyclic (left and right) R-N-act (under the natural diagonal actions s(alpha, b) = (s alpha, sb), (alpha, b)s = (alphas, bs)). We also construct a finitely presented monoid S such that S x S is a cyclic left and right S-act, and study further interesting properties of diagonal acts and their relationship with power monoids.