Reductions of additive sets, sets of uniqueness and pyramids

Abstract In this paper we introduce the notion of reduction for any subset S of the 3-dimensional box B ( p , q , r ) = [ p ] × [ q ] × [ r ]. This yields an equivalence relation in B ( p , q , r ). Our main theorem states that S is additive if and only if any reduction of S is additive. It follows that a set is additive if and only if the irreducible set in its equivalence class is additive. In this way, out of each additive irreducible set, we produce lots of examples of additive sets and therefore of sets of uniqueness. We also introduce a class of subsets of B ( p , q , r ) we call pyramids. They do not have 2-bad configurations and do have essentially one kind of 3-bad configurations. We use them to give examples and counterexamples. Pyramids have been studied since MacMahon under the name of plane partitions.