Direct decompositions in free commutative monoids

Let N be the positive integers; let C be a subset closed under taking divisors; and let A and B be subsets of C such that every member of AB (= {ab: a?A, b?B) is uniquely representable in the form ab, and also AB contains C. Given C, all such pairs (A, B) are found. The result is obtained in a slightly more general setting, and the pairs are replaced by arbitrarily large families.