SETS UNIQUELY DETERMINED BY PROJECTIONS ON AXES .2. DISCRETE CASE

A subset S of N(n) = {1, 2,..., N}n is a discrete set of uniqueness if it is the only subset of N(n) with projections P1,..., P(n), where P(i)(j) = \{(x1,..., x(n) is-an-element-of S: x(i) = j}\. Also, S is additive if there are real valued functions f1,..., f(n) on N such that, for all (x1,..., x(n)) is-an-element-of N(n), (x1,..., x(n)) is-an-element-of S if and only if SIGMA (i)f(i)(x(i)) greater-than-or-equal-to 0. Sets of uniqueness and additive sets are characterized by the absence of certain configurations in the lattice N(n). The characterization shows that every additive set is a set of uniqueness. If n = 2, every set of uniqueness is also additive. However, when n greater-than-or-equal-to 3, there are sets of uniqueness that are not additive.