Representation functions of additive bases for abelian semigroups

Abstract

Let X = S \oplus G, where S is a countable abelian semigroup and G is a countably infinite abelian group such that {2g : g in G} is infinite. Let pi: X \to G be the projection map defined by pi(s,g) = g for all x =(s,g) in X. Let f:X \to N_0 cup infty be any map such that the set pi(f^{-1}(0)) is a finite subset of G. Then there exists a set B contained in X such that r_B(x) = f(x) for all x in X, where the representation function r_B(x) counts the number of sets {x',x''} contained in B such that x' \neq x'' and x'+x''=x. In particular, every function f from the integers Z into N_0 \cup infty such that f^{-1}(0) is finite is the representation function of an asymptotic basis for Z.