Characterizing additive systems

An additive system is a collection of sets that gives a unique way to represent either all nonnegative integers, or all nonnegative integers up to some maximum. A structure theorem of de Bruijn gives a certain form for an additive system of infinite size. This form is not, in general, unique. We improve de Bruijn's theorem by finding a unique form for an additive system of arbitrary size. Our proof gives a concrete construction that allows us to test easily whether a collection of sets is an additive system. We also show how to determine most of the structure of an additive system if we are only given its union.