On diagonal acts of monoids

Abstract

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