Let 〈A:S〉 be the A-closure of a set S of nonnegative integers where A is a finite set of affine functions each one defined on the integers. Suppose every element of A has the form α(x)=mex+a defined for all x with e−1, a, m−2 fixed nonnegative integers where only e and a depend on α. That is, every element of A has a multiplier which is a power of one and the same number m. A set S of positive integers is said to be an m-recognizable set just when the m-ary strings representing elements of S are recognized by some finite automaton. Also, suppose S is an m-recognizable set. Then 〈A:S〉 is also m-recognizable. Roughly speaking, this means that if S has a certain kind of simple description in terms of the base-m strings which represent elements of S, then a similar description can be found for the A-closure of S.
