Complementing sets of $n$-tuples of integers

Abstract

Let S, A1, A2,*, A, be finite nonempty sets of n-tuples of integers such that, if ai E Ai for i=1, 2, * * *, p, then al+a2 +. +a, E S, and such that every s E S has a unique representation as a sum s=al+a2+' * +a, with ai E Ai. If S is the cartesian product of n sets of integers, then each Ai is also the cartesian product of n sets of integers, and conversely. Let S, A1, A2, * * *, A. be sets of n-tuples of integers. Define addition of n-tuples componentwise. Then S is the sum of A1, A2,. * * , A., denoted S=A1+A2+ * *+Ap, if S={a1+a2+ * +a|jaj E Ai for i=1, 2, ... ,p}. If S is the sum of A1, A2, ... , A,, and if for each s E S there exist unique n-tuples ai E Ai such that s=a1+a2+. * +ap, then A1, A2, * . *, A. are called complementing sets for S, denoted S_A1+ A2+. * +AP. A set of n-tuples is proper if it is the cartesian product of n sets of integers. For positive integers u and v, let S={O, 1, 2, ... , u} x {0, 1, 2,. * , v}. If A1 and A2 are subsets of S such that SA Secondary 10A45, 10J99, 05A15.