Representation functions of additive bases for abelian semigroups

Abstract

A subset of an abelian semigroup is called an asymptotic basis for the semigroup if every element of the semigroup with at most finitely many exceptions can be represented as the sum of two distinct elements of the basis. The representation function of the basis counts the number of representations of an element of the semigroup as the sum of two distinct elements of the basis. Suppose there is given function from the semigroup into the set of nonnegative integers together with infinity such that this function has only finitely many zeros. It is proved that for a large class of countably infinite abelian semigroups, there exists a basis whose representation function is exactly equal to the given function for every element in the semigroup.