On a set-theoretic invariant

Let a_1,…,a_m be positive real numbers. Besser and Moree considered weighted numbers of -1,+1 solutions of the linear inequality |a_i-a_j| < e_ka_k < a_i+a_j, with e_k=-1 of 1 and k running over the integers 1,…,m with i and j skipped. They introduced some invariants and near invariants related to this situation (invariant meaning here: not depending on the choice of i and j). The main result of their paper is extended here to a much more general setting, namely that of certain maps from finite sets to {-1,1}. Some applications are given.