Bases and nonbases of square-free integers

A basis is a set A of nonnegative integers such that every sufficiently large integer n can be represented in the form n = a_i + a_j with a_i, a_i ∈ A. If A is a basis, but no proper subset of A is a basis, then A is a minimal basis. A nonbasis is a set of nonnegative integers that is not a basis, and a nonbasis A is maximal if every proper superset of A is a basis. In this paper, minimal bases consisting of square-free numbers are constructed, and also bases of square-free numbers no subset of which is minimal. Maximal nonbases of square-free numbers do not exist. However, nonbases of square-free numbers that are maximal with respect to the set of square-free numbers are constructed, and also nonbases of square-free numbers that are not contained in any nonbasis of square-free numbers maximal with respect to the square-free numbers.