TY - JOUR
T1 - Characterizing additive systems
AU - Maltenfort, Michael Brian
PY - 2017/2/1
Y1 - 2017/2/1
N2 - 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.
AB - 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.
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U2 - 10.4169/amer.math.monthly.124.2.132
DO - 10.4169/amer.math.monthly.124.2.132
M3 - Article
AN - SCOPUS:85020707031
VL - 124
SP - 132
EP - 148
JO - American Mathematical Monthly
JF - American Mathematical Monthly
SN - 0002-9890
IS - 2
ER -