THE CLOSURE UNDER DIVISION AND A CHARACTERIZATION OF THE RECOGNIZABLE Z-SUBSETS

摘要

We show that the family of recognizable Z-subsets of A~* is closed under (integer) division by a positive integer. The technique that we use to prove this result is constructive and, by generalizing this construction, we obtain a characterization of recognizable Z-subsets of A~+ as a sum of finitely many simple Z-subsets of A~+. We also show that the family of recognizable Z-subsets of A~* is not closed under division by a negative integer, or under taking the remainder of the division by an integer of absolute value greater than 1.