Characterizing Additive Systems. Issue 2 (1st February 2017)

• Record Type: Journal Article
• Authors: Maltenfort, Michael
• Is Part Of: American Mathematical Monthly. Volume 124:Issue 2(2017)
• Page Start: 132
• Page End: 148
• Publication Date: 2017-02-01
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.4169/amer.math.monthly.124.2.132
• Languages: English
• ISSNs: 0002-9890

Abstract

Abstract: 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.